<![CDATA[Hardy-Ramanujan Journal - RSS]]>
https://hrj.episciences.org
Sun, 28 Nov 2021 07:02:19 +0100https://hrj.episciences.org/img/episciences_sign_50x50.png<![CDATA[Hardy-Ramanujan Journal - RSS]]>
https://hrj.episciences.org
Zend_Feedenhttp://blogs.law.harvard.edu/tech/rss<![CDATA[Congruences modulo powers of 5 for the rank parity function]]>
https://hrj.episciences.org/7424
Thu, 06 May 2021 14:52:27 +0200<![CDATA[A localized Erdős-Kac theorem]]>
https://hrj.episciences.org/7433
Thu, 06 May 2021 14:29:56 +0200<![CDATA[A universal identity for theta functions of degree eight and applications]]>
https://hrj.episciences.org/7432
Thu, 06 May 2021 14:26:17 +0200<![CDATA[One level density of low-lying zeros of quadratic Hecke L-functions to prime moduli]]>
https://hrj.episciences.org/7461
Thu, 06 May 2021 14:16:41 +0200<![CDATA[Distribution of generalized mex-related integer partitions]]>
https://hrj.episciences.org/7425
Thu, 06 May 2021 14:08:34 +0200<![CDATA[Arithmetical Fourier transforms and Hilbert space: Restoration of the lost legacy]]>
https://hrj.episciences.org/7426
Thu, 06 May 2021 14:06:03 +0200<![CDATA[(q-)Supercongruences hit again]]>
https://hrj.episciences.org/7427
Thu, 06 May 2021 14:01:11 +0200<![CDATA[Partition-theoretic formulas for arithmetic densities, II]]>
https://hrj.episciences.org/7428
Thu, 06 May 2021 13:57:52 +0200<![CDATA[Ramanujan's Beautiful Integrals]]>
https://hrj.episciences.org/7429
Thu, 06 May 2021 13:54:25 +0200<![CDATA[Bounds for d-distinct partitions]]>
https://hrj.episciences.org/7430
Thu, 06 May 2021 13:50:24 +0200<![CDATA[A heuristic guide to evaluating triple-sums]]>
https://hrj.episciences.org/7431
Thu, 06 May 2021 13:45:16 +0200<![CDATA[A Reinforcement Learning Based Algorithm to Find a Triangular Graham Partition]]>
https://hrj.episciences.org/7434
Thu, 06 May 2021 13:27:33 +0200<![CDATA[Multiplicatively dependent vectors with coordinates algebraic numbers]]>
https://hrj.episciences.org/6603
Mon, 29 Jun 2020 11:46:52 +0200<![CDATA[Some New Congruences for l-Regular Partitions Modulo 13, 17, and 23]]>
https://hrj.episciences.org/6495
Thu, 21 May 2020 10:39:02 +0200<![CDATA[Density modulo 1 of a sequence associated to some multiplicative functions evaluated at polynomial arguments]]>
https://hrj.episciences.org/5650
Thu, 21 May 2020 10:20:07 +0200<![CDATA[On some Lambert-like series]]>
https://hrj.episciences.org/6461
Wed, 20 May 2020 21:57:40 +0200<![CDATA[Divisibility of Selmer groups and class groups]]>
https://hrj.episciences.org/6460
Wed, 20 May 2020 21:56:53 +0200<![CDATA[A transference inequality for rational approximation to points in geometric progression]]>
https://hrj.episciences.org/6489
Wed, 20 May 2020 21:49:02 +0200<![CDATA[Linear forms in logarithms and exponential Diophantine equations]]>
https://hrj.episciences.org/6458
Wed, 20 May 2020 21:44:16 +0200<![CDATA[Polynomial representations of GL(m|n)]]>
https://hrj.episciences.org/6459
Wed, 20 May 2020 21:42:30 +0200<![CDATA[On the Galois group of Generalised Laguerre polynomials II]]>
https://hrj.episciences.org/6457
Wed, 20 May 2020 21:40:43 +0200<![CDATA[Class Numbers of Quadratic Fields]]>
https://hrj.episciences.org/6488
Wed, 20 May 2020 21:37:23 +0200<![CDATA[The Barban-Vehov Theorem in Arithmetic Progressions]]>
https://hrj.episciences.org/5118
Wed, 23 Jan 2019 15:41:30 +0100<![CDATA[Explicit abc-conjecture and its applications]]>
https://hrj.episciences.org/5117
Wed, 23 Jan 2019 15:40:55 +0100<![CDATA[Integral points on circles]]>
https://hrj.episciences.org/5116
