Hardy-Ramanujan Journal - Latest Publications Latest papers https://hrj.episciences.org/img/episciences_sign_50x50.png episciences.org https://hrj.episciences.org Thu, 06 Oct 2022 01:34:15 +0000 episciences.org https://hrj.episciences.org Hardy-Ramanujan Journal Hardy-Ramanujan Journal Generating Functions for Certain Weighted Cranks Sun, 09 Jan 2022 19:41:54 +0000 https://doi.org/10.46298/hrj.2022.8922 https://doi.org/10.46298/hrj.2022.8922 Bandyopadhyay, Shreejit Yee, Ae, Bandyopadhyay, Shreejit Yee, Ae, 0 A survey on t-core partitions Sun, 09 Jan 2022 19:27:16 +0000 https://doi.org/10.46298/hrj.2022.8928 https://doi.org/10.46298/hrj.2022.8928 Cho, Hyunsoo Kim, Byungchan Nam, Hayan Sohn, Jaebum Cho, Hyunsoo Kim, Byungchan Nam, Hayan Sohn, Jaebum 0 A Dynamical Proof of the Prime Number Theorem Sun, 09 Jan 2022 19:25:35 +0000 https://doi.org/10.46298/hrj.2022.8924 https://doi.org/10.46298/hrj.2022.8924 Mcnamara, Redmond Mcnamara, Redmond 0 The last chapter of the Disquisitiones of Gauss Sun, 09 Jan 2022 19:19:30 +0000 https://doi.org/10.46298/hrj.2022.8925 https://doi.org/10.46298/hrj.2022.8925 Anderson, Laura Chahal, Jasbir S Top, Jaap Anderson, Laura Chahal, Jasbir S Top, Jaap 0 Proof of the functional equation for the Riemann zeta-function Sun, 09 Jan 2022 19:11:54 +0000 https://doi.org/10.46298/hrj.2022.7663 https://doi.org/10.46298/hrj.2022.7663 Mehta, Jay Zhu, P. -Y Mehta, Jay Zhu, P. -Y 0 Note on Artin's Conjecture on Primitive Roots 1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $ p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R. Murty and S. Srinivasan \cite{Murty-Srinivasan} showed that if $\displaystyle \sum_{p < x} \frac 1 {f_a(p)} = O(x^{1/4})$ then Artin's conjecture is true for $a.$ We relate the Murty-Srinivasan condition to sums involving the cyclotomic periods from the subfields of $\mathbb Q(e^{2\pi i /p})$ corresponding to the subgroups $ \subseteq \mathbb F_p^*.$]]> Sun, 09 Jan 2022 19:06:40 +0000 https://doi.org/10.46298/hrj.2022.7664 https://doi.org/10.46298/hrj.2022.7664 Sitaraman, Sankar Sitaraman, Sankar 1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $ p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R. Murty and S. Srinivasan \cite{Murty-Srinivasan} showed that if $\displaystyle \sum_{p < x} \frac 1 {f_a(p)} = O(x^{1/4})$ then Artin's conjecture is true for $a.$ We relate the Murty-Srinivasan condition to sums involving the cyclotomic periods from the subfields of $\mathbb Q(e^{2\pi i /p})$ corresponding to the subgroups $ \subseteq \mathbb F_p^*.$]]> 0 Filter integrals for orthogonal polynomials Sun, 09 Jan 2022 18:59:14 +0000 https://doi.org/10.46298/hrj.2022.8926 https://doi.org/10.46298/hrj.2022.8926 Amdeberhan, T Duncan, Adriana Moll, Victor H Sharma, Vaishavi Amdeberhan, T Duncan, Adriana Moll, Victor H Sharma, Vaishavi 0 Partition Identities for Two-Color Partitions Sun, 09 Jan 2022 18:40:13 +0000 https://doi.org/10.46298/hrj.2022.8929 https://doi.org/10.46298/hrj.2022.8929 Andrews, George E Andrews, George E 0 Quantum q-series identities Sun, 09 Jan 2022 17:18:00 +0000 https://doi.org/10.46298/hrj.2022.8930 https://doi.org/10.46298/hrj.2022.8930 Lovejoy, Jeremy Lovejoy, Jeremy 0 Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product Sun, 09 Jan 2022 17:11:29 +0000 https://doi.org/10.46298/hrj.2022.8931 https://doi.org/10.46298/hrj.2022.8931 Wang, Liuquan Wang, Liuquan 0 Explicit Values for Ramanujan's Theta Function ϕ(q) Sun, 09 Jan 2022 16:57:02 +0000 https://doi.org/10.46298/hrj.2022.8923 https://doi.org/10.46298/hrj.2022.8923 Berndt, Bruce C Rebák, Örs Berndt, Bruce C Rebák, Örs 0 A variant of the Hardy-Ramanujan theorem 0$, as $x\to \infty$, $$\sum_{[p-1,q-1]\leq x} {1 \over [p-1, q-1]} \sim C \log x, $$ where the summation is over primes $p,q\leq x$ such that the least common multiple $[p-1,q-1]$ is less than or equal to $x$.]]> Sun, 09 Jan 2022 16:48:25 +0000 https://doi.org/10.46298/hrj.2022.8343 https://doi.org/10.46298/hrj.2022.8343 Murty, M. Ram Kumar Murty, V Murty, M. Ram Kumar Murty, V 0$, as $x\to \infty$, $$\sum_{[p-1,q-1]\leq x} {1 \over [p-1, q-1]} \sim C \log x, $$ where the summation is over primes $p,q\leq x$ such that the least common multiple $[p-1,q-1]$ is less than or equal to $x$.]]> 0 Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours Sun, 09 Jan 2022 16:29:59 +0000 https://doi.org/10.46298/hrj.2022.8927 https://doi.org/10.46298/hrj.2022.8927 Deka Baruah, Nayandeep Das, Hirakjyoti Deka Baruah, Nayandeep Das, Hirakjyoti 0 Power partitions and a generalized eta transformation property Sun, 09 Jan 2022 16:19:17 +0000 https://doi.org/10.46298/hrj.2022.8932 https://doi.org/10.46298/hrj.2022.8932 Zagier, Don Zagier, Don 0 Congruences modulo powers of 5 for the rank parity function Thu, 06 May 2021 12:52:27 +0000 https://doi.org/10.46298/hrj.2021.7424 https://doi.org/10.46298/hrj.2021.7424 Chen, Dandan Chen, Rong Garvan, Frank Chen, Dandan Chen, Rong Garvan, Frank 0 A localized Erdős-Kac theorem Thu, 06 May 2021 12:29:56 +0000 https://doi.org/10.46298/hrj.2021.7433 https://doi.org/10.46298/hrj.2021.7433 Dixit, Anup B Ram Murty, M Dixit, Anup B Ram Murty, M 0 A universal identity for theta functions of degree eight and applications Thu, 06 May 2021 12:26:17 +0000 https://doi.org/10.46298/hrj.2021.7432 https://doi.org/10.46298/hrj.2021.7432 Liu, Zhi-Guo Liu, Zhi-Guo 0 One level density of low-lying zeros of quadratic Hecke L-functions to prime moduli Thu, 06 May 2021 12:16:41 +0000 https://doi.org/10.46298/hrj.2021.7461 https://doi.org/10.46298/hrj.2021.7461 Gao, Peng Zhao, Liangyi Gao, Peng Zhao, Liangyi 0 Distribution of generalized mex-related integer partitions Thu, 06 May 2021 12:08:34 +0000 https://doi.org/10.46298/hrj.2021.7425 https://doi.org/10.46298/hrj.2021.7425 Chakraborty, Kalyan Ray, Chiranjit Chakraborty, Kalyan Ray, Chiranjit 0 Arithmetical Fourier transforms and Hilbert space: Restoration of the lost legacy Thu, 06 May 2021 12:06:03 +0000 https://doi.org/10.46298/hrj.2021.7426 https://doi.org/10.46298/hrj.2021.7426 Feng, J.-W Kanemitsu, S Kuzumaki, T Feng, J.-W Kanemitsu, S Kuzumaki, T 0 (q-)Supercongruences hit again Thu, 06 May 2021 12:01:11 +0000 https://doi.org/10.46298/hrj.2021.7427 https://doi.org/10.46298/hrj.2021.7427 Zudilin, Wadim Zudilin, Wadim 0 Partition-theoretic formulas for arithmetic densities, II Thu, 06 May 2021 11:57:52 +0000 https://doi.org/10.46298/hrj.2021.7428 https://doi.org/10.46298/hrj.2021.7428 Ono, Ken Schneider, Robert Wagner, Ian Ono, Ken Schneider, Robert Wagner, Ian 0 Ramanujan's Beautiful Integrals Thu, 06 May 2021 11:54:25 +0000 https://doi.org/10.46298/hrj.2021.7429 https://doi.org/10.46298/hrj.2021.7429 Berndt, Bruce, Dixit, Atul Berndt, Bruce, Dixit, Atul 0 Bounds for d-distinct partitions Thu, 06 May 2021 11:50:24 +0000 https://doi.org/10.46298/hrj.2021.7430 https://doi.org/10.46298/hrj.2021.7430 Kang, Soon-Yi Kim, Young, Kang, Soon-Yi Kim, Young, 0 A heuristic guide to evaluating triple-sums Thu, 06 May 2021 11:45:16 +0000 https://doi.org/10.46298/hrj.2021.7431 https://doi.org/10.46298/hrj.2021.7431 Mortenson, Eric, Mortenson, Eric, 0 A Reinforcement Learning Based Algorithm to Find a Triangular Graham Partition Thu, 06 May 2021 11:27:33 +0000 https://doi.org/10.46298/hrj.2021.7434 https://doi.org/10.46298/hrj.2021.7434 Byungchan, Kim Byungchan, Kim 0 Multiplicatively dependent vectors with coordinates algebraic numbers Mon, 