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Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of representations, and play an important role in the arithmetic theory of quadratic forms. In a 1938 paper, Siegel made a comment to the effect that the modularity of his zeta functions would be proved with the help of a suitable converse theorem. In the present paper, we accomplish Siegel's original plan by using a Weil-type converse theorem for Maass forms, which has appeared recently. It is also shown that "half" of Siegel's zeta functions correspond to holomorphic modular forms. [ABSTRACT FROM AUTHOR]

We study a sequence of constants known as the Bendersky-Adamchik constants which appear quite naturally in number theory and generalize the classical Glaisher-Kinkelin constant. Our main initial purpose is to elucidate the close relation between the logarithm of these constants and the Ramanujan summation of certain divergent series. In addition, we also present a remarkable, and previously unknown, expansion of the logarithm of these constants in convergent series involving the Bernoulli numbers of the second kind. [ABSTRACT FROM AUTHOR]

We show that the Generalized Riemann Hypothesis for Dirichlet L-functions is a consequence of certain conjectural properties of the zeros of the Riemann zeta function. Conversely, we prove that the zeros of ζ (s) satisfy those properties under GRH. [ABSTRACT FROM AUTHOR]

We prove that the Riemann zeta-function ζ (σ + i t) has no zeros in the region σ ≥ 1 - 1 / (55.241 (log | t |) 2 / 3 (log log | t |) 1 / 3 ) for | t | ≥ 3 . In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region σ ≥ 1 - 1 / (5.558691 log | t |) for | t | ≥ 2 . We also provide new bounds that are useful for intermediate values of | t | . Combined, our results improve the largest known zero-free region within the critical strip for 3 · 10 12 ≤ | t | ≤ exp (64.1) and | t | ≥ exp (1000) . [ABSTRACT FROM AUTHOR]

This note gives some applications of the zeta distribution to number theory. First, a number of probabilistic proofs of the equality between the Riemann zeta functions and some number-theoretic functions are provided. Second, some problems of probability with respect to positive integers considered in the literature are discussed in the zeta distribution case. In contrast to usual heuristic arguments of probabilistic approach in number theory where the uniform distribution is assumed on positive integers, the proofs here are rigorous by utilizing the zeta distributions on positive integers. [ABSTRACT FROM AUTHOR]

We unify in a large class of additive functions the results obtained in the first part of this work. The proof rests on series involving the Riemann zeta function and certain sums of primes which may have their own interest. [ABSTRACT FROM AUTHOR]

For a number field K , the Euler–Kronecker constant γ K associated to K is an arithmetic invariant the size and nature of which is linked to some of the deepest questions in number theory. This theme was given impetus by Ihara who obtained bounds, both unconditional as well as under GRH for Dedekind zeta functions. In this note, we study the analogous constants associated to the narrow ray class fields of an imaginary quadratic field. Our goal for studying such families is twofold. First to show that for such families, the conditional bounds obtained by Ihara can be improved on the average, again under GRH for Dedekind zeta functions. Further, our family of number fields are non-abelian while such average bounds have earlier been studied for cyclotomic fields. The technical part of our work is to make the dependence of these upper bounds on the ambient number field explicit. Such explicit dependence is essential to further our objective. [ABSTRACT FROM AUTHOR]

It is known that multiple zeta values whose weight and depth are of opposite parity can be written in terms of multiple zeta values of lower depth. This theorem is called parity result. Multiple zeta values are special values of the multiple polylogarithms and the parity result is generalized to functional relations satisfied by the multiple polylogarithms. In this paper, we consider q- and elliptic generalizations of the parity result. As a main result of this paper, we establish parity result for functions L k (a , α ; p , q) , which can be considered to be common deformations of q- and elliptic multiple polylogarithms. By taking the trigonometric and classical limits in the main theorem, we obtain q- and elliptic analogues of the parity result. [ABSTRACT FROM AUTHOR]