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)$}.]]>0$, $\gcd(m,n^2)$ 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)$}.]]>Wed, 23 Jan 2019 15:40:18 +0100<![CDATA[A note on some congruences involving arithmetic functions]]>
https://hrj.episciences.org/5115
Wed, 23 Jan 2019 15:39:48 +0100<![CDATA[A remark on cube-free numbers in Segal-Piatestki-Shapiro sequences]]>
https://hrj.episciences.org/5114
Wed, 23 Jan 2019 15:38:59 +0100<![CDATA[Set Equidistribution of subsets of (Z/nZ) *]]>
https://hrj.episciences.org/5113
Wed, 23 Jan 2019 15:38:20 +0100<![CDATA[Hybrid level aspect subconvexity for GL(2) × GL(1) Rankin-Selberg L-Functions]]>
https://hrj.episciences.org/5112
Wed, 23 Jan 2019 15:36:30 +0100<![CDATA[Two applications of number theory to discrete tomography]]>
https://hrj.episciences.org/5111
Wed, 23 Jan 2019 15:35:46 +0100<![CDATA[On certain sums over ordinates of zeta-zeros II]]>
https://hrj.episciences.org/5110
0} {\gamma}^{-s}$ 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} {\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 15:35:00 +0100<![CDATA[The Zeta Mahler measure of $(z^n − 1)/(z − 1)$]]>
https://hrj.episciences.org/5109
Wed, 23 Jan 2019 15:34:11 +0100<![CDATA[An elementary property of correlations]]>
https://hrj.episciences.org/5108
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. 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 15:33:18 +0100<![CDATA[When are Multiples of Polygonal Numbers again Polygonal Numbers?]]>
https://hrj.episciences.org/5107
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.]]> 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 15:32:36 +0100<![CDATA[On-Regular Bipartitions Modulo $m$]]>
https://hrj.episciences.org/5106
Wed, 23 Jan 2019 15:31:19 +0100<![CDATA[On an identity of Ramanujan]]>
https://hrj.episciences.org/5105
Wed, 23 Jan 2019 15:30:10 +0100<![CDATA[On the Wintner-Ingham-Segal summability method]]>
https://hrj.episciences.org/5104
Wed, 23 Jan 2019 15:28:48 +0100<![CDATA[Fluctuation of the primitive of Hardy's function]]>
https://hrj.episciences.org/5103
Wed, 23 Jan 2019 15:27:05 +0100<![CDATA[On Pillai's problem with Pell numbers and powers of 2]]>
https://hrj.episciences.org/5102
Wed, 23 Jan 2019 15:24:02 +0100<![CDATA[A Theorem of Fermat on Congruent Number Curves]]>
https://hrj.episciences.org/5101
Wed, 23 Jan 2019 15:19:01 +0100<![CDATA[On the Riesz means of δ k (n)]]>
https://hrj.episciences.org/2058
Thu, 11 Jan 2018 17:05:07 +0100<![CDATA[Dual Ramanujan-Fourier series]]>
https://hrj.episciences.org/2541
Thu, 11 Jan 2018 17:04:40 +0100<![CDATA[Non-vanishing of Dirichlet series without Euler products]]>
https://hrj.episciences.org/4027
1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).]]> 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).]]>Thu, 11 Jan 2018 17:03:50 +0100<![CDATA[Book review : Lectures on the Riemann Zeta Function, by Henryk Iwaniec]]>
https://hrj.episciences.org/2650
Wed, 18 Jan 2017 18:31:05 +0100<![CDATA[Contributions of Ramachandra to the Theory of the Riemann Zeta-Function]]>
https://hrj.episciences.org/2633
Mon, 09 Jan 2017 14:34:48 +0100<![CDATA[A note on Hardy's theorem]]>
https://hrj.episciences.org/2634
Mon, 09 Jan 2017 14:34:04 +0100<![CDATA[Sieve functions in arithmetic bands]]>
https://hrj.episciences.org/2635
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. 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 14:33:26 +0100<![CDATA[Ramanujan-Fourier series of certain arithmetic functions of two variables]]>
https://hrj.episciences.org/2636
Mon, 09 Jan 2017 14:32:35 +0100<![CDATA[Remarks on the impossibility of a Siegel-Shidlovskii like theorem for G-functions]]>
https://hrj.episciences.org/1357