29 Jun 2020 09:46:52 +0000 https://doi.org/10.46298/hrj.2020.6603 https://doi.org/10.46298/hrj.2020.6603 Stewart, C, Stewart, C, 0 Some New Congruences for l-Regular Partitions Modulo 13, 17, and 23 Thu, 21 May 2020 08:39:02 +0000 https://doi.org/10.46298/hrj.2020.5827 https://doi.org/10.46298/hrj.2020.5827 Abinash, S Kathiravan, T Srilakshmi, K Abinash, S Kathiravan, T Srilakshmi, K 0 Density modulo 1 of a sequence associated to some multiplicative functions evaluated at polynomial arguments Thu, 21 May 2020 08:20:07 +0000 https://doi.org/10.46298/hrj.2020.5650 https://doi.org/10.46298/hrj.2020.5650 Nasiri-Zare, Mohammad Nasiri-Zare, Mohammad 0 On some Lambert-like series Wed, 20 May 2020 19:57:40 +0000 https://doi.org/10.46298/hrj.2020.6461 https://doi.org/10.46298/hrj.2020.6461 Agarwal, P Kanemitsu, S Kuzumaki, T Agarwal, P Kanemitsu, S Kuzumaki, T 0 Divisibility of Selmer groups and class groups Wed, 20 May 2020 19:56:53 +0000 https://doi.org/10.46298/hrj.2020.6460 https://doi.org/10.46298/hrj.2020.6460 Banerjee, Kalyan Chakraborty, Kalyan Hoque, Azizul Banerjee, Kalyan Chakraborty, Kalyan Hoque, Azizul 0 A transference inequality for rational approximation to points in geometric progression Wed, 20 May 2020 19:49:02 +0000 https://doi.org/10.46298/hrj.2020.5819 https://doi.org/10.46298/hrj.2020.5819 Champagne, Jérémy Roy, Damien Champagne, Jérémy Roy, Damien 0 Linear forms in logarithms and exponential Diophantine equations Wed, 20 May 2020 19:44:16 +0000 https://doi.org/10.46298/hrj.2020.6458 https://doi.org/10.46298/hrj.2020.6458 Tijdeman, Rob Tijdeman, Rob 0 Polynomial representations of GL(m|n) Wed, 20 May 2020 19:42:30 +0000 https://doi.org/10.46298/hrj.2020.6459 https://doi.org/10.46298/hrj.2020.6459 Z. Flicker, Yuval Z. Flicker, Yuval 0 On the Galois group of Generalised Laguerre polynomials II Wed, 20 May 2020 19:40:43 +0000 https://doi.org/10.46298/hrj.2020.6457 https://doi.org/10.46298/hrj.2020.6457 Laishram, Shanta G. Nair, Saranya Shorey, T. N. Laishram, Shanta G. Nair, Saranya Shorey, T. N. 0 Class Numbers of Quadratic Fields Wed, 20 May 2020 19:37:23 +0000 https://doi.org/10.46298/hrj.2020.6488 https://doi.org/10.46298/hrj.2020.6488 Bhand, Ajit Ram Murty, M Bhand, Ajit Ram Murty, M 0 The Barban-Vehov Theorem in Arithmetic Progressions Wed, 23 Jan 2019 14:41:30 +0000 https://doi.org/10.46298/hrj.2019.5118 https://doi.org/10.46298/hrj.2019.5118 Kumar Murty , V Kumar Murty , V 0 Explicit abc-conjecture and its applications Wed, 23 Jan 2019 14:40:55 +0000 https://doi.org/10.46298/hrj.2019.5117 https://doi.org/10.46298/hrj.2019.5117 Kwok , Chi Chim Nair , Saranya G. Shorey , T. N. Kwok , Chi Chim Nair , Saranya G. Shorey , T. N. 0 Integral points on circles 0$, $\gcd(m,n^2)$ squarefree and $a,b\in\mathbb Q$ does not exceed $r(m)/4$, where $r(m)$ is the number of representations of $m$ as the sum of two squares, unless $n|2$ and $n\cdot (a,b)\in\mathbb Z^2$; then $ N\leq r(m)$}.]]> Wed, 23 Jan 2019 14:40:18 +0000 https://doi.org/10.46298/hrj.2019.5116 https://doi.org/10.46298/hrj.2019.5116 Schinzel , A Skalba , M Schinzel , A Skalba , M 0$, $\gcd(m,n^2)$ squarefree and $a,b\in\mathbb Q$ does not exceed $r(m)/4$, where $r(m)$ is the number of representations of $m$ as the sum of two squares, unless $n|2$ and $n\cdot (a,b)\in\mathbb Z^2$; then $ N\leq r(m)$}.]]> 0 A note on some congruences involving arithmetic functions Wed, 23 Jan 2019 14:39:48 +0000 https://doi.org/10.46298/hrj.2019.5115 https://doi.org/10.46298/hrj.2019.5115 Sándor , József Sándor , József 0 A remark on cube-free numbers in Segal-Piatestki-Shapiro sequences Wed, 23 Jan 2019 14:38:59 +0000 https://doi.org/10.46298/hrj.2019.5114 https://doi.org/10.46298/hrj.2019.5114 Deshouillers , Jean-Marc Deshouillers , Jean-Marc 0 Set Equidistribution of subsets of (Z/nZ) * Wed, 23 Jan 2019 14:38:20 +0000 https://doi.org/10.46298/hrj.2019.5113 https://doi.org/10.46298/hrj.2019.5113 Chattopadhyay , Jaitra Kumar , Veekesh Thangadurai , R Chattopadhyay , Jaitra Kumar , Veekesh Thangadurai , R 0 Hybrid level aspect subconvexity for GL(2) × GL(1) Rankin-Selberg L-Functions Wed, 23 Jan 2019 14:36:30 +0000 https://doi.org/10.46298/hrj.2019.5112 https://doi.org/10.46298/hrj.2019.5112 Aggarwal , Keshav Jo , Yeongseong Nowland , Kevin Aggarwal , Keshav Jo , Yeongseong Nowland , Kevin 0 Two applications of number theory to discrete tomography Wed, 23 Jan 2019 14:35:46 +0000 https://doi.org/10.46298/hrj.2019.5111 https://doi.org/10.46298/hrj.2019.5111 Tijdeman , Rob Tijdeman , Rob 0 On certain sums over ordinates of zeta-zeros II 0} {\gamma}^{-s}$ to the left of the line $\Re{s} = −1 $ is investigated, and its Laurent expansion at the pole s = 1 is obtained. Estimates for the second moment on the critical line $\int_{1}^{T} {| G (\frac{1}{2} + it) |}^2 dt $ are revisited. This paper is a continuation of work begun by the second author in [Iv01].]]> Wed, 23 Jan 2019 14:35:00 +0000 https://doi.org/10.46298/hrj.2019.5110 https://doi.org/10.46298/hrj.2019.5110 Bondarenko , Andriy Ivić , Aleksandar Saksman , Eero Seip , Kristian Bondarenko , Andriy Ivić , Aleksandar Saksman , Eero Seip , Kristian 0} {\gamma}^{-s}$ to the left of the line $\Re{s} = −1 $ is investigated, and its Laurent expansion at the pole s = 1 is obtained. Estimates for the second moment on the critical line $\int_{1}^{T} {| G (\frac{1}{2} + it) |}^2 dt $ are revisited. This paper is a continuation of work begun by the second author in [Iv01].]]> 0 The Zeta Mahler measure of $(z^n − 1)/(z − 1)$ Wed, 23 Jan 2019 14:34:11 +0000 https://doi.org/10.46298/hrj.2019.5109 https://doi.org/10.46298/hrj.2019.5109 Biswas , Arunabha Murty , M Ram Biswas , Arunabha Murty , M Ram 0 An elementary property of correlations 0. Assuming Delange Hypothesis for the correlation, we get the "Ramanujan exact explicit formula", a kind of finite shift-Ramanujan expansion. A noteworthy case is when f = g = Λ, the von Mangoldt function; so $C_{\Lamda, \Lambda} (N, 2k)$, for natural k, corresponds to 2k-twin primes; under the assumption of Delange Hypothesis, we easily obtain the proof of Hardy-Littlewood Conjecture for this case.]]> Wed, 23 Jan 2019 14:33:18 +0000 https://doi.org/10.46298/hrj.2019.5108 https://doi.org/10.46298/hrj.2019.5108 Coppola , Giovanni Coppola , Giovanni 0. Assuming Delange Hypothesis for the correlation, we get the "Ramanujan exact explicit formula", a kind of finite shift-Ramanujan expansion. A noteworthy case is when f = g = Λ, the von Mangoldt function; so $C_{\Lamda, \Lambda} (N, 2k)$, for natural k, corresponds to 2k-twin primes; under the assumption of Delange Hypothesis, we easily obtain the proof of Hardy-Littlewood Conjecture for this case.]]> 0 When are Multiples of Polygonal Numbers again Polygonal Numbers? 1, the relation ∆ = m∆' is satisfied by infinitely many pairs of triangular numbers ∆, ∆'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit of Q(√ m), finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP' , P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P , P' , P'' of polygonal numbers, and give examples of such solutions.]]> Wed, 23 Jan 2019 14:32:36 +0000 https://doi.org/10.46298/hrj.2019.5107 https://doi.org/10.46298/hrj.2019.5107 Chahal , Jasbir , Griffin , Michael Priddis , Nathan Chahal , Jasbir , Griffin , Michael Priddis , Nathan 1, the relation ∆ = m∆' is satisfied by infinitely many pairs of triangular numbers ∆, ∆'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit of Q(√ m), finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP' , P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P , P' , P'' of polygonal numbers, and give examples of such solutions.]]