This work characterizes the vanishing of the Fourier coefficients of all CM (Complex Multiplication) eta quotients. As consequences, we recover Serre’s characterization about that of $\eta(12z)^{2}$ and recent results of Chang on the pth coefficients of $\eta(4z)^{6}$ and $\eta(6z)^{4}$. Moreover, we generalize the results on the cases of weight 1 to the setting of binary quadratic forms.

An explicit formula forthe mean value of $\vert L(1,\chi )\vert ^2$ is known, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Lately, the authors obtained that the mean value of $\vert L(1,\chi )\vert ^2$ is asymptotic to $\pi ^2/6$, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\ \ \pmod {2d}$ which are trivial on a subgroup H of odd order d of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$, provided that $d\ll \frac {\log p}{\log \log p}$. Bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Here, for a given integer $d_0>1$, we consider the same questions for the nonprimitive odd Dirichlet characters $\chi '$ modulo $d_0p$ induced by the odd primitive characters $\chi $ modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi ')\vert ^2$ is asymptotic to $\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order $d\ll \frac {\log p}{\log \log p}$. As a consequence, we improve the previous bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$. Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.

In this note, we use Dedekind’s eta function to prove a congruence relation between the number of representations by binary quadratic forms of discriminant $-31$ and Fourier coefficients of a weight $16$ cusp form. Our result is analogous to the classical result concerning Ramanujan’s tau function and binary quadratic forms of discriminant $-23$ .

The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$, for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters $(m,t)\in (0,\infty )$, initially proposed by Terry Gannon. We show that the minimisers of the lattice theta function are the maximisers of $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms.

In this article, we give new proofs of two of Ramanujan’s $1/\unicode[STIX]{x1D70B}$ formulae

$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70B}}=\frac{2\sqrt{2}}{99^{2}}\mathop{\sum }_{m=0}^{\infty }(26390m+1103)\frac{(4m)!}{396^{4m}(m!)^{4}}\end{eqnarray}$$

$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70B}}=\frac{2}{84^{2}}\mathop{\sum }_{m=0}^{\infty }(21460m+1123)\frac{(-1)^{m}(4m)!}{(84\sqrt{2})^{4m}(m!)^{4}}\end{eqnarray}$$

$1/\unicode[STIX]{x1D70B}$

Let $\eta \left( z \right)\,\left( z\,\in \,\mathbb{C},\,\operatorname{Im}\left( z \right)\,>\,0 \right)$ denote the Dedekind eta function. We use a recent product-to-sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotients

$$f\left( z \right)\,:=\,{{\eta }^{a\left( {{m}_{1}} \right)}}\,\left( {{m}_{1}}z \right)\,.\,.\,.\,{{\eta }^{a\left( {{m}_{r}} \right)}}\,\left( {{m}_{r}}z \right)\,=\,\sum\limits_{n=1}^{\infty }{c\left( n \right){{e}^{2\pi inz}},\,\,\,z\,\in \,\mathbb{C},\,\operatorname{Im}\left( z \right)\,>\,0,}$$

such that the Fourier coefficients $c\left( n \right)$ vanish for all positive integers $n$ in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if $f\left( z \right)\,=\,{{\eta }^{4}}\left( z \right){{\eta }^{9}}\left( 4z \right){{\eta }^{-2}}\left( 8z \right)$ we have $c\left( n \right)\,=\,0$ for all $n$ in each of the arithmetic progressions ${{\{16k\,+\,14\}}_{k\ge 0}},\,{{\{64k\,+\,56\}}_{k\ge 0}},\,{{\{256k,\,224\}}_{k\ge 0}},\,{{\{1024k\,+\,869\}}_{k\ge 0}},\,.\,.\,.$

We describe an ${{E}_{8}}$ correspondence for the multiplicative eta-products of weight at least 4.

Let q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with 1 ≤ a ≤ q − 1, it is clear that there exists one and only one b with 0 ≤ b ≤ q − 1 such that ab ≡ c (mod q). Let N(c, q) denotes the number of all solutions of the congruence equation ab ≡ c (mod q) for 1 ≤ a, b ≤ q − 1 in which a and b are of opposite parity, where b is defined by the congruence equation bb ≡ 1(modq). The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involving (N(c, q) − φ(q)) and Ramanujan's sum, and give two exact computational formulae.

In this paper, we investigate a conjecture due to S. and P. Chowla and its generalization by Milnor. These are related to the delicate question of non-vanishing of $L$-functions associated to periodic functions at integers greater than 1. We report on some progress in relation to these conjectures. In a different vein, we link them to a conjecture of Zagier on multiple zeta values and also to linear independence of polylogarithms.

We prove an observation associated with η3(τ)η3(7τ) which is found on page 54 of Ramanujan’s Lost Notebook (S. Ramanujan, The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988)). We then study functions of the type η3(aτ)η3(bτ) with a+b=8.