Thu, 01 Jan 2015 08:00:00 +0100<![CDATA[Algebraic independence results on the generating Lambert series of the powers of a fixed integer]]>
https://hrj.episciences.org/1358
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.]]> 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 08:00:00 +0100<![CDATA[Effective irrationality measures for quotients of logarithms of rational numbers]]>
https://hrj.episciences.org/1359
Thu, 01 Jan 2015 08:00:00 +0100<![CDATA[Baker's Explicit abc-Conjecture and Waring's problem]]>
https://hrj.episciences.org/1360
Thu, 01 Jan 2015 08:00:00 +0100<![CDATA[Mean Values of Certain Multiplicative Functions and Artin's Conjecture on Primitive Roots]]>
https://hrj.episciences.org/1316
Wed, 01 Jan 2014 08:00:00 +0100<![CDATA[On the Galois groups of generalized Laguerre Polynomials]]>
https://hrj.episciences.org/1317
Wed, 01 Jan 2014 08:00:00 +0100<![CDATA[New developments on the twin prime problem and generalizations]]>
https://hrj.episciences.org/1318
Wed, 01 Jan 2014 08:00:00 +0100<![CDATA[On congruences for certain sums of E. Lehmer's type]]>
https://hrj.episciences.org/1356
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]]> 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 08:00:00 +0100<![CDATA[On the Half Line: K Ramachandra.]]>
https://hrj.episciences.org/172
Tue, 01 Jan 2013 08:00:00 +0100<![CDATA[The work of K. Ramachandra in algebraic number theory]]>
https://hrj.episciences.org/173
Tue, 01 Jan 2013 08:00:00 +0100<![CDATA[On Ramachandra's Contributions to Transcendental Number Theory]]>
https://hrj.episciences.org/174
Tue, 01 Jan 2013 08:00:00 +0100<![CDATA[Mathematical Reminiscences: How to Keep the Pot Boiling]]>
https://hrj.episciences.org/175
Tue, 01 Jan 2013 08:00:00 +0100<![CDATA[K. Ramachandra : Reminiscences of his Students.]]>
https://hrj.episciences.org/176
Tue, 01 Jan 2013 08:00:00 +0100<![CDATA[K. Ramachandra: Reminiscences of his Friends.]]>
https://hrj.episciences.org/177
Tue, 01 Jan 2013 08:00:00 +0100<![CDATA[On the class number formula of certain real quadratic fields ]]>
https://hrj.episciences.org/178
Tue, 01 Jan 2013 08:00:00 +0100<![CDATA[On the Riesz means of $\frac{n}{\phi(n)}$]]>
https://hrj.episciences.org/179
Tue, 01 Jan 2013 08:00:00 +0100<![CDATA[Ramanujan series for arithmetical functions]]>
https://hrj.episciences.org/180
Tue, 01 Jan 2013 08:00:00 +0100<![CDATA[I am fifty-five years old]]>
https://hrj.episciences.org/181
Tue, 01 Jan 2013 08:00:00 +0100<![CDATA[The combinatorics of moment calculations]]>
https://hrj.episciences.org/168
Fri, 01 Jan 2010 08:00:00 +0100<![CDATA[An estimate for the Mellin transform of powers of Hardy's function.]]>
https://hrj.episciences.org/169
Fri, 01 Jan 2010 08:00:00 +0100<![CDATA[On the Mellin transforms of powers of Hardy's function.]]>
https://hrj.episciences.org/170
Fri, 01 Jan 2010 08:00:00 +0100<![CDATA[Some diophantine problems concerning equal sums of integers and their cubes]]>
https://hrj.episciences.org/171
Fri, 01 Jan 2010 08:00:00 +0100<![CDATA[On mean value results for the Riemann zeta-function in short intervals.]]>
https://hrj.episciences.org/164
Thu, 01 Jan 2009 08:00:00 +0100<![CDATA[Alternating Euler sums at the negative integers.]]>
https://hrj.episciences.org/165
Thu, 01 Jan 2009 08:00:00 +0100<![CDATA[Finite expressions for higher derivatives of the Dirichlet L-function and the Deninger R-function]]>
https://hrj.episciences.org/166
Thu, 01 Jan 2009 08:00:00 +0100<![CDATA[Little flowers to Srinivasa Ramanujan.]]>
https://hrj.episciences.org/167
Thu, 01 Jan 2009 08:00:00 +0100<![CDATA[Ideal solutions of the Tarry-Escott problem of degree eleven with applications to sums of thirteenth powers.]]>
