On Dirichlet series with identical beginnings

It is shown that if Vinogradov's conjecture is false for a Dirichlet character (mod q), then ζ(s) and L(s) are very similar in regions of the critical strip where ζ(s), L(s) are small. In particular, ζ(s) = L(s + h(s)) (where h(s) → 0) in such regions.     

An adelic Hankel summation formula

In this paper, the convergence of the Euler product of the Hecke zeta-function ζ(s,χ) is proved on the line R(s)=1 with s≠1. A certain functional identity between ζ(s,χ) and ζ(2−s,χ) is found. An analogue of Tate's adelic Poisson summation is obtained for the global Hankel transformation, which is constructed in Li (2010) [7].     

On some averages at the zeros of the derivatives of the Riemann zeta-function

In this article we study two problems raised by a work of Conrey and Ghosh from 1989. Let ζ(k)(s) be the k-th derivative of the Riemann zeta-function, and χ(s) be factor in the functional equation of the Riemann zeta-function. We calculate the average values of ζ(j) and χ at the nontrivial zeros of ζ(k).     

On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ ²(s)ζ(2s?1)ζ ^M (2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R _s > σ₀, with some fixed σ₀ < 1/2.     

Torsion points on the jacobian of a Fermat quotient

Let p≥5 be a prime, ζ a primitive pth root of unity and λ=1−ζ. For 1≤s≤p−2, the smooth projective model C_p,s of the affine curve v^p=u^s(1−u) is a curve of genus (p−1)/2 whose jacobian J_p,s has complex multiplication by the ring of integers of the cyclotomic field Q(ζ). In 1981, Greenberg determined the field of rationality of the p-torsion subgroup of J_p,s and moreover he proved that the λ³-torsion points of J_p,s are all rational over Q(ζ). In this paper we determine quite explicitly the λ³-torsion points of J_p,1 for p=5 and p=7, as well as some further p-torsion points which have interesting arithmetical applications, notably to the complementary laws of Kummer’s reciprocity for pth powers.     

Self-approximation of Hurwitz zeta-functions with rational parameter

In this paper, we show the self-approximation property for Hurwitz zeta-functions with rational parameters. Namely, we prove that ζ(s?+?iατ, a/b) approximates uniformly ζ(s?+?iβτ, a/b) for infinitely many real τ , where α, β are arbitrary real numbers linearly independent over $ \mathbb{Q} $ , and s is in a compact set lying in the open right half of the critical strip.     

On the dimension of solution spaces of a noncommutative sigma model in the case of uniton number 2

This is primarily an overview article on some results and problems involving the classical Hardy function Z(t):= ζ(1/2 + it)(χ(1/2 + it))^?1/2, ζ(s) = χ(s)ζ(1 ? s). In particular, we discuss the first and third moments of Z(t) (with and without shifts) and the distribution of its positive and negative values. A new result involving the distribution of its values is presented.     

Müntz formula and zero free regions for the Riemann zeta function

A formula first derived by Müntz which relates the Riemann zeta function ζ times the Mellin transform of a test function f and the Mellin transform of the theta transform of f is exploited, together with other analytic techniques, to construct zero free regions for ζ(s) with s in the critical strip. Among these are regions with a shape independent of Res.     

The density of zeros of the Riemann zeta-function in the critical strip

A new proof of Ingam’s theorem on the density of zeros of the Riemann zeta-function in the critical strip is given basing on an idea of H. Bohr and F. Carlson. Multiplication of segments of the Dirichlet series for the functions ζ(s) and 1/ζ(s) is used, which permits to simplify the proof.     

Linear twists of L-functions of degree 2 from the Selberg class

In 1983, S. M. Voronin obtain an analytic continuation to the entire complex plane for some twists of L-functions associated with holomorphic modular forms on SL(2, Z). In this paper, we extend Voronin’s result to L-functions of degree 2 from the extended Selberg class.     

Erratum to: “The Mellin Transform of Hardy’s Function is Entire”

We prove that an appropriately modified Mellin transform of the Hardy function Z(x) is an entire function. The proof is based on the fact that the function (2^1?s ? 1)ζ(s) is entire.     

