Hardy-Ramanujan Journal |

This is a faithful reproduction of Ramanujan's Quarterly Reports from a handwritten copy by T.A. Satagopan in 1925.

In this paper, we investigate the mean square estimate for the logarithmic derivative of the Godement--Jacquet $L$-function $L_f(s)$ assuming the Riemann hypothesis for $L_f(s)$ and Rudnick--Sarnak conjecture.

Let $K$ be a number field of degree $n$, $A$ be its ring of integers, and $A_n$ (resp. $K_n$) be the set of elements of $A$ ( resp. $K$) which are primitive over $\mathbb Q$. For any $\gamma \in {K_n}$, let $F_{\gamma} (x)$ be the unique irreducible polynomial in $\mathbb Z[x]$, such that its leading coefficient is positive and $F_{\gamma} ({\gamma}) = 0$. Let $i(\gamma)=\gcd_{x\in\mathbb Z}F_{\gamma}(x)$, $i(K)=\lcm_{\theta\in{A_n}}i(\theta)$ and $\hat{\imath}(K) = \lcm_{\gamma\in{K_n}}i(\gamma)$. For any $\gamma \in {K_n}$, there exists a unique pair $(\theta,d)$, where $\theta\in A_n$ and $d$ is a positive integer such that $\gamma=\theta/d$ and $\theta\not\equiv 0\pmod{p}$ for any prime divisor $p$ of $d$. In this paper, we study the possible values of $\nu_{p}(d)$ when $p | i(\gamma)$. We introduce and study a new invariant of $K$ defined using $\nu_{p}(d)$, when $\gamma$ describes $K_n$. In the last theorem of this paper, we establish a generalisation of a theorem of MacCluer.

Let $\omega_y(n)$ denote the number of distinct prime divisors of $n$ less than $y$. Suppose $y_n$ is an increasing sequence of positive real numbers satisfying $\log y_n = o(\log\log n)$. In this paper, we prove an Erdös-Kac theorem for the distribution of $\omega_{y_n}(p+a)$, where $p$ runs over all prime numbers and $a$ is a fixed integer. We also highlight the connection between the distribution of $\omega_y(p-1)$ and Ihara's conjectures on Euler-Kronecker constants.

We consider a $q$-analog $r_2(n, q)$ of the number of representations of an integer as a sum of two squares $r_2(n)$. This $q$-analog is generated by the expansion of a product that was studied by Kronecker and Jordan. We generalize Jacobi's two squares formula from $r_2(n)$ to $r_2(n, q)$. We characterize the signs in the coefficients of $r_2(n, q)$ using the prime factors of $n$. We use $r_2(n, q)$ to characterize the integers which are the length of the hypotenuse of a primitive Pythagorean triangle.

In this note, we exhibit a weakly holomorphic modular form for use in constructing a Fourier eigenfunction in four dimensions. Such auxiliary functions may be of use to the D4 checkerboard lattice and the four dimensional sphere packing problem.

The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces of cusp forms in terms of weighted sums of Hurwitz-Kronecker class numbers. For cusp forms on $\text {\rm SL}_2(\mathbb{Z}),$ Zagier proved these formulas by cleverly making use of the weight 3/2 nonholomorphic Eisenstein series he discovered in the 1970s. The holomorphic part of this form, its so-called {\it mock modular form}, is the generating function for these class numbers. In this expository note we revisit Zagier's method, and we show how to obtain such formulas for congruence subgroups, working out the details for $\Gamma_0(2)$ and $\Gamma_0(4).$ The trace formulas fall out naturally from the computation of the Rankin-Cohen brackets of Zagier's mock modular form with specific theta functions.

In this paper we resolve a question by Bringmann, Lovejoy, and Rolen on a new vector-valued $U$-type function. We obtain an expression for a corresponding family of Hecke--Appell-type sums in terms of mixed mock modular forms; that is, we express the sum in terms of Appell functions and theta functions. This $U$-type function appears from considering the special polynomials related to generating functions for the partitions occurring in Gordon’s generalization of the Rogers--Ramanujan identities.

In this paper we give a generalization of Iseki's formula and use it to prove the transformation law of $\theta_1(z,\tau)$.

A modular relation of the form $F(\alpha, w)=F(\beta, iw)$, where $i=\sqrt{-1}$ and $\alpha\beta=1$, is obtained. It involves the generalized digamma function $\psi_w(a)$ which was recently studied by the authors in their work on developing the theory of the generalized Hurwitz zeta function $\zeta_w(s, a)$. The limiting case $w\to0$ of this modular relation is a famous result of Ramanujan on page $220$ of the Lost Notebook. We also obtain asymptotic estimate of a general integral involving the Riemann function $\Xi(t)$ as $\alpha\to\infty$. Not only does it give the asymptotic estimate of the integral occurring in our modular relation as a corollary but also some known results.