> 0 On-Regular Bipartitions Modulo $m$ Wed, 23 Jan 2019 14:31:19 +0000 https://doi.org/10.46298/hrj.2019.5106 https://doi.org/10.46298/hrj.2019.5106 Somashekara , D D Vidya , K N Somashekara , D D Vidya , K N 0 On an identity of Ramanujan Wed, 23 Jan 2019 14:30:10 +0000 https://doi.org/10.46298/hrj.2019.5105 https://doi.org/10.46298/hrj.2019.5105 Motohashi , Yoichi Motohashi , Yoichi 0 On the Wintner-Ingham-Segal summability method Wed, 23 Jan 2019 14:28:48 +0000 https://doi.org/10.46298/hrj.2019.5104 https://doi.org/10.46298/hrj.2019.5104 Kanemitsu , S Kuzumaki , T Tanigawa , Y Kanemitsu , S Kuzumaki , T Tanigawa , Y 0 Fluctuation of the primitive of Hardy's function Wed, 23 Jan 2019 14:27:05 +0000 https://doi.org/10.46298/hrj.2019.5103 https://doi.org/10.46298/hrj.2019.5103 Jutila , Matti Jutila , Matti 0 On Pillai's problem with Pell numbers and powers of 2 Wed, 23 Jan 2019 14:24:02 +0000 https://doi.org/10.46298/hrj.2019.5102 https://doi.org/10.46298/hrj.2019.5102 Ouamar Hernane , Mohand Luca , Florian Rihane , Salah , Togbé , Alain Ouamar Hernane , Mohand Luca , Florian Rihane , Salah , Togbé , Alain 0 A Theorem of Fermat on Congruent Number Curves Wed, 23 Jan 2019 14:19:01 +0000 https://doi.org/10.46298/hrj.2019.5101 https://doi.org/10.46298/hrj.2019.5101 Halbeisen , Lorenz Hungerbühler , Norbert Halbeisen , Lorenz Hungerbühler , Norbert 0 On the Riesz means of δ k (n) Thu, 11 Jan 2018 16:05:07 +0000 https://doi.org/10.46298/hrj.2018.2058 https://doi.org/10.46298/hrj.2018.2058 Singh , Saurabh Singh , Saurabh 0 Dual Ramanujan-Fourier series Thu, 11 Jan 2018 16:04:40 +0000 https://doi.org/10.46298/hrj.2018.2541 https://doi.org/10.46298/hrj.2018.2541 Ushiroya , Noboru Ushiroya , Noboru 0 Non-vanishing of Dirichlet series without Euler products 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).]]> Thu, 11 Jan 2018 16:03:50 +0000 https://doi.org/10.46298/hrj.2018.4027 https://doi.org/10.46298/hrj.2018.4027 Banks , William D. Banks , William D. 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).]]> 0 Book review : Lectures on the Riemann Zeta Function, by Henryk Iwaniec Wed, 18 Jan 2017 17:31:05 +0000 https://doi.org/10.46298/hrj.2017.2650 https://doi.org/10.46298/hrj.2017.2650 Perelli, Alberto Perelli, Alberto 0 Contributions of Ramachandra to the Theory of the Riemann Zeta-Function Mon, 09 Jan 2017 13:34:48 +0000 https://doi.org/10.46298/hrj.2017.2633 https://doi.org/10.46298/hrj.2017.2633 Sankaranarayanan, A. Sankaranarayanan, A. 0 A note on Hardy's theorem Mon, 09 Jan 2017 13:34:04 +0000 https://doi.org/10.46298/hrj.2017.2634 https://doi.org/10.46298/hrj.2017.2634 Sangale, Usha K. Sangale, Usha K. 0 Sieve functions in arithmetic bands 0. Here, we study the distribution of f over the so-called short arithmetic bands 1≤a≤H {n ∈ (N, 2N ] : n ≡ a (mod q)}, with H = o(N), and give applications to both the correlations and to the so-called weighted Selberg integrals of f , on which we have concentrated our recent research.]]> Mon, 09 Jan 2017 13:33:26 +0000 https://doi.org/10.46298/hrj.2017.2635 https://doi.org/10.46298/hrj.2017.2635 Coppola, Giovanni Laporta, Maurizio Coppola, Giovanni Laporta, Maurizio 0. Here, we study the distribution of f over the so-called short arithmetic bands 1≤a≤H {n ∈ (N, 2N ] : n ≡ a (mod q)}, with H = o(N), and give applications to both the correlations and to the so-called weighted Selberg integrals of f , on which we have concentrated our recent research.]]> 0 Ramanujan-Fourier series of certain arithmetic functions of two variables Mon, 09 Jan 2017 13:32:35 +0000 https://doi.org/10.46298/hrj.2017.2636 https://doi.org/10.46298/hrj.2017.2636 Ushiroya, Noboru Ushiroya, Noboru 0 Remarks on the impossibility of a Siegel-Shidlovskii like theorem for G-functions Thu, 01 Jan 2015 07:00:00 +0000 https://doi.org/10.46298/hrj.2015.1357 https://doi.org/10.46298/hrj.2015.1357 Rivoal, T Rivoal, T 0 Algebraic independence results on the generating Lambert series of the powers of a fixed integer 1 a fixed integer, at non-zero algebraic points in the unit disk is studied. Whereas the case of multiplicatively independent points has been resolved some time ago, a particularly interesting case of multiplicatively dependent points is considered here, and similar results are obtained for more general functions. The main tool is Mahler's method reducing the investigation of the algebraic independence of numbers (over Q) to the one of functions (over the rational function field) if these satisfy certain types of functional equations.]]> Thu, 01 Jan 2015 07:00:00 +0000 https://doi.org/10.46298/hrj.2015.1358 https://doi.org/10.46298/hrj.2015.1358 Bundschuh, Peter Väänänen, Keijo Bundschuh, Peter Väänänen, Keijo 1 a fixed integer, at non-zero algebraic points in the unit disk is studied. Whereas the case of multiplicatively independent points has been resolved some time ago, a particularly interesting case of multiplicatively dependent points is considered here, and similar results are obtained for more general functions. The main tool is Mahler's method reducing the investigation of the algebraic independence of numbers (over Q) to the one of functions (over the rational function field) if these satisfy certain types of functional equations.]]> 0 Effective irrationality measures for quotients of logarithms of rational numbers Thu, 01 Jan 2015 07:00:00 +0000 https://doi.org/10.46298/hrj.2015.1359 https://doi.org/10.46298/hrj.2015.1359 Bugeaud, Yann Bugeaud, Yann 0 Baker's Explicit abc-Conjecture and Waring's problem Thu, 01 Jan 2015 07:00:00 +0000 https://doi.org/10.46298/hrj.2015.1360 https://doi.org/10.46298/hrj.2015.1360 Laishram, Shanta Laishram, Shanta 0 Mean Values of Certain Multiplicative Functions and Artin's Conjecture on Primitive Roots Wed, 01 Jan 2014 07:00:00 +0000 https://doi.org/10.46298/hrj.2014.1316 https://doi.org/10.46298/hrj.2014.1316 Sitaraman, Sankar Sitaraman, Sankar 0 On the Galois groups of generalized Laguerre Polynomials Wed, 01 Jan 2014 07:00:00 +0000 https://doi.org/10.46298/hrj.2014.1317 https://doi.org/10.46298/hrj.2014.1317 Laishram, Shanta Laishram, Shanta 0 New developments on the twin prime problem and generalizations Wed, 01 Jan 2014 07:00:00 +0000 https://doi.org/10.46298/hrj.2014.1318 https://doi.org/10.46298/hrj.2014.1318 Murty, M. Ram Murty, M. Ram 0 On congruences for certain sums of E. Lehmer's type 1 be an odd natural number and let r (1 < r < n) be a natural number relatively prime to n. Denote by χn the principal character modulo n. In Section 3 we prove some new congruences for the sums T r,k (n) = n r ] i=1 (χn(i) i k) (mod n s+1) for s ∈ {0, 1, 2}, for all divisors r of 24 and for some natural numbers k.We obtain 82 new congruences for T r,k (n), which generalize those obtained in [Ler05], [Leh38] and [Sun08] if n = p is an odd prime. Section 4 is an appendix by the second and third named authors. It contains some new congruences for the sums Ur(n) = n]]> Wed, 01 Jan 2014 07:00:00 +0000 https://doi.org/10.46298/hrj.2014.1356 https://doi.org/10.46298/hrj.2014.1356 Kanemitsu, Shigeru Kuzumaki, Takako Urbanowicz, Jerzy Kanemitsu, Shigeru Kuzumaki, Takako