The mellin transform of Hardy’s function is entire

We prove that an appropriately modified Mellin transform of the Hardy function Z(x) Is en entire function. The proof is based on the fact that the function (2^1?s ? 1)ζ(s) is integer.     

Transformation formulae for Dirichlet polynomials

Formulae of Voronoi-Atkinson type are proved for Dirichlet polynomials related to the Dirichlet series ζ²(s) = Σd(n)n^?s or ?(s) = Σa(n)n^?s, where the a(n) are the Fourier coefficients of a cusp form, a typical example being a(n) = τ(n), the Ramanujan function. Applications are given to a formula of Atkinson (Acta Math.81 (1949), 353–376) for the mean square of $\text{|ζ(}\text{1}\text{2}\text{+ it)|}$ and to the differences between consecutive zeros of ?(s) on the critical line in the case when all the a(n) are real.     

Dirichlet series related to the Riemann zeta function

A study is made of the function H(s, z) defined by analytic continuation of the Dirichlet series H(s, z) = Σ_n=1^∞n^?sΣ_m=1ⁿm^?z, where s and z are complex variables. For each fixed z it is shown that H(s, z) exists in the entire s-plane as a meromorphic function of s, and its poles and residues are determined. Also, for each fixed s ≠ 1 it is shown that H(s, z) exists in the entire z-plane as a meromorphic function of z, and again its poles and residues are determined. Two different representations of H(s, z) are given from which a reciprocity law, H(s, z) + H(z, s) = ζ(s) ζ(z) + ζ(s + z), is deduced. For each integer q ≥ 0 the function values H(s, ?q) and H(?q, s) are expressed in terms of the Riemann zeta function. Similar results are also obtained for the Dirichlet series T(s, z) = Σ_n=1^∞n^?sΣ_m=1ⁿm^?z (m + n)^?1. Applications include identities previously obtained by Ramanujan, Williams, and Rao and Sarma.     

On some functional relations between Mordell-Tornheim double L-functions and Dirichlet L-functions

In the past decade, many relation formulas for the multiple zeta values, further for the multiple L-values at positive integers have been discovered. Recently Matsumoto suggested that it is important to reveal whether those relations are valid only at integer points, or valid also at other values. Indeed the famous Euler formula for ζ(2k) can be regarded as a part of the functional equation of ζ(s). In this paper, we give certain analytic functional relations between the Mordell-Tornheim double L-functions and the Dirichlet L-functions of conductor 3 and 4. These can be regarded as continuous generalizations of the known discrete relations between the Mordell-Tornheim L-values and the Dirichlet L-values of conductor 3 and 4 at positive integers.     

The distribution and average order of the coefficients of Dedekind ζ functions

Let K/$\text{Q}$ be an algebraic number field and ζ_K(s) be the associated Dedekind ζ function. A quantitative estimate is proved which shows that the average order of the coefficients of ζ_k^m(s) (for m ∈ $\text{Z}$⁺) arises from infrequent occurrences of very large values of these coefficients. This leads to new Ω-estimates for the associated error terms, improving results of Szegö and Walfisz.     

Integral representations of the Legendre chi function

We, by making use of elementary arguments, deduce integral representations of the Legendre chi function χs(z) valid for |z|<1 and Res>1. Our earlier established results on the integral representations for the Riemann zeta function ζ(2n+1) and the Dirichlet beta function β(2n), n∈N, are a direct consequence of these representations.     

Generalized Brouncker’s continued fractions and their logarithmic derivatives

In this paper, we study the continued fraction y(s,r) which satisfies the equation y(s,r)y(s+2r,r)=(s+1)(s+2r?1) for $r > \frac{1}{2}$ . This continued fraction is a generalization of the Brouncker’s continued fraction b(s). We extend the formulas for the first and the second logarithmic derivatives of b(s) to the case of y(s,r). The asymptotic series for y(s,r) at ∞ are also studied. The generalizations of some Ramanujan’s formulas are presented.     

Series involving zeta and related functions

Using Padé approximation to the exponential function, we obtain new identities involving values of the Rieman zeta function at integers. Applications to series associated with zeta numbers are proved. In particular, expansion of ζ(3) (resp. ζ(5)) in terms of ζ(4j + 2) (resp. ζ(4j)) are proved.