Related papers: On some problems involving Hardy's function

The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed.

The Laplace transform of $|\zeta(1/2+it)|$ is investigated, for which a precise expression is obtained, valid in a certain region in the complex plane. The method of proof is based on complex integration and spectral theory of the…

In this paper we characterize the validity of the Hardy-type inequality \begin{equation*} \left\|\left\|\int_s^{\infty}h(z)dz\right\|_{p,u,(0,t)}\right\|_{q,w,\infty}\leq c \,\|h\|_{1,v,\infty} \end{equation*} where $0<p< \infty$, $0<q\leq…

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions $\frac{1}{2} + i(a n + b)$. It reveals noticeable relation between the discrete moments and the continuous moment of the…

This study deals with certain harmonic zeta functions, one of them occurs in the study of the multiplication property of the harmonic Hurwitz zeta function. The values at the negative even integers are found and Laurent expansions at poles…

We improve the classical discrete Hardy inequality for $ 1<p<\infty $ for functions on the natural numbers. For integer values of $ p $ the Hardy weight is an absolutely monotonic function.

This paper begins with a re-examination of the Riemann-Siegel Integral, which first discovered amongst by Bessel-Hagen in 1926 and expanded upon by C. L. Siegel on his 1932 account of Riemanns unpublished work on the zeta function. By…

The elementary geometric properties of Jacob's ladders of the second order lead to a class of new asymptotic formulae for short and microscopic parts of the Hardy-Littlewood integral of $|\zeta(1/2+it)|^4$. These formulae cannot be obtained…

Building on the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, and by defining the Zeta and related functions including the Hurwitz Zeta function…

An explicit subconvex bound for the Riemann zeta function $\zeta(s)$ on the critical line $s=1/2+it$ is proved. Previous subconvex bounds relied on an incorrect version of the Kusmin-Landau lemma. After accounting for the needed correction…

The secondary zeta function $Z(s)=\sum_{n=1}^\infty\alpha_n^{-s}$, where $\rho_n=\frac12+i\alpha_n$ are the zeros of zeta with $\Im(\rho)>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis…

Four Jacobi settings are considered in the context of Hardy's inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy's inequality is proved for the type…

Starting with a functional difficulty appeared in the paper \cite{vz00} by V\'azquez and Zuazua, we obtain new insights into the Hardy Inequality and the evolution problem associated to it by means of a reformulation of the problem.…

We study the moments $M_k(T;\alpha) = \int_T^{2T} |\zeta(s,\alpha)|^{2k}\,dt$ of the Hurwitz zeta function $\zeta(s,\alpha)$ on the critical line, $s = 1/2 + it$ with a rational shift $\alpha \in \mathbb Q$. We conjecture, in analogy with…

Some new results on power moments of the integral $$ J_k(t,G) = {1\over\sqrt{\pi}G} \int_{-\infty}^\infty |\zeta(1/2 + it + iu)|^{2k}{\rm e}^{-(u/G)^2}du \qquad(t \asymp T, T^\epsilon \le G \ll T, k\in\N) $$ are obtained when $k=1$. These…

In this article we prove an explicit sub-Weyl bound for the Riemann zeta function $\zeta(s)$ on the critical line $s = 1/2 + it$. In particular, we show that $|\zeta(1/2 + it)| \le 66.7\, t^{27/164}$ for $t \ge 3$. Combined, our results…

In this paper we introduce the iterations of the Jacob's ladder and the new type of integral containing certain product of the factors $|\zeta|^2$ corresponding to the components of some disconnected set of the critical line. Next, we…

We describe in detail three distinct families of generalized zeta functions built over the (nontrivial) zeros of a rather general arithmetic zeta or L-function, extending the scope of two earlier works that treated the Riemann zeros only.…

Let $d_{\alpha, \beta}(n)=\sum\limits_{\substack{n=kl \alpha l<k\leq\beta l}}1$ be the number of ways of factoring n into two almost equal integers. For rational numbers $0<\alpha <\beta $, we consider the following Zeta function…

We study the value-distribution of the Hurwitz zeta-function with algebraic irrational parameter $\zeta(s;\alpha)=\sum_{n\geq_0}(n+\alpha)^{-s}$. In particular, we prove effective denseness results of the Hurwitz zeta-function and its…