In his lost notebook, Ramanujan noted down many elegant identities involving divisor functions and the modified K-Bessel function, and some of them are connected with the Fourier series expansion of the non-holomorphic Eisenstein series. Recently, Cohen established interesting generalizations of some of the identities of Ramanujan. In this paper, we study Ramanujan and Cohen-type identities associated to a generalized divisor function and the modified K-Bessel function. In the process, we extend a result of Chandrasekharan and Narasimhan and some identities of Cohen. Furthermore, we obtain a new identity for odd zeta values that can be thought of as a Bessel function analogue of Ramanujan's famous formula for odd zeta values. [ABSTRACT FROM AUTHOR]

For any real β 0 ∈ [ 1 2 , 1) , let GRH [ β 0 ] be the assertion that for every Dirichlet character χ and all zeros ρ = β + i γ of L (s , χ) , one has β ⩽ β 0 (in particular, GRH [ 1 2 ] is the Generalized Riemann Hypothesis). In this paper, we show that the validity of GRH [ 9 10 ] depends only on certain distributional properties of the zeros of the Riemann zeta function ζ (s) . No conditions are imposed on the zeros of nonprincipal Dirichlet L-functions. [ABSTRACT FROM AUTHOR]

We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then obtained by making corrections at singular points. We report on an implementation and provide some examples in MAGMA which demonstrate an improvement over Tuitman's algorithm. [ABSTRACT FROM AUTHOR]

We proof a Garrett–Böcherer decomposition of a vector-valued Siegel Eisenstein series E l , 0 2 of genus 2 transforming with the Weil representation of Sp 2 (Z) on the group ring C [ (L ′ / L) 2 ] . We show that the standard zeta function associated to a vector-valued common eigenform f for the Weil representation can be meromorphically continued to the whole s-plane and that it satisfies a functional equation. The proof is based on an integral representation of this zeta function in terms of f and E l , 0 2 . [ABSTRACT FROM AUTHOR]

An interpolation function of the multiple harmonic sum is defined. Its basic properties, asymptotic behaviors and derivatives, which play fundamental roles in the study of the multiple polylogarithm and multiple zeta values, are investigated. [ABSTRACT FROM AUTHOR]

We study the elliptic curves given by y 2 = x 3 + b x + t 3 n + 1 over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L-function by relating it to the zeta function of a certain superelliptic curve u 3 + b u = v 3 n + 1 . In this way, using the Néron–Tate height on the Mordell–Weil group, we obtain lattices in dimension 2 · 3 n for every n ≥ 1 , which improve on the currently best known sphere packing densities in dimensions 162 (case n = 4 ) and 486 (case n = 5 ). For n = 3 , the construction has the same packing density as the best currently known sphere packing in dimension 54, and for n = 1 it has the same density as the lattice E 6 in dimension 6. [ABSTRACT FROM AUTHOR]

For β > 1 a real algebraic integer (the base), the finite alphabets A ⊂ Z which realize the identity Q (β) = Per A (β) , where Per A (β) is the set of complex numbers which are (β , A) -eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and natural alphabets are defined. The natural alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base β and Lehmer's problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in Q (β) , generalizing Schmidt's theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases γ s : = γ n , m 1 , ... , m s such that γ s - 1 is the unique root in (0, 1) of an almost Newman polynomial of the type - 1 + x + x n + x m 1 + ⋯ + x m s , n ≥ 3 , s ≥ 1 , m 1 - n ≥ n - 1 , m q + 1 - m q ≥ n - 1 for all q ≥ 1 . For β > 1 a reciprocal algebraic integer close to one, the poles of modulus < 1 of the dynamical zeta function of the β -shift ζ β (z) are shown, under some assumptions, to be zeroes of the minimal polynomial of β . [ABSTRACT FROM AUTHOR]