Urbanowicz, Jerzy 1 be an odd natural number and let r (1 < r < n) be a natural number relatively prime to n. Denote by χn the principal character modulo n. In Section 3 we prove some new congruences for the sums T r,k (n) = n r ] i=1 (χn(i) i k) (mod n s+1) for s ∈ {0, 1, 2}, for all divisors r of 24 and for some natural numbers k.We obtain 82 new congruences for T r,k (n), which generalize those obtained in [Ler05], [Leh38] and [Sun08] if n = p is an odd prime. Section 4 is an appendix by the second and third named authors. It contains some new congruences for the sums Ur(n) = n]]> 0 On the Half Line: K Ramachandra. Tue, 01 Jan 2013 07:00:00 +0000 https://doi.org/10.46298/hrj.2013.172 https://doi.org/10.46298/hrj.2013.172 Sinha, Nilotpal Kanti Sinha, Nilotpal Kanti 0 The work of K. Ramachandra in algebraic number theory Tue, 01 Jan 2013 07:00:00 +0000 https://doi.org/10.46298/hrj.2013.173 https://doi.org/10.46298/hrj.2013.173 Murty, M. Ram Murty, M. Ram 0 On Ramachandra's Contributions to Transcendental Number Theory Tue, 01 Jan 2013 07:00:00 +0000 https://doi.org/10.46298/hrj.2013.174 https://doi.org/10.46298/hrj.2013.174 Waldschmidt, Michel Waldschmidt, Michel 0 Mathematical Reminiscences: How to Keep the Pot Boiling Tue, 01 Jan 2013 07:00:00 +0000 https://doi.org/10.46298/hrj.2013.175 https://doi.org/10.46298/hrj.2013.175 Ramachandra, K Ramachandra, K 0 K. Ramachandra : Reminiscences of his Students. Tue, 01 Jan 2013 07:00:00 +0000 https://doi.org/10.46298/hrj.2013.176 https://doi.org/10.46298/hrj.2013.176 Sankaranarayanan, A Narlikar, Mangala J Srinivas, K Bhat, K G Sankaranarayanan, A Narlikar, Mangala J Srinivas, K Bhat, K G 0 K. Ramachandra: Reminiscences of his Friends. Tue, 01 Jan 2013 07:00:00 +0000 https://doi.org/10.46298/hrj.2013.177 https://doi.org/10.46298/hrj.2013.177 Murthy, M. P Waldschmidt, Michel Soundararajan, K Vaidya, Prabhakar Jutila, Matti Murthy, M. P Waldschmidt, Michel Soundararajan, K Vaidya, Prabhakar Jutila, Matti 0 On the class number formula of certain real quadratic fields Tue, 01 Jan 2013 07:00:00 +0000 https://doi.org/10.46298/hrj.2013.178 https://doi.org/10.46298/hrj.2013.178 Chakraborty, K Kanemitsu, S Kuzumaki, T Chakraborty, K Kanemitsu, S Kuzumaki, T 0 On the Riesz means of $\frac{n}{\phi(n)}$ Tue, 01 Jan 2013 07:00:00 +0000 https://doi.org/10.46298/hrj.2013.179 https://doi.org/10.46298/hrj.2013.179 Sankaranarayanan, A Singh, Saurabh Kumar Sankaranarayanan, A Singh, Saurabh Kumar 0 Ramanujan series for arithmetical functions Tue, 01 Jan 2013 07:00:00 +0000 https://doi.org/10.46298/hrj.2013.180 https://doi.org/10.46298/hrj.2013.180 Murty, M. Ram Murty, M. Ram 0 I am fifty-five years old Tue, 01 Jan 2013 07:00:00 +0000 https://doi.org/10.46298/hrj.2013.181 https://doi.org/10.46298/hrj.2013.181 Ramachandra, K Ramachandra, K 0 The combinatorics of moment calculations Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/hrj.2010.168 https://doi.org/10.46298/hrj.2010.168 Montgomery, hugh L Montgomery, hugh L 0 An estimate for the Mellin transform of powers of Hardy's function. Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/hrj.2010.169 https://doi.org/10.46298/hrj.2010.169 Jutila, Matti Jutila, Matti 0 On the Mellin transforms of powers of Hardy's function. Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/hrj.2010.170 https://doi.org/10.46298/hrj.2010.170 Ivić, Aleksandar Ivić, Aleksandar 0 Some diophantine problems concerning equal sums of integers and their cubes Fri, 01 Jan 2010 07:00:00 +0000 https://doi.org/10.46298/hrj.2010.171 https://doi.org/10.46298/hrj.2010.171 Choudhry, Ajai Choudhry, Ajai 0 On mean value results for the Riemann zeta-function in short intervals. Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/hrj.2009.164 https://doi.org/10.46298/hrj.2009.164 Ivić, Aleksandar Ivić, Aleksandar 0 Alternating Euler sums at the negative integers. Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/hrj.2009.165 https://doi.org/10.46298/hrj.2009.165 Boyadzhiev, Khristo Gadiyar, H. Gopalkrishna Padma, R Boyadzhiev, Khristo Gadiyar, H. Gopalkrishna Padma, R 0 Finite expressions for higher derivatives of the Dirichlet L-function and the Deninger R-function Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/hrj.2009.166 https://doi.org/10.46298/hrj.2009.166 Chakraborty, K Kanemitsu, S Kuzumaki, T Chakraborty, K Kanemitsu, S Kuzumaki, T 0 Little flowers to Srinivasa Ramanujan. Thu, 01 Jan 2009 07:00:00 +0000 https://doi.org/10.46298/hrj.2009.167 https://doi.org/10.46298/hrj.2009.167 Ramachandra, K Ramachandra, K 0 Ideal solutions of the Tarry-Escott problem of degree eleven with applications to sums of thirteenth powers. Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/hrj.2008.161 https://doi.org/10.46298/hrj.2008.161 Choudhry, Ajai Wroblewski, Jaroslaw Choudhry, Ajai Wroblewski, Jaroslaw 0 Arithmetical investigations of particular Wynn power series 1$), and if $\beta$ is a unit in $\mathbb{Q}(q)$ with $|\beta|\le1$ but no other conjugates in the open unit disc. Our applications concern meromorphic functions defined in $|z|<|u|^{a\ell}$ by power series $\sum_{n\ge1}z^n/(\prod_{0\le\lambda<\ell}R_{a(n+\lambda)+b})$, where $R_m:=gu^m+hv^m$ with non-zero $u,v,g,h$ satisfying $|u|>|v|, R_m\ne0$ for any $m\ge1$, and $a,b+1,\ell$ are positive rational integers. Clearly, the case where $R_m$ are the Fibonacci or Lucas numbers is of particular interest. It should be noted that power series of the above type were first studied by Wynn from the analytical point of view.]]> Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/hrj.2008.162 https://doi.org/10.46298/hrj.2008.162 Bundschuh, Peter Bundschuh, Peter 1$), and if $\beta$ is a unit in $\mathbb{Q}(q)$ with $|\beta|\le1$ but no other conjugates in the open unit disc. Our applications concern meromorphic functions defined in $|z|<|u|^{a\ell}$ by power series $\sum_{n\ge1}z^n/(\prod_{0\le\lambda<\ell}R_{a(n+\lambda)+b})$, where $R_m:=gu^m+hv^m$ with non-zero $u,v,g,h$ satisfying $|u|>|v|, R_m\ne0$ for any $m\ge1$, and $a,b+1,\ell$ are positive rational integers. Clearly, the case where $R_m$ are the Fibonacci or Lucas numbers is of particular interest. It should be noted that power series of the above type were first studied by Wynn from the analytical point of view.]]> 0 On Kronecker's limit formula and the hypergeometric function. Tue, 01 Jan 2008 07:00:00 +0000 https://doi.org/10.46298/hrj.2008.163 https://doi.org/10.46298/hrj.2008.163 Kanemitsu, S Tanigawa, Y Tsukada, H Kanemitsu, S Tanigawa, Y Tsukada, H 0 Carmichael number with three prime factors. Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/hrj.2007.156 https://doi.org/10.46298/hrj.2007.156 Heath-Brown , D R Heath-Brown , D R 0 The number of imaginary quadratic fields with a given class number Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/hrj.2007.157 https://doi.org/10.46298/hrj.2007.157 Soundararajan, K Soundararajan, K 0 Rational triangles with the same perimeter and the same area. Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/hrj.2007.158 https://doi.org/10.46298/hrj.2007.158 Choudhry, Ajai Choudhry, Ajai 0 Contributions to the theory of the Hurwitz zeta-function Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/hrj.2007.159 https://doi.org/10.46298/hrj.2007.159 Kanemitsu, S Tanigawa, Y Tsukada, H Yoshimoto, M Kanemitsu, S Tanigawa, Y Tsukada, H Yoshimoto, M 0 On the Barban-Davenport Halberstam theorem: XIX \frac{1}{12}\{\Gamma(C)+o(1)\}Q^2+O(x \log_{-A}x)$ when $x/Q\rightarrow\infty$ where $\Gamma(C)$ is an explicitly defined function of $C$.]]