https://hrj.episciences.org/161
Tue, 01 Jan 2008 08:00:00 +0100<![CDATA[Arithmetical investigations of particular Wynn power series]]>
https://hrj.episciences.org/162
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.
]]>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 08:00:00 +0100<![CDATA[On Kronecker's limit formula and the hypergeometric function.]]>
https://hrj.episciences.org/163
Tue, 01 Jan 2008 08:00:00 +0100<![CDATA[Carmichael number with three prime factors.]]>
https://hrj.episciences.org/156
Mon, 01 Jan 2007 08:00:00 +0100<![CDATA[The number of imaginary quadratic fields with a given class number]]>
https://hrj.episciences.org/157
Mon, 01 Jan 2007 08:00:00 +0100<![CDATA[Rational triangles with the same perimeter and the same area.]]>
https://hrj.episciences.org/158
Mon, 01 Jan 2007 08:00:00 +0100<![CDATA[Contributions to the theory of the Hurwitz zeta-function]]>
https://hrj.episciences.org/159
Mon, 01 Jan 2007 08:00:00 +0100<![CDATA[On the Barban-Davenport Halberstam theorem: XIX]]>
https://hrj.episciences.org/160
\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 $C$.]]>\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 08:00:00 +0100<![CDATA[On polynomials that equal binary cubic forms.]]>
https://hrj.episciences.org/153
Sun, 01 Jan 2006 08:00:00 +0100<![CDATA[On the periodic Hurwitz zeta-function.]]>
https://hrj.episciences.org/154
Sun, 01 Jan 2006 08:00:00 +0100<![CDATA[A remark on a theorem of A. E. Ingham.]]>
https://hrj.episciences.org/155
\frac{5}{8})$ is any constant. Further, we point out that even this stronger form can be proved without using the functional equation of $\zeta(s)$.]]>\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 08:00:00 +0100<![CDATA[A lower bound concerning subset sums which do not cover all the residues modulo $p$.]]>
https://hrj.episciences.org/85
\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 $$ \sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$$
]]>\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 08:00:00 +0100<![CDATA[Further Variations on the Six Exponentials Theorem.]]>
https://hrj.episciences.org/86
Sat, 01 Jan 2005 08:00:00 +0100<![CDATA[On an exponential sum involving the Möbius function]]>
https://hrj.episciences.org/151
Sat, 01 Jan 2005 08:00:00 +0100<![CDATA[A lower bound concerning subset sums which do not cover all the residues modulo $p$.]]>
https://hrj.episciences.org/152
\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 $$ \sum_{a\in\mathcal A}\Vert\frac{as}{p}\Vert>1+Kp^{-1/2}.$$]]>\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 08:00:00 +0100<![CDATA[Hardy-Littlewood first approximation theorem for quasi $L$-functions]]>
https://hrj.episciences.org/148
Thu, 01 Jan 2004 08:00:00 +0100<![CDATA[Mean square of the Hurwitz zeta-function and other remarks]]>
https://hrj.episciences.org/149
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}$$ for every $\varepsilon>0$ and every $T\geq T_0(\varepsilon)$. Here, as usual, $01)$$ 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, $0Thu, 01 Jan 2004 08:00:00 +0100<![CDATA[A generalization of Bochner's formula]]>
https://hrj.episciences.org/150
Thu, 01 Jan 2004 08:00:00 +0100<![CDATA[Mean-square upper bound of Hecke $L$-functions on the critical line.]]>
https://hrj.episciences.org/147
Wed, 01 Jan 2003 08:00:00 +0100<![CDATA[Some problems of Analytic number theory IV]]>
https://hrj.episciences.org/145
Tue, 01 Jan 2002 08:00:00 +0100<![CDATA[On totally reducible binary forms: II.]]>
https://hrj.episciences.org/146
0$ and $\eta_l>0$.