In this article we derive analytic and Fourier aspects of a Kronecker limit formula for second-order Eisenstein series. Let Γ be any Fuchsian group of the first kind which acts on the hyperbolic upper half-space H such that the quotient Γ\H has finite volume yet is non-compact. Associated to each cusp of Γ\H, there is a classically studied first-order non-holomorphic Eisenstein series E(s, z) which is defined by a generalized Dirichlet series that converges for Re(s) > 1. The Eisenstein series E(s, z) admits a meromorphic continuation with a simple pole at s = 1. Classically, Kronecker’s limit formula is the study of the constant term 1 (z) in the Laurent expansion of E(s, z) at s = 1. A number of authors recently have studied what is known as the second-order Eisenstein series E*(s, z), which is formed by twisting the Dirichlet series that defines the series E(s, z) by periods of a given cusp form f. In the work we present here, we study an analogue of Kronecker’s limit formula in the setting of the second-order Eisenstein series E* (s, z), meaning we determine the constant term 2(z) in the Laurent expansion of E*(s, z) at its first pole, which is also at s = 1. To begin our investigation, we prove a bound for the Fourier coefficients associated to the first-order Kronecker limit function 1. We then define two families of convolution Dirichlet series, denoted by and with m ∈ ℕ, which are formed by using the Fourier coefficients of 1 and the weight two cusp form f. We prove that for all m, and admit a meromorphic continuation and are holomorphic at s = 1. Turning our attention to the second-order Kronecker limit function 2, we first express 2 as a solution to various differential equations. Then we obtain its complete Fourier expansion in terms of the cusp form f, the Fourier coefficients of the first-order Kronecker limit function 1, and special values (1) and (1) of the convolution Dirichlet series. Finally, we prove a bound for the special values (1) and (1) which then implies a bound for the Fourier coefficients of 2. Our analysis leads to certain natural questions concerning the holomorphic projection operator, and we conclude this paper by examining certain numerical examples and posing questions for future study.

Let $s:\Q\,{\longrightarrow}\,\Q$ be the Dedekind sum, given by $s(h/k)=\sum_{\nu=1}^{k-1}({\nu/k}\,{-}\,{1/2})(\{{h\nu/k}\}\,{-}\,{1/2})$ when $\gcd(h,k)\,{=}\,1$. Then for every rational $\alpha\,{\ne}\,1/12$ there are infinitely many rational $x$ such that $s(x)\,{=}\,\alpha x$. Also, the fixed points of $s$ are dense in the real line.

Habib Muzaffar of Carleton University has pointed out to the authors that in their paper $\left[ \text{A} \right]$ only the result

$${{\pi }_{K,d}}\left( x \right)\,+\,{{\pi }_{{{K}^{-1}},d}}\left( x \right)\,=\,\frac{1}{h\left( d \right)}\,\frac{x}{\log x}\,+\,{{O}_{K,d}}\left( \frac{x}{{{\log }^{2}}x} \right)$$

follows from the prime ideal theorem with remainder for ideal classes, and not the stronger result

$${{\pi }_{K,d}}\left( x \right)\,=\,\frac{1}{2h\left( d \right)}\,\frac{x}{\log \,x}\,+\,{{O}_{K,d}}\left( \frac{x}{{{\log }^{2}}\,x} \right)$$

stated in Lemma 5.2. This necessitates changes in Sections 5 and 6 of $\left[ \text{A} \right]$. The main results of the paper are not affected by these changes. It should also be noted that, starting on page 177 of $\left[ \text{A} \right]$, each and every occurrence of $o\left( s-1 \right)$ should be replaced by $o\left( 1 \right)$.

Sections 5 and 6 of $\left[ \text{A} \right]$ have been rewritten to incorporate the abovementioned correction and are given below. They should replace the original Sections 5 and 6 of $\left[ \text{A} \right]$.

Let $d$ be the discriminant of an imaginary quadratic field. Let $a$, $b$, $c$ be integers such that

$${{b}^{2}}-4ac=d,a>0,\gcd (a,b,c)=1.$$

The value of $\left| \eta ((b+\sqrt{d})/\left. 2a) \right| \right.$ is determined explicitly, where $\eta \left( z \right)$ is Dedekind’s eta function

$$\eta (z)\,=\,{{e}^{\pi iz/12}}\,\prod\limits_{m=1}^{\infty }{(1-{{e}^{2\pi imz}})\,\,\,(im(z)>0).}$$

Let ((x)) = x −⌊x⌋−1/2 be the swatooth function. If a, b, c and e are positive integeral, then the integral or ((ax)) ((bx)) ((cx)) ((ex)) over the unit interval involves Apolstol's generalized Dedekind sums. By expressing this integral as a lattice-point sum we obtain an elementary method for its evaluation. We also give an elementary proof of the reciprocity law for the third generalized Dedekind sum.

For each integer r we define the sequence pr(n) by We note that p-1(n) = p(n), the ordinary partition function. On account of this some authors set r = — k to make positive values of k correspond to positive powers of the generating function for p(n): We follow this convention here. In [3], Atkin proved the following theorem.