> Mon, 01 Jan 2007 07:00:00 +0000 https://doi.org/10.46298/hrj.2007.160 https://doi.org/10.46298/hrj.2007.160 Hooley, C Hooley, C \frac{1}{12}\{\Gamma(C)+o(1)\}Q^2+O(x \log_{-A}x)$ when $x/Q\rightarrow\infty$ where $\Gamma(C)$ is an explicitly defined function of $C$.]]> 0 On polynomials that equal binary cubic forms. Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/hrj.2006.153 https://doi.org/10.46298/hrj.2006.153 Hooley, C Hooley, C 0 On the periodic Hurwitz zeta-function. Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/hrj.2006.154 https://doi.org/10.46298/hrj.2006.154 Javtokas, A Laurinčikas, A Javtokas, A Laurinčikas, A 0 A remark on a theorem of A. E. Ingham. \frac{5}{8})$ is any constant. Further, we point out that even this stronger form can be proved without using the functional equation of $\zeta(s)$.]]> Sun, 01 Jan 2006 07:00:00 +0000 https://doi.org/10.46298/hrj.2006.155 https://doi.org/10.46298/hrj.2006.155 Bhat, K G Ramachandra, K Bhat, K G Ramachandra, K \frac{5}{8})$ is any constant. Further, we point out that even this stronger form can be proved without using the functional equation of $\zeta(s)$.]]> 0 A lower bound concerning subset sums which do not cover all the residues modulo $p$. \sqrt{2}$ and let $p$ be a prime number. J-M. Deshouillers and G. A. Freiman proved that a subset $\mathcal A$ of $\mathbb{Z}/p\mathbb{Z}$, with cardinality larger than $c\sqrt{p}$ and such that its subset sums do not cover $\mathbb{Z}/p\mathbb{Z}$ has an isomorphic image which is rather concentrated; more precisely, there exists $s$ prime to $p$ such that $$\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert < 1+O(p^{-1/4}\ln p),$$ where the constant implied in the ``O'' symbol depends on $c$ at most. We show here that there exist a $K$ depending on $c$ at most, and such sets $\mathcal A$, such that for all $s$ prime to $p$ one has $$ \sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$$]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/hrj.2005.85 https://doi.org/10.46298/hrj.2005.85 Deshouillers, Jean-Marc Deshouillers, Jean-Marc \sqrt{2}$ and let $p$ be a prime number. J-M. Deshouillers and G. A. Freiman proved that a subset $\mathcal A$ of $\mathbb{Z}/p\mathbb{Z}$, with cardinality larger than $c\sqrt{p}$ and such that its subset sums do not cover $\mathbb{Z}/p\mathbb{Z}$ has an isomorphic image which is rather concentrated; more precisely, there exists $s$ prime to $p$ such that $$\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert < 1+O(p^{-1/4}\ln p),$$ where the constant implied in the ``O'' symbol depends on $c$ at most. We show here that there exist a $K$ depending on $c$ at most, and such sets $\mathcal A$, such that for all $s$ prime to $p$ one has $$ \sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$$]]> 0 Further Variations on the Six Exponentials Theorem. Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/hrj.2005.86 https://doi.org/10.46298/hrj.2005.86 Waldschmidt, Michel Waldschmidt, Michel 0 On an exponential sum involving the Möbius function Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/hrj.2005.151 https://doi.org/10.46298/hrj.2005.151 Maier, H. Sankaranarayanan, A Maier, H. Sankaranarayanan, A 0 A lower bound concerning subset sums which do not cover all the residues modulo $p$. \sqrt{2}$ and let $p$ be a prime number. J-M. Deshouillers and G. A. Freiman proved that a subset $\mathcal A$ of $\mathbb{Z}/p\mathbb{Z}$, with cardinality larger than $c\sqrt{p}$ and such that its subset sums do not cover $\mathbb{Z}/p\mathbb{Z}$ has an isomorphic image which is rather concentrated; more precisely, there exists $s$ prime to $p$ such that $$\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert < 1+O(p^{-1/4}\ln p),$$ where the constant implied in the ``O'' symbol depends on $c$ at most. We show here that there exist a $K$ depending on $c$ at most, and such sets $\mathcal A$, such that for all $s$ prime to $p$ one has $$ \sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$$]]> Sat, 01 Jan 2005 07:00:00 +0000 https://doi.org/10.46298/hrj.2005.152 https://doi.org/10.46298/hrj.2005.152 Deshouillers, Jean-Marc Deshouillers, Jean-Marc \sqrt{2}$ and let $p$ be a prime number. J-M. Deshouillers and G. A. Freiman proved that a subset $\mathcal A$ of $\mathbb{Z}/p\mathbb{Z}$, with cardinality larger than $c\sqrt{p}$ and such that its subset sums do not cover $\mathbb{Z}/p\mathbb{Z}$ has an isomorphic image which is rather concentrated; more precisely, there exists $s$ prime to $p$ such that $$\sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert < 1+O(p^{-1/4}\ln p),$$ where the constant implied in the ``O'' symbol depends on $c$ at most. We show here that there exist a $K$ depending on $c$ at most, and such sets $\mathcal A$, such that for all $s$ prime to $p$ one has $$ \sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$$]]> 0 Hardy-Littlewood first approximation theorem for quasi $L$-functions Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/hrj.2004.148 https://doi.org/10.46298/hrj.2004.148 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 0 Mean square of the Hurwitz zeta-function and other remarks 1)$$ and its analytic continuation. %In fact $$\zeta(s,a)=\sum_{n=0}^{\infty}\left((n+a)^{-s}-\int_{n}^{n+1}\frac{du}{(u+a)^s} \right)+\frac{a^{1-s}}{s-1}$$ gives the analytic continuation to $(\sigma>0)$. A repetition of this several times shows that $$\zeta-\frac{a^{1-s}}{s-1}$$ can be continued as an entire function to the whole plane. In $Re(s)\geq-1,\,t\geq2,\,\zeta(s,a)-a^{-s}=O(t^3)$ and by the functional equation (see \S2) it is $$O\left(\left(\frac{\vert s\vert}{2\pi}\right)^{\frac{1}{2}-Re(s)}\right)$$ in $Re(s)\leq-1,\,t\geq2$. From these facts In this paper, we deduce an `Approximate function equation' (see \S3), which is a generalisation of the approximate functional equation for $\zeta(s)$. Combining this with an important theorem due to van-der-Corput, we prove $$T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\zeta(\frac{1}{2}+it)-a^{-\frac{1}{2}-it}\vert^2 dt <\!\!\!< (\log T)^3$$ uniformly in $a(0< a\leq1)$. From this we deduce similar results for quasi $L$-functions and more general functions. %Let $a_1, a_2,\ldots$, be any periodic sequence of complex numbers for which the sum over a period is zero. Let $b_1, b_2,\ldots$ be any sequence of complex numbers for which $\sum_{j=2}^{n}\vert b_j-b_{j-1}\vert+\vert b_n\vert\leq n^{\varepsilon}$ for every $\varepsilon>0$ and every $n\geq n_0(\varepsilon)$. Then we prove $$T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\sum_{n=1}^{\infty}\frac{a_nb_n}{(n+a)^{\frac{1}{2}+it}}\vert^2\,dt\leq T^{\varepsilon}$$ for every $\varepsilon>0$ and every $T\geq T_0(\varepsilon)$. Here, as usual, $0 Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/hrj.2004.149 https://doi.org/10.46298/hrj.2004.149 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 1)$$ and its analytic continuation. %In fact $$\zeta(s,a)=\sum_{n=0}^{\infty}\left((n+a)^{-s}-\int_{n}^{n+1}\frac{du}{(u+a)^s} \right)+\frac{a^{1-s}}{s-1}$$ gives the analytic continuation to $(\sigma>0)$. A repetition of this several times shows that $$\zeta-\frac{a^{1-s}}{s-1}$$ can be continued as an entire function to the whole plane. In $Re(s)\geq-1,\,t\geq2,\,\zeta(s,a)-a^{-s}=O(t^3)$ and by the functional equation (see \S2) it is $$O\left(\left(\frac{\vert s\vert}{2\pi}\right)^{\frac{1}{2}-Re(s)}\right)$$ in $Re(s)\leq-1,\,t\geq2$. From these facts In this paper, we deduce an `Approximate function equation' (see \S3), which is a generalisation of the approximate functional equation for $\zeta(s)$. Combining this with an important theorem due to van-der-Corput, we prove $$T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\zeta(\frac{1}{2}+it)-a^{-\frac{1}{2}-it}\vert^2 dt <\!