]]>0$ and $\eta_l>0$.
]]>Tue, 01 Jan 2002 08:00:00 +0100<![CDATA[On some Ramanujan's P-Q Identities]]>
https://hrj.episciences.org/143
Mon, 01 Jan 2001 08:00:00 +0100<![CDATA[On the values of the Riemann zeta-function at rational arguments]]>
https://hrj.episciences.org/144
Mon, 01 Jan 2001 08:00:00 +0100<![CDATA[Notes on the Riemann zeta-function-IV]]>
https://hrj.episciences.org/141
Sat, 01 Jan 2000 08:00:00 +0100<![CDATA[On a problem of Ivi\'c.]]>
https://hrj.episciences.org/142
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$, 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 08:00:00 +0100<![CDATA[Disproof of some conjectures of K.Ramachandra]]>
https://hrj.episciences.org/137
Fri, 01 Jan 1999 08:00:00 +0100<![CDATA[On generalised Carmichael numbers.]]>
https://hrj.episciences.org/138
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.]]>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 08:00:00 +0100<![CDATA[Notes on the Riemann zeta Function-III]]>
https://hrj.episciences.org/139
C$ for every fixed $C>0$. Also, we study the gaps between the ordinates of the consecutive poles of $F(s)$.
]]>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 08:00:00 +0100<![CDATA[Notes on the Riemann zeta-function-IV]]>
https://hrj.episciences.org/140
C$ for every fixed $C>0$. Also we study the gaps between the numbers $p_2$ arranged in the non-decreasing order.]]>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 08:00:00 +0100<![CDATA[On a method of Davenport and Heilbronn I.]]>
https://hrj.episciences.org/136
Thu, 01 Jan 1998 08:00:00 +0100<![CDATA[On the Barban-Davenport-Halberstam theorem : X]]>
https://hrj.episciences.org/182
Thu, 01 Jan 1998 08:00:00 +0100<![CDATA[On the zeros of a class of generalised Dirichlet series-XVIII (a few remarks on littlewood's theorem and Totchmarsh points)]]>
https://hrj.episciences.org/134
Wed, 01 Jan 1997 08:00:00 +0100<![CDATA[On the zeros of a class of generalised Dirichlet series-XIX]]>
https://hrj.episciences.org/135
Wed, 01 Jan 1997 08:00:00 +0100<![CDATA[Ramanujan's lattice point problem, prime number theory and other remarks.]]>
https://hrj.episciences.org/133
Mon, 01 Jan 1996 08:00:00 +0100<![CDATA[A large value theorem for $\zeta(s)$.]]>
https://hrj.episciences.org/130
Sun, 01 Jan 1995 08:00:00 +0100<![CDATA[On Riemann zeta-function and allied questions-II]]>
https://hrj.episciences.org/131
Sun, 01 Jan 1995 08:00:00 +0100<![CDATA[Is the binomial coefficient $2n \choose n$ square free?]]>
https://hrj.episciences.org/132
4$.
]]>4$.
