Related papers: Mean square of zeta function, circle problem and d…
To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special…
The purpose of this paper is to prove that the so-called Quasi-Riemann Hypothesis for the Zeta-function implies the Riemann Hypothesis
The aim of the present article is to reveal a structure shared by two basic zeta-functions in their fourth power moments through the view point of representation theory of Lie groups, relying specifically upon the Kirillov model. It might…
We conclude the rigorous analysis of a previous paper concerning the relation between the (Euclidean) point-splitting approach and the local $\zeta$-function procedure to renormalize physical quantities at one-loop in (Euclidean) QFT in…
Some problems involving the classical Hardy function $$ Z(t) := \zeta(1/2+it)\bigl(\chi(1/2+it)\bigr)^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s) $$ are discussed. In particular we discuss the odd moments of $Z(t)$, the distribution of its…
Sums of squares of $|\zeta(1/2+it)|$ over short intervals are investigated. Known upper bounds for the fourth and twelfth moment of $|\zeta(1/2+it)|$ are derived. A discussion concerning other possibilities for the estimation of higher…
We obtain an asymptotic formula for the second discrete moment of the Riemann zeta function over the arithmetic progression $\frac{1}{2} + in$. It shows that the first main term is equal to that of the continuous mean value.
Let ${\cal Z}_1(s) = \int_1^\infty |\zeta({1\over2}+ix)|^2x^{-s}{\rm d}x (\sigma = \Re s > 1)$. A result concerning analytic continuation of ${\cal Z}_1(s)$ to $\bf C$ is proved, and also a result relating the order of ${\cal Z}_1(\sigma +…
We discuss modifications in the integral representation of the Riemann zeta-function that lead to generalizations of the Riemann functional equation that preserves the symmetry $s\to (1-s)$ in the critical strip. By modifying one integral…
We define certain higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums, and show how to compute them effectively using a generalization of the continued-fraction algorithm. We present two applications.…
Ap\'ery's remarkable discovery of rapidly converging continued fractions with small coefficients for $\zeta(2)$ and $\zeta(3)$ has led to a flurry of important activity in an incredible variety of different directions. Our purpose is to…
We study a certain class of arithmetic functions that appeared in Klurman's classification of $\pm 1$ multiplicative functions with bounded partial sums, c.f., Comp. Math. 153 (8), 2017, pp. 1622-1657. These functions are periodic and…
We first study the mean value of certain restricted divisor sums involving the Chowla-Walum sums, improving in particular a recent estimate given by Iannucci. The aim of the second part of this work is the generalization of the previous…
This paper contains a small improvement to the explicit bounds on the growth of the function $S(T)$. It is shown how more substantial improvements are possible if one has better explicit bounds on the growth of $|\zeta(\frac{1}{2}+it)|$.
The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each…
Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{{\rm sa}}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a…
The motion in the complex plane of the zeros to various zeta functions is investigated numerically. First the Hurwitz zeta function is considered and an accurate formula for the distribution of its zeros is suggested. Then functions which…
The main goal of this article is to present several quadratic refinements and reverses of the well known Heinz inequality, for numbers and matrices, where the refining term is a quadratic function in the mean parameters. The proposed idea…
Several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation if the important results of [11]. Also, a relation derived…
In the paper, we give partition-theoretic results for the coefficients of some mock theta functions and prove their congruence properties. Some recurrence relations connecting the coefficients of the mock theta functions with certain…