Let ζ (s) and Z(t) be the Riemann zeta function and Hardy's function respectively. We show asymptotic formulas for ∫ 0 T Z (t) ζ (1 / 2 + i t) d t and ∫ 0 T Z 2 (t) ζ (1 / 2 + i t) d t . Furthermore we derive an upper bound for ∫ 0 T Z 3 (t) χ α (1 / 2 + i t) d t for - 1 / 2 < α < 1 / 2 , where χ (s) is the function which appears in the functional equation of the Riemann zeta function: ζ (s) = χ (s) ζ (1 - s) . [ABSTRACT FROM AUTHOR]

We study a refinement of the symmetric multiple zeta value, called the t-adic symmetric multiple zeta value, by considering its finite truncation. More precisely, two kinds of regularizations (harmonic and shuffle) give two kinds of the t-adic symmetric multiple zeta values, thus we introduce two kinds of truncations correspondingly. Then we show that our truncations tend to the corresponding t-adic symmetric multiple zeta values, and satisfy the harmonic and shuffle relations, respectively. This gives a new proof of the double shuffle relations for t-adic symmetric multiple zeta values, first proved by Jarossay. In order to prove the shuffle relation, we develop the theory of truncated t-adic symmetric multiple zeta values associated with 2-colored rooted trees. Finally, we discuss a refinement of Kaneko–Zagier's conjecture and the t-adic symmetric multiple zeta values of Mordell–Tornheim type. [ABSTRACT FROM AUTHOR]

We prove a formula relating the Mahler measure of an infinite family of three-variable polynomials to a combination of the Riemann zeta function at s = 3 and special values of the Bloch–Wigner dilogarithm by evaluating a regulator. The evaluation requires two different applications of Jensen's formula and analyzing the integral in two different planes, as opposed to the more common strategy of using only one plane. The degrees of the monomials involving one of the variables are allowed to vary freely, leading to an interesting application of the Boyd–Lawton formula. [ABSTRACT FROM AUTHOR]

Inspired by the Mordell–Tornheim double zeta function, in this paper, we introduce the Eisenstein series of Mordell–Tornheim type. Based on the theory of Cohen series and partial fraction decomposition, we explicitly compute these series. We end by discussing several identities about products of Eisenstein series and their Fourier expansions. [ABSTRACT FROM AUTHOR]

We investigate the relationship between L-functions satisfying certain functional equations, summation formulas of Ferrar–Suzuki type and Maass forms of integral and half-integral weight. Summation formulas of Ferrar–Suzuki type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of SL(2). Our goal is converse theorems for automorphic distributions and Maass forms of level N characterizing them by analytic properties of the associated L-functions. As an application of our converse theorems, we construct Maass forms from the two-variable zeta functions related to quadratic forms studied by Peter and the fourth author. [ABSTRACT FROM AUTHOR]

We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special cases coincide with zeta values at positive integer arguments. [ABSTRACT FROM AUTHOR]

In this series we examine the calculation of the 2 kth moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along [ T, 2 T] of a Dirichlet polynomial of length up to $$T^3$$ with divisor functions as coefficients. [ABSTRACT FROM AUTHOR]

We exploit transformations relating generalized q-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as π, and to connect sums over partitions to the Riemann zeta function, multiple zeta values, and other number-theoretic objects. [ABSTRACT FROM AUTHOR]

We provide a q-analogue of Euler's formula for ζ(2k) for k∈Z+. The result generalizes a recent result of Sun who obtained q-analogues of ζ(2)=π2/6 and ζ(4)=π4/90. This also extends an earlier result of the present author who obtained a q-analogue of ζ(6)=π6/945. [ABSTRACT FROM AUTHOR]

New expressions are given for the Fourier expansions of non-holomorphic Eisenstein series with weight k. Among other applications, this leads to non-holomorphic analogs of formulas of Ramanujan, Grosswald and Berndt containing Eichler integrals of holomorphic Eisenstein series. [ABSTRACT FROM AUTHOR]