]]>Sun, 01 Jan 1995 08:00:00 +0100<![CDATA[Estimate of sums of Dirichlet series]]>
https://hrj.episciences.org/128
Sat, 01 Jan 1994 08:00:00 +0100<![CDATA[On some over primes]]>
https://hrj.episciences.org/129
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$,
\[
{\sum_{p\leq n}}^* \; \frac{\log p}{p} = \frac{1}{2} \log n + O\Big((\log n)^{\frac{5}{6}+\delta}\Big),
\]
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 08:00:00 +0100<![CDATA[On sets of coprime integers in intervals]]>
https://hrj.episciences.org/126
Fri, 01 Jan 1993 08:00:00 +0100<![CDATA[The number of primes in a short interval.]]>
https://hrj.episciences.org/127
Fri, 01 Jan 1993 08:00:00 +0100<![CDATA[A Chebychev's type of prime number theorem in a short interval II.]]>
https://hrj.episciences.org/124
0$, where $p_n$ denotes the $n$th prime.]]>0$, where $p_n$ denotes the $n$th prime.]]>Wed, 01 Jan 1992 08:00:00 +0100<![CDATA[Some of my forgotten problems in number theory.]]>
https://hrj.episciences.org/125
Wed, 01 Jan 1992 08:00:00 +0100<![CDATA[Proof of some conjectures on the mean-value of Titchmarsh series-II ]]>
https://hrj.episciences.org/121
Tue, 01 Jan 1991 08:00:00 +0100<![CDATA[On the zeros of a class of generalised Dirichlet series-VIII]]>
https://hrj.episciences.org/122
\!\!\!>T^{1-\varepsilon}$.
]]>\!\!\!>T^{1-\varepsilon}$.
]]>Tue, 01 Jan 1991 08:00:00 +0100<![CDATA[On the zeros of a class of generalised Dirichlet series-IX]]>
https://hrj.episciences.org/123
Tue, 01 Jan 1991 08:00:00 +0100<![CDATA[Proof of some conjectures on the mean-value of Titchmarsh series I.]]>
https://hrj.episciences.org/118
Mon, 01 Jan 1990 08:00:00 +0100<![CDATA[Proof of some conjectures on the mean-value of titchmarsh series with applications to Titchmarsh's phenomenon]]>
https://hrj.episciences.org/119
Mon, 01 Jan 1990 08:00:00 +0100<![CDATA[On the frequency of Titchmarsh's phenomenon for $\zeta(s)$ IX.]]>
https://hrj.episciences.org/120
Mon, 01 Jan 1990 08:00:00 +0100<![CDATA[A Lemma in complex function theory I]]>
https://hrj.episciences.org/108
Sun, 01 Jan 1989 08:00:00 +0100<![CDATA[A Lemma in complex function theory II]]>
https://hrj.episciences.org/109
0$ is any real number.]]>0$ is any real number.]]>Sun, 01 Jan 1989 08:00:00 +0100<![CDATA[A trivial remark on Goldbach conjecture]]>
https://hrj.episciences.org/111
Sun, 01 Jan 1989 08:00:00 +0100<![CDATA[An $\Omega$-result related to $r_4(n)$.]]>
https://hrj.episciences.org/113
Sun, 01 Jan 1989 08:00:00 +0100<![CDATA[Some local-convexity theorems for the zeta-function-like analytic functions]]>
https://hrj.episciences.org/103
Fri, 01 Jan 1988 08:00:00 +0100<![CDATA[Large values of some zeta-functions near the line $\sigma = 1.$]]>
https://hrj.episciences.org/104
Fri, 01 Jan 1988 08:00:00 +0100<![CDATA[On $n$ numbers on a circle]]>
https://hrj.episciences.org/105
Fri, 01 Jan 1988 08:00:00 +0100<![CDATA[A remark on $\zeta(1+it).$]]>
https://hrj.episciences.org/101
Thu, 01 Jan 1987 08:00:00 +0100<![CDATA[Srinivasa Ramanujan (The inventor of the circle method) (22-12-1887 to 26-4-1920)]]>
https://hrj.episciences.org/102
Thu, 01 Jan 1987 08:00:00 +0100<![CDATA[On the frequency of Titchmarsh's phenomenon for $\zeta(s)$ IV.]]>
https://hrj.episciences.org/115
Wed, 01 Jan 1986 08:00:00 +0100<![CDATA[Hybrid mean square of $L$-functions.]]>
https://hrj.episciences.org/116
Wed, 01 Jan 1986 08:00:00 +0100<![CDATA[The number of finite non-isomorphic abelian groups in mean square.]]>