\!\!< (\log T)^3$$ uniformly in $a(0< a\leq1)$. From this we deduce similar results for quasi $L$-functions and more general functions. %Let $a_1, a_2,\ldots$, be any periodic sequence of complex numbers for which the sum over a period is zero. Let $b_1, b_2,\ldots$ be any sequence of complex numbers for which $\sum_{j=2}^{n}\vert b_j-b_{j-1}\vert+\vert b_n\vert\leq n^{\varepsilon}$ for every $\varepsilon>0$ and every $n\geq n_0(\varepsilon)$. Then we prove $$T^{-\frac{1}{3}}\int_{T}^{T+T^{\frac{1}{3}}} \vert\sum_{n=1}^{\infty}\frac{a_nb_n}{(n+a)^{\frac{1}{2}+it}}\vert^2\,dt\leq T^{\varepsilon}$$ for every $\varepsilon>0$ and every $T\geq T_0(\varepsilon)$. Here, as usual, $0 0 A generalization of Bochner's formula Thu, 01 Jan 2004 07:00:00 +0000 https://doi.org/10.46298/hrj.2004.150 https://doi.org/10.46298/hrj.2004.150 Kanemitsu, S Tanigawa, Y Tsukada, H Kanemitsu, S Tanigawa, Y Tsukada, H 0 Mean-square upper bound of Hecke $L$-functions on the critical line. Wed, 01 Jan 2003 07:00:00 +0000 https://doi.org/10.46298/hrj.2003.147 https://doi.org/10.46298/hrj.2003.147 Sankaranarayanan, A Sankaranarayanan, A 0 Some problems of Analytic number theory IV Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/hrj.2002.145 https://doi.org/10.46298/hrj.2002.145 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 0 On totally reducible binary forms: II. 0$ and $\eta_l>0$.]]> Tue, 01 Jan 2002 07:00:00 +0000 https://doi.org/10.46298/hrj.2002.146 https://doi.org/10.46298/hrj.2002.146 Hooley, C Hooley, C 0$ and $\eta_l>0$.]]> 0 On some Ramanujan's P-Q Identities Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/hrj.2001.143 https://doi.org/10.46298/hrj.2001.143 Madhusudhan, H S Mahadeva Naika, M S Vasuki, K R Madhusudhan, H S Mahadeva Naika, M S Vasuki, K R 0 On the values of the Riemann zeta-function at rational arguments Mon, 01 Jan 2001 07:00:00 +0000 https://doi.org/10.46298/hrj.2001.144 https://doi.org/10.46298/hrj.2001.144 Kanemitsu, S Tanigawa, Y Yoshimoto, M Kanemitsu, S Tanigawa, Y Yoshimoto, M 0 Notes on the Riemann zeta-function-IV Sat, 01 Jan 2000 07:00:00 +0000 https://doi.org/10.46298/hrj.2000.141 https://doi.org/10.46298/hrj.2000.141 Balasubramanian, R Ramachandra, K Sankaranarayanan, A Srinivas, K Balasubramanian, R Ramachandra, K Sankaranarayanan, A Srinivas, K 0 On a problem of Ivi\'c. 0$, Ivi\'c proved that $\sum_{T<\gamma\leq2T} \vert\zeta(\frac{1}{2}+i\gamma)\vert^2 <\!\!\!<_{\varepsilon} (T(\log T)^2\log\log T)^{3/2+\varepsilon},$ where the implicit constant depends only on $\varepsilon$. In this paper, this result is improved by (i) replacing $\vert\zeta(\frac{1}{2}+i\gamma)\vert^2$ by $\max\vert\zeta(s)\vert^2$, where the maximum is taken over all $s=\sigma+it$ in the rectangle $\frac{1}{2}-A/\log T\leq\sigma\leq2,\, \vert t-\gamma\vert\leq B(\log\log T)/\log T$ with some fixed positive constants $A, B,$ and (ii) replacing the upper bound by $T(\log T)^2\log\log T$. The method of proof differs completely from Ivi\'c's approach.]]> Sat, 01 Jan 2000 07:00:00 +0000 https://doi.org/10.46298/hrj.2000.142 https://doi.org/10.46298/hrj.2000.142 Ramachandra, K Ramachandra, K 0$, Ivi\'c proved that $\sum_{T<\gamma\leq2T} \vert\zeta(\frac{1}{2}+i\gamma)\vert^2 <\!\!\!<_{\varepsilon} (T(\log T)^2\log\log T)^{3/2+\varepsilon},$ where the implicit constant depends only on $\varepsilon$. In this paper, this result is improved by (i) replacing $\vert\zeta(\frac{1}{2}+i\gamma)\vert^2$ by $\max\vert\zeta(s)\vert^2$, where the maximum is taken over all $s=\sigma+it$ in the rectangle $\frac{1}{2}-A/\log T\leq\sigma\leq2,\, \vert t-\gamma\vert\leq B(\log\log T)/\log T$ with some fixed positive constants $A, B,$ and (ii) replacing the upper bound by $T(\log T)^2\log\log T$. The method of proof differs completely from Ivi\'c's approach.]]> 0 Disproof of some conjectures of K.Ramachandra Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/hrj.1999.137 https://doi.org/10.46298/hrj.1999.137 anderson, johan anderson, johan 0 On generalised Carmichael numbers. 1$ is square-free. We also discuss generalised Carmichael numbers which have one or two prime factors. Finally, we consider the Jeans numbers, i.e. the set of odd numbers $n$ which satisfy the equation $a^n\equiv a \mod n$ only for $a=2$, and the corresponding generalizations. We give a stochastic argument which supports the conjecture that infinitely many Jeans numbers exist which are squares.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/hrj.1999.138 https://doi.org/10.46298/hrj.1999.138 Halbeisen, L Hungerbühler, N Halbeisen, L Hungerbühler, N 1$ is square-free. We also discuss generalised Carmichael numbers which have one or two prime factors. Finally, we consider the Jeans numbers, i.e. the set of odd numbers $n$ which satisfy the equation $a^n\equiv a \mod n$ only for $a=2$, and the corresponding generalizations. We give a stochastic argument which supports the conjecture that infinitely many Jeans numbers exist which are squares.]]> 0 Notes on the Riemann zeta Function-III C$ for every fixed $C>0$. Also, we study the gaps between the ordinates of the consecutive poles of $F(s)$.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/hrj.1999.139 https://doi.org/10.46298/hrj.1999.139 Balasubramanian, R Ramachandra, K Sankaranarayanan, A Srinivas, K Balasubramanian, R Ramachandra, K Sankaranarayanan, A Srinivas, K C$ for every fixed $C>0$. Also, we study the gaps between the ordinates of the consecutive poles of $F(s)$.]]> 0 Notes on the Riemann zeta-function-IV C$ for every fixed $C>0$. Also we study the gaps between the numbers $p_2$ arranged in the non-decreasing order.]]> Fri, 01 Jan 1999 07:00:00 +0000 https://doi.org/10.46298/hrj.1999.140 https://doi.org/10.46298/hrj.1999.140 Balasubramanian, R Ramachandra, K Sankaranarayanan, A Balasubramanian, R Ramachandra, K Sankaranarayanan, A C$ for every fixed $C>0$. Also we study the gaps between the numbers $p_2$ arranged in the non-decreasing order.]]> 0 On a method of Davenport and Heilbronn I. Thu, 01 Jan 1998 07:00:00 +0000 https://doi.org/10.46298/hrj.1998.136 https://doi.org/10.46298/hrj.1998.136 Ramachandra, K Ramachandra, K 0 On the Barban-Davenport-Halberstam theorem : X Thu, 01 Jan 1998 07:00:00 +0000 https://doi.org/10.46298/hrj.1998.182 https://doi.org/10.46298/hrj.1998.182 Hooley, C Hooley, C 0 On the zeros of a class of generalised Dirichlet series-XVIII (a few remarks on littlewood's theorem and Totchmarsh points) Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/hrj.1997.134 https://doi.org/10.46298/hrj.1997.134 Balasubramanian, R Ramachandra, K Sankaranarayanan, A Balasubramanian, R Ramachandra, K Sankaranarayanan, A 0 On the zeros of a class of generalised Dirichlet series-XIX Wed, 01 Jan 1997 07:00:00 +0000 https://doi.org/10.46298/hrj.1997.135 https://doi.org/10.46298/hrj.1997.135 Ramachandra, K Ramachandra, K 0 Ramanujan's lattice point problem, prime number theory and other remarks. Mon, 01 Jan 1996 07:00:00 +0000 https://doi.org/10.46298/hrj.1996.133 https://doi.org/10.46298/hrj.1996.133 Ramachandra, K Sankaranarayanan, A Srinivas, K Ramachandra, K Sankaranarayanan, A Srinivas, K 0 A large value theorem for $\zeta(s)$. Sun, 01 Jan 1995 07:00:00 +0000 https://doi.org/10.46298/hrj.1995.130 https://doi.org/10.46298/hrj.1995.130 Ramachandra, K Ramachandra, K 0 On Riemann zeta-function and allied questions-II Sun, 01 Jan 1995 07:00:00 +0000 https://doi.org/10.46298/hrj.1995.131 https://doi.org/10.46298/hrj.1995.131 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 0 Is the binomial coefficient $2n \choose n$ square free? 4$.]]