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.     

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.     

Addison-type series representation for the Stieltjes constants

The Stieltjes constants γk(a) appear in the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(s,a) about its only pole at s=1. We generalize a technique of Addison for the Euler constant γ=γ₀(1) to show its application to finding series representations for these constants. Other generalizations of representations of γ are given.     

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.     

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.     

Applied hypernumbers: computational concepts

The key importance of hypernumbers in enlarging and fruitfully generalizing (as distinct from abstraction of a sterile sort) algebra, function theory and computation is discussed, with specific examples and theorems. The rich serendipity of hypernumber research is shown in the author's recent findings; for example, those generalizing the Bernoulli numbers for any real, complex, or countercomplex index s, as B_s= -s!2cos(πs/2) ζ(s)/(2π)^s, where ζ is Riemann's Zeta function; whence, e.g., B₀=1, B₂=1/6, B_1/2=hf;ζ(hf;), and B₃/B₅=-(80π²)^-1ζ(3)/ζ(5), results like the last two being unknown and unobtainable before. As in APL computer language, the symbol “!” is used to denote Gauss' π function: the factorial of unrestricted argument.     

Probabilistic properties of number fields

In general a bound on number theoretic invariants under the Generalized Riemann Hypothesis (GRH) for the Dedekind zeta function of a number field K   is much stronger than an unconditional one. In this article, we consider three invariants; the residue of _ζK(s)${\zeta }_K(s)$ at s=1$s=1$, the logarithmic derivative of Artin L-function attached to K   at s=1$s=1$, and the smallest prime which does not split completely in K. We obtain bounds on them just as good as the bounds under GRH except for a density zero set of number fields.     

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.     

Hyperbolic stochastic differential equations: Absolute continuity of the LKW of the solution at a fixed point

LetW be the Wiener process onT=[0, 1]². Consider the stochastic integral equation $$\begin{gathered} X_\zeta = x_0 + \int_{R_\zeta } {a_1 (\zeta \prime )X(s\prime ,dt\prime )ds\prime + } \int_{R_\zeta } {a_2 (\zeta \prime )X(ds\prime ,t\prime )dt\prime } \hfill \\ + \int_{R_\zeta } {a_3 (X_{\zeta \prime , } \zeta \prime )W(ds\prime ,dt\prime ) + } \int_{R_\zeta } {a_4 (X_{\zeta \prime , } \zeta \prime )ds\prime ,dt\prime ,} \hfill \\ \end{gathered} $$ whereR _ζ =(s, t) ∈ T, andx ₀ ∈ ?. Under some assumptions on the coefficients a_i, the existence and uniqueness of a solution for this stochastic integral equation is already known (see [6]). In this paper we present some sufficient conditions for the law ofX _ζ to have a density.     

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.     

Lacunary interpolation by quartic splines on uniform meshes

The interpolation of a discrete set of data on the interval [0, 1], representing the first and the second derivatives (except at 0) of a smooth function f is investigated via quartic C²-splines. Error bounds in the uniform norm for ∥s⁽ⁱ⁾ − f⁽ⁱ⁾∥, i=0(1)2, if f ∈ C^l[0, 1], l=3, 5 and ⁽³⁾ ∈ BV[0, 1], together with computational examples will also be presented.     

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.     

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.     

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.     

A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials

A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given.     

The joint universality theorem for a pair of Hurwitz zeta functions

In this paper we investigate the joint functional distribution for a pair of Hurwitz zeta functions ζ(s,αj) (j=1,2) in the case that real transcendental numbers α₁ and α₂ satisfy α₂∈Q(α₁). Especially we establish the joint universality theorem for these zeta functions.     

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.     

Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem

In this paper we study generalised prime systems for which the integer counting function NP(x) is asymptotically well behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and . For such systems, the associated zeta function ζP(s) is holomorphic for . We prove that for , for any ε>0, and also for ε=0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in NkP(x)−Ress=1(ζPk(s)xs/s), which is O(xθ) for some θ<1. Letting αk denote the infimum of such θ, we show that .