Related papers: Differentiating polynomials, and zeta(2)

This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation…

We investigate the distribution of the zeros of partial sums of the Riemann zeta-function, sum_{n\leq X}n^{-s}, estimating the number of zeros up to height T, the number of zeros to the right of a given vertical line, and other aspects of…

This is part II of our examination of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations.

In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative…

Some aspects of the multiplicative anomaly of zeta determinants are investigated. A rather simple approach is adopted and, in particular, the question of zeta function factorization, together with its possible relation with the…

We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1…

This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…

For every $m\in\mathbb{N}$, we establish the convergence of the averaged distributions of the zeros of the $m$-th order derivatives $(f^n)^{(m)}$ of the iterated polynomials $f^n$ of a polynomial $f\in\mathbb{C}[z]$ of degree $>1$ towards…

The aim of this work is an analytic investigation of differential equations producing mirror maps as well as giving new examples of mirror maps; one of these examples is related to (rational approximations to) $\zeta(4)$. We also indicate…

We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann's zeta function.

There exists an infinite series of ratios by which one can derive the Riemann zeta function $\zeta(s)$ from Catalan numbers and central binomial coefficients which appear in the terms of the series. While admittedly the derivation is not…

We investigate the behavior of fractional derivatives of polynomials. In particular, we consider the locations and the asymptotic behaviour of their zeros and give bounds for their Mahler measure.

Commenting on an observation of Prof. Edwards, this note presents a method of evaluation of $\zeta(2n)$ that follows easily from Riemann's own representation of the zeta function.

Using Parseval's identity for the Fourier coefficients of $x^k$, we provide a new proof that $\zeta(2k)=\dfrac{(-1)^{k+1}B_{2k}(2\pi)^{2k}}{2(2k)!}$.

We employ mean value estimates of Weyl sums in order to obtain discrete second moments of the Riemann Zeta-function with respect to polynomials near the vertical line $1+i\mathbb{R}$.

Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a…

Historically known as the Basel problem, evaluating the Riemann zeta function at two has resulted in numerous proofs, many of which have been generalized to compute the function's values at even positive integers. We apply Parseval's…

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

In this paper we study the distribution of the non-trivial zeros of the Riemann zeta-function $\zeta(s)$ (and other L-functions) using Montgomery's pair correlation approach. We use semidefinite programming to improve upon numerous…

We prove an explicit integral formula for computing the product of two shifted Riemann zeta functions everywhere in the complex plane. We show that this formula implies the existence of infinite families of exact exponential sum identities…