> Sun, 01 Jan 1995 07:00:00 +0000 https://doi.org/10.46298/hrj.1995.132 https://doi.org/10.46298/hrj.1995.132 Velammal, G Velammal, G 4$.]]> 0 Estimate of sums of Dirichlet series Sat, 01 Jan 1994 07:00:00 +0000 https://doi.org/10.46298/hrj.1994.128 https://doi.org/10.46298/hrj.1994.128 Shituo, Lou Qi, Yao Shituo, Lou Qi, Yao 0 On some over primes 0$, \[ {\sum_{p\leq n}}^* \; \frac{\log p}{p} = \frac{1}{2} \log n + O\Big((\log n)^{\frac{5}{6}+\delta}\Big), \] where (*) restricts the summation to those primes $p$, which satisfy $n = kp+r$ for some integers $k$ and $r$, $p/2 < r < p$. This result is connected with questions concerning prime divisors of binomial coefficients.]]> Sat, 01 Jan 1994 07:00:00 +0000 https://doi.org/10.46298/hrj.1994.129 https://doi.org/10.46298/hrj.1994.129 Sander, J W Sander, J W 0$, \[ {\sum_{p\leq n}}^* \; \frac{\log p}{p} = \frac{1}{2} \log n + O\Big((\log n)^{\frac{5}{6}+\delta}\Big), \] where (*) restricts the summation to those primes $p$, which satisfy $n = kp+r$ for some integers $k$ and $r$, $p/2 < r < p$. This result is connected with questions concerning prime divisors of binomial coefficients.]]> 0 On sets of coprime integers in intervals Fri, 01 Jan 1993 07:00:00 +0000 https://doi.org/10.46298/hrj.1993.126 https://doi.org/10.46298/hrj.1993.126 Erdös, Paul Sárközy, András Erdös, Paul Sárközy, András 0 The number of primes in a short interval. Fri, 01 Jan 1993 07:00:00 +0000 https://doi.org/10.46298/hrj.1993.127 https://doi.org/10.46298/hrj.1993.127 Shituo, Lou Qi, Yao Shituo, Lou Qi, Yao 0 A Chebychev's type of prime number theorem in a short interval II. 0$, where $p_n$ denotes the $n$th prime.]]> Wed, 01 Jan 1992 07:00:00 +0000 https://doi.org/10.46298/hrj.1992.124 https://doi.org/10.46298/hrj.1992.124 Shituo, Lou Qi, Yao Shituo, Lou Qi, Yao 0$, where $p_n$ denotes the $n$th prime.]]> 0 Some of my forgotten problems in number theory. Wed, 01 Jan 1992 07:00:00 +0000 https://doi.org/10.46298/hrj.1992.125 https://doi.org/10.46298/hrj.1992.125 Erdös, Paul Erdös, Paul 0 Proof of some conjectures on the mean-value of Titchmarsh series-II Tue, 01 Jan 1991 07:00:00 +0000 https://doi.org/10.46298/hrj.1991.121 https://doi.org/10.46298/hrj.1991.121 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 0 On the zeros of a class of generalised Dirichlet series-VIII \!\!\!>T^{1-\varepsilon}$.]]> Tue, 01 Jan 1991 07:00:00 +0000 https://doi.org/10.46298/hrj.1991.122 https://doi.org/10.46298/hrj.1991.122 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K \!\!\!>T^{1-\varepsilon}$.]]> 0 On the zeros of a class of generalised Dirichlet series-IX Tue, 01 Jan 1991 07:00:00 +0000 https://doi.org/10.46298/hrj.1991.123 https://doi.org/10.46298/hrj.1991.123 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 0 Proof of some conjectures on the mean-value of Titchmarsh series I. Mon, 01 Jan 1990 07:00:00 +0000 https://doi.org/10.46298/hrj.1990.118 https://doi.org/10.46298/hrj.1990.118 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 0 Proof of some conjectures on the mean-value of titchmarsh series with applications to Titchmarsh's phenomenon Mon, 01 Jan 1990 07:00:00 +0000 https://doi.org/10.46298/hrj.1990.119 https://doi.org/10.46298/hrj.1990.119 Ramachandra, K Ramachandra, K 0 On the frequency of Titchmarsh's phenomenon for $\zeta(s)$ IX. Mon, 01 Jan 1990 07:00:00 +0000 https://doi.org/10.46298/hrj.1990.120 https://doi.org/10.46298/hrj.1990.120 Ramachandra, K Ramachandra, K 0 A Lemma in complex function theory I Sun, 01 Jan 1989 07:00:00 +0000 https://doi.org/10.46298/hrj.1989.108 https://doi.org/10.46298/hrj.1989.108 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 0 A Lemma in complex function theory II 0$ is any real number.]]> Sun, 01 Jan 1989 07:00:00 +0000 https://doi.org/10.46298/hrj.1989.109 https://doi.org/10.46298/hrj.1989.109 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 0$ is any real number.]]> 0 A trivial remark on Goldbach conjecture Sun, 01 Jan 1989 07:00:00 +0000 https://doi.org/10.46298/hrj.1989.111 https://doi.org/10.46298/hrj.1989.111 Ramachandra, K Ramachandra, K 0 An $\Omega$-result related to $r_4(n)$. Sun, 01 Jan 1989 07:00:00 +0000 https://doi.org/10.46298/hrj.1989.113 https://doi.org/10.46298/hrj.1989.113 Adhikari, Sukumar Das Balasubramanian, R Sankaranarayanan, A Adhikari, Sukumar Das Balasubramanian, R Sankaranarayanan, A 0 Some local-convexity theorems for the zeta-function-like analytic functions Fri, 01 Jan 1988 07:00:00 +0000 https://doi.org/10.46298/hrj.1988.103 https://doi.org/10.46298/hrj.1988.103 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 0 Large values of some zeta-functions near the line $\sigma = 1.$ Fri, 01 Jan 1988 07:00:00 +0000 https://doi.org/10.46298/hrj.1988.104 https://doi.org/10.46298/hrj.1988.104 Ivić, Aleksandar Ivić, Aleksandar 0 On $n$ numbers on a circle Fri, 01 Jan 1988 07:00:00 +0000 https://doi.org/10.46298/hrj.1988.105 https://doi.org/10.46298/hrj.1988.105 Schinzel, Andrzej Misiurewicz, Michal Schinzel, Andrzej Misiurewicz, Michal 0 A remark on $\zeta(1+it).$ Thu, 01 Jan 1987 07:00:00 +0000 https://doi.org/10.46298/hrj.1987.101 https://doi.org/10.46298/hrj.1987.101 Ramachandra, K Ramachandra, K 0 Srinivasa Ramanujan (The inventor of the circle method) (22-12-1887 to 26-4-1920) Thu, 01 Jan 1987 07:00:00 +0000 https://doi.org/10.46298/hrj.1987.102 https://doi.org/10.46298/hrj.1987.102 Ramachandra, K Ramachandra, K 0 On the frequency of Titchmarsh's phenomenon for $\zeta(s)$ IV. Wed, 01 Jan 1986 07:00:00 +0000 https://doi.org/10.46298/hrj.1986.115 https://doi.org/10.46298/hrj.1986.115 Balasubramanian, R Balasubramanian, R 0 Hybrid mean square of $L$-functions. Wed, 01 Jan 1986 07:00:00 +0000 https://doi.org/10.46298/hrj.1986.116 https://doi.org/10.46298/hrj.1986.116 Narlikar, Mangala J Narlikar, Mangala J 0 The number of finite non-isomorphic abelian groups in mean square. Wed, 01 Jan 1986 07:00:00 +0000 https://doi.org/10.46298/hrj.1986.117 https://doi.org/10.46298/hrj.1986.117 Ivić, Aleksandar Ivić, Aleksandar 0 On Waring's Problem: $g(4)\leq20.