https://hrj.episciences.org/117
Wed, 01 Jan 1986 08:00:00 +0100<![CDATA[On Waring's Problem: $g(4)\leq20.$]]>
https://hrj.episciences.org/114
Tue, 01 Jan 1985 08:00:00 +0100<![CDATA[On the equation $a(x^m-1)/(x-1)=b(y^n-1)/(y-1)$: II]]>
https://hrj.episciences.org/100
Sun, 01 Jan 1984 08:00:00 +0100<![CDATA[On the algebraic differential equations satisfied by some elliptic function I
]]>
https://hrj.episciences.org/106
Sun, 01 Jan 1984 08:00:00 +0100<![CDATA[On algebraic differential equations satisfied by some elliptic functions II]]>
https://hrj.episciences.org/107
Sun, 01 Jan 1984 08:00:00 +0100<![CDATA[A remark on goldbach's problem II ]]>
https://hrj.episciences.org/110
Sun, 01 Jan 1984 08:00:00 +0100<![CDATA[On infinitude of primes]]>
https://hrj.episciences.org/112
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 $q$.]]>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 08:00:00 +0100<![CDATA[Mean-value of the Riemann zeta-function and other remarks III]]>
https://hrj.episciences.org/96
\!\!\!>_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.
]]>\!\!\!>_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 08:00:00 +0100<![CDATA[The greatest square free factor of a binary recursive sequence]]>
https://hrj.episciences.org/97
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.]]>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 08:00:00 +0100<![CDATA[A note to a paper by Ramachandra on transctndental numbers]]>
https://hrj.episciences.org/98
Sat, 01 Jan 1983 08:00:00 +0100<![CDATA[Primes between $p_n+1$ and $p_{n+1}^2-1.$]]>
https://hrj.episciences.org/99
Sat, 01 Jan 1983 08:00:00 +0100<![CDATA[On a question of Ramachandra]]>
https://hrj.episciences.org/95
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.
]]>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 08:00:00 +0100<![CDATA[Progress Towards a conjecture on the mean-value of titchmarsh series-II ]]>
https://hrj.episciences.org/91
Thu, 01 Jan 1981 08:00:00 +0100<![CDATA[Some problems of analytic number theory III]]>
https://hrj.episciences.org/92
Thu, 01 Jan 1981 08:00:00 +0100<![CDATA[On series integrals and continued fractions I]]>
https://hrj.episciences.org/93
Thu, 01 Jan 1981 08:00:00 +0100<![CDATA[Addendum and corrigendum to my paper - "One more proof of Siegel's theorem"]]>
https://hrj.episciences.org/94
Thu, 01 Jan 1981 08:00:00 +0100<![CDATA[Some remarks on the mean value of the riemann zeta-function and other Dirichlet series-II ]]>
https://hrj.episciences.org/88
1$.
]]>1$.
]]>Tue, 01 Jan 1980 08:00:00 +0100<![CDATA[One more proof of Siegel's theorem]]>
https://hrj.episciences.org/89
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$ 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 08:00:00 +0100<![CDATA[On a theorem of Erdos and Szemeredi]]>
https://hrj.episciences.org/90
\!\!> 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.
]]>\!\!> 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 08:00:00 +0100<![CDATA[On warning's problem : g (4) < 21]]>
https://hrj.episciences.org/643
Mon, 01 Jan 1979 08:00:00 +0100<![CDATA[Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series 1]]>
https://hrj.episciences.org/87
(\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},$$ where $C$ is a constant depending only on $\delta$.]]> (\log H)^{k^2}(\log\log H)^{-C}$$ 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 08:00:00 +0100