Related papers: On a function of Marcel Riesz
We derive and prove a new formulation of the Lerch zeta function as a fractional derivative of an elementary function. We demonstrate how this formulation interacts very naturally with basic known properties of Lerch zeta, and use the…
A relationship between the Riemann zeta function and a density on integer sets is explored. Several properties of the examined density are derived.
Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory…
In this paper we explore a family of congruences over $\N^\ast$ from which one builds a sequence of symmetric matrices related to the Mertens function. From the results of numerical experiments, we formulate a conjecture about the growth of…
A new integral representation for the Riemann zeta function is derived. This representation covers the important region of the complex plane where the real part of the argument of the function lies between 0 and 1. Using this…
Assuming the Riemann hypothesis, we obtain a formula for the mean value of the $k$-derivative of $\zeta'/\zeta$, depending on the pair correlation of zeros of the Riemann zeta-function. This formula allows us to obtain new equivalences to…
In this work, Miller Ross function with bicomplex arguments has been introduced. Various properties of this function including recurrence relations, integral representations and differential relations are established. Furthermore, the…
Beginning from the formal resolution of Riemann Zeta function, by using the formula of inner product between two infinite-dimensional vectors in the complex space, the author proved the world's baffling problem -- Riemann hypothesis raised…
We introduce a screw function corresponding to the Riemann zeta-function and study its properties from various aspects. Typical results are several equivalent conditions for the Riemann hypothesis in terms of the screw function. One of them…
We consider some properties of integrals considered by Hardy and Koshliakov, and which have also been further extended recently by Dixit. We establish a new general integral formula from some observations about the digamma function. We also…
This paper describes some validated numerics aspects of Riemann zeta function, Dirichlet L-functions, Dedekind zeta functions and Hasse-Weil L-functions.
In this paper a new integral for the remainder of $\pi(x)$ is obtained. It is proved that there is an infinite set of the formulae containing miscellaneous parts of this integral.
We generalize a size-biased distribution related to the Riemann xi function using the work of Ferrar. Some analysis and properties of this more general distribution are offered as well.
We consider the alternating Riemann zeta function $\zeta^*(s)= \sum^{\infty} _{ n=1} \frac{(-1)^{n-1}}{n^s}$, which converges if $Re (s)>0 .$ By using Rouche's theorem, the Bolzano-Weierstrass theorem and by method of contradiction we…
In this article, we study the distribution of values of Dirichlet $L$-functions, the distribution of values of the random models for Dirichlet $L$-functions, and the discrepancy between these two kinds of distributions. For each question,…
Approximation in measure is employed to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann Hypothesis are…
A multidimensional generalization of the Bernstein class of functions and the properties of functions of the introduced class are examined. In particular, a new proof of the integral representation of Bernstein functions of many variables…
There are many analytic functions $U(t)$ satisfying $Z(t)=2\Re\bigl\{ e^{i\vartheta(t)}U(t)\bigr\}$. Here, we consider an entire function $\mathop{\mathcal L}(s)$ such that $U(t)=\mathop{\mathcal L}(\frac12+it)$ is one of the simplest among…
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
Using a concept of filter we propose one generalization of Riemann integral, that is integration with respect to filter. We study this problem, demonstrate different properties and phenomena of filter integration.