$ Tue, 01 Jan 1985 07:00:00 +0000 https://doi.org/10.46298/hrj.1985.114 https://doi.org/10.46298/hrj.1985.114 Balasubramanian, R Balasubramanian, R 0 On the equation $a(x^m-1)/(x-1)=b(y^n-1)/(y-1)$: II Sun, 01 Jan 1984 07:00:00 +0000 https://doi.org/10.46298/hrj.1984.100 https://doi.org/10.46298/hrj.1984.100 Shorey, Tarlok Nath Shorey, Tarlok Nath 0 On the algebraic differential equations satisfied by some elliptic function I Sun, 01 Jan 1984 07:00:00 +0000 https://doi.org/10.46298/hrj.1984.106 https://doi.org/10.46298/hrj.1984.106 Chowla, P Chowla, S Chowla, P Chowla, S 0 On algebraic differential equations satisfied by some elliptic functions II Sun, 01 Jan 1984 07:00:00 +0000 https://doi.org/10.46298/hrj.1984.107 https://doi.org/10.46298/hrj.1984.107 Chowla, P Chowla, S Chowla, P Chowla, S 0 A remark on goldbach's problem II Sun, 01 Jan 1984 07:00:00 +0000 https://doi.org/10.46298/hrj.1984.110 https://doi.org/10.46298/hrj.1984.110 Srinivasan, S Srinivasan, S 0 On infinitude of primes 1)$ and $k (>1)$ be given integers. In this paper we prove that $e_K(q)\equiv0 \mod k^{[m]}$ for infinitely many primes $q$, where $m=c_k\log\log q$ for a certain $c_k>0$ and $e_K(q)$ denotes the exponent of $K$ modulo $q$. In particular, $q\equiv1 \mod k$ for infinitely many primes $q$.]]> Sun, 01 Jan 1984 07:00:00 +0000 https://doi.org/10.46298/hrj.1984.112 https://doi.org/10.46298/hrj.1984.112 Srinivasan, S Srinivasan, S 1)$ and $k (>1)$ be given integers. In this paper we prove that $e_K(q)\equiv0 \mod k^{[m]}$ for infinitely many primes $q$, where $m=c_k\log\log q$ for a certain $c_k>0$ and $e_K(q)$ denotes the exponent of $K$ modulo $q$. In particular, $q\equiv1 \mod k$ for infinitely many primes $q$.]]> 0 Mean-value of the Riemann zeta-function and other remarks III \!\!\!>_k (\log H_0/q_n)^{k^2}$, where $p_m/q_m$ is the $m$th convergent of the continued fraction expansion of $k$, and $n$ is the unique integer such that $q_nq_{n+1}\geq \log\log H_0 > q_nq_{n-1}$. Section 4 of part III discusses lower bounds of mean values of Titchmarsh series.]]> Sat, 01 Jan 1983 07:00:00 +0000 https://doi.org/10.46298/hrj.1983.96 https://doi.org/10.46298/hrj.1983.96 Ramachandra, K Ramachandra, K \!\!\!>_k (\log H_0/q_n)^{k^2}$, where $p_m/q_m$ is the $m$th convergent of the continued fraction expansion of $k$, and $n$ is the unique integer such that $q_nq_{n+1}\geq \log\log H_0 > q_nq_{n-1}$. Section 4 of part III discusses lower bounds of mean values of Titchmarsh series.]]> 0 The greatest square free factor of a binary recursive sequence n$ (and $u_n\neq0$); this implies a lower bound for $\log Q[u_n]$ of the form $C(\log m)^2(\log\log m)^{-1}$, thereby improving on an earlier result of C. L. Stewart.]]> Sat, 01 Jan 1983 07:00:00 +0000 https://doi.org/10.46298/hrj.1983.97 https://doi.org/10.46298/hrj.1983.97 Shorey, Tarlok Nath Shorey, Tarlok Nath n$ (and $u_n\neq0$); this implies a lower bound for $\log Q[u_n]$ of the form $C(\log m)^2(\log\log m)^{-1}$, thereby improving on an earlier result of C. L. Stewart.]]> 0 A note to a paper by Ramachandra on transctndental numbers Sat, 01 Jan 1983 07:00:00 +0000 https://doi.org/10.46298/hrj.1983.98 https://doi.org/10.46298/hrj.1983.98 Ramachandra, K Srinivasan, S Ramachandra, K Srinivasan, S 0 Primes between $p_n+1$ and $p_{n+1}^2-1.$ Sat, 01 Jan 1983 07:00:00 +0000 https://doi.org/10.46298/hrj.1983.99 https://doi.org/10.46298/hrj.1983.99 Venugopalan, A. Venugopalan, A. 0 On a question of Ramachandra 1$, and the sum on the left runs over all primes $p$. This paper is devoted to proving the following theorem: If $1/2<\sigma<1$, then $$\max_k(\sum_{n\leq N} a_k(n)^2n^{-2\sigma})^{1/2k}\approx (\log N)^{1-\sigma}/\log\log N$$ and $$(\sum_{n=1}^{\infty} a_k(n)^2n^{-2\sigma})^{1/2k} \approx k^{1-\sigma}/(\log k)^{\sigma}.$$ The constants implied by the $\approx$ sign may depend upon $\sigma$. This theorem has applications to the Riemann zeta function.]]> Fri, 01 Jan 1982 07:00:00 +0000 https://doi.org/10.46298/hrj.1982.95 https://doi.org/10.46298/hrj.1982.95 Montgomery, hugh L Montgomery, hugh L 1$, and the sum on the left runs over all primes $p$. This paper is devoted to proving the following theorem: If $1/2<\sigma<1$, then $$\max_k(\sum_{n\leq N} a_k(n)^2n^{-2\sigma})^{1/2k}\approx (\log N)^{1-\sigma}/\log\log N$$ and $$(\sum_{n=1}^{\infty} a_k(n)^2n^{-2\sigma})^{1/2k} \approx k^{1-\sigma}/(\log k)^{\sigma}.$$ The constants implied by the $\approx$ sign may depend upon $\sigma$. This theorem has applications to the Riemann zeta function.]]> 0 Progress Towards a conjecture on the mean-value of titchmarsh series-II Thu, 01 Jan 1981 07:00:00 +0000 https://doi.org/10.46298/hrj.1981.91 https://doi.org/10.46298/hrj.1981.91 Ramachandra, K Ramachandra, K 0 Some problems of analytic number theory III Thu, 01 Jan 1981 07:00:00 +0000 https://doi.org/10.46298/hrj.1981.92 https://doi.org/10.46298/hrj.1981.92 Balasubramanian, R Ramachandra, K Balasubramanian, R Ramachandra, K 0 On series integrals and continued fractions I Thu, 01 Jan 1981 07:00:00 +0000 https://doi.org/10.46298/hrj.1981.93 https://doi.org/10.46298/hrj.1981.93 Ramachandra, K Ramachandra, K 0 Addendum and corrigendum to my paper - "One more proof of Siegel's theorem" Thu, 01 Jan 1981 07:00:00 +0000 https://doi.org/10.46298/hrj.1981.94 https://doi.org/10.46298/hrj.1981.94 Ramachandra, K Ramachandra, K 0 Some remarks on the mean value of the riemann zeta-function and other Dirichlet series-II 1$.]]> Tue, 01 Jan 1980 07:00:00 +0000 https://doi.org/10.46298/hrj.1980.88 https://doi.org/10.46298/hrj.1980.88 Ramachandra, K Ramachandra, K 1$.]]> 0 One more proof of Siegel's theorem 0$ for which $\chi_1(n)\cdot\chi_2(n)=-1$ and, moreover, if $L(1,\chi_1)\leq10^{-40}(\log k_1)^{-1}$, then $L(1,\chi_2)>10^{-4} (\log k_2){-1}\cdot(\log k_1)^{-2}k_2^{-40000L(1,\chi_1)}$. From this the result of T. Tatuzawa on Siegel's theorem follows.]]> Tue, 01 Jan 1980 07:00:00 +0000 https://doi.org/10.46298/hrj.1980.89 https://doi.org/10.46298/hrj.1980.89 Ramachandra, K Ramachandra, K 0$ for which $\chi_1(n)\cdot\chi_2(n)=-1$ and, moreover, if $L(1,\chi_1)\leq10^{-40}(\log k_1)^{-1}$, then $L(1,\chi_2)>10^{-4} (\log k_2){-1}\cdot(\log k_1)^{-2}k_2^{-40000L(1,\chi_1)}$. From this the result of T. Tatuzawa on Siegel's theorem follows.]]> 0 On a theorem of Erdos and Szemeredi \!\!> h,$ where $h \geq x^{\theta}.$ Refining some of Szemeredi's ideas, it is proved in this paper that %if 0 < < 1, and $\sum\frac{1}{b_i}<\infty$, then $$Q(x+h) - Q(x) >\!\!> h,$$ where $x\geq h \geq x^{\theta}$ and $\theta >\frac{1}{2}$ is any constant. %In the later part, using the ideas of Jutila, Brun and I. M. Vinogradov, a stronger version (Theorem 2) is proved.]]> Tue, 01 Jan 1980 07:00:00 +0000 https://doi.org/10.46298/hrj.1980.90 https://doi.org/10.46298/hrj.1980.90 Narlikar, Mangala J Narlikar, Mangala J \!\!> h,$ where $h \geq x^{\theta}.$ Refining some of Szemeredi's ideas, it is proved in this paper that %if 0 < < 1, and $\sum\frac{1}{b_i}<\infty$, then $$Q(x+h) - Q(x) >\!\!> h,$$ where $x\geq h \geq x^{\theta}$ and $\theta >\frac{1}{2}$ is any constant. %In the later part, using the ideas of Jutila, Brun and I. M. Vinogradov, a stronger version (Theorem 2) is proved.]]> 0 On warning's problem : g (4) < 21 Mon, 01 Jan 1979 07:00:00 +0000 https://doi.org/10.46298/hrj.1979.643 https://doi.org/10.46298/hrj.1979.643 Balasubramanian, R Balasubramanian, R 0 Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series 1 (\log H)^{k^2}(\log\log H)^{-C}$$ and $$(1/H)\int_{T}^{T+H} \vert\zeta'(\frac{1}{2}+it)\vert dt > (\log H)^{5/4}(\log\log H)^{-C},$$ where $C$ is a constant depending only on $\delta$.]]> Sun, 01 Jan 1978 07:00:00 +0000 https://doi.org/10.46298/hrj.1978.87 https://doi.org/10.46298/hrj.1978.87 Ramachandra, K Ramachandra, K (\log H)^{k^2}(\log\log H)^{-C}$$ and $$(1/H)\int_{T}^{T+H} \vert\zeta'(\frac{1}{2}+it)\vert dt > (\log H)^{5/4}(\log\log H)^{-C},$$ where $C$ is a constant depending only on $\delta$.]]> 0