arXiv subjects

B. Candelpergher

Publications and source records attributed to B. Candelpergher.

4 recordsLinked to original sources

arXiv2025

A new expansion of the Riemann zeta function

This article presents polynomial expansions for the Dirichlet eta function and Riemann zeta function that are convergent in the critical strip. To do this we introduce a family of hypergeometric polynomials, whose roots lie on the line $\{\Re(s)=1/2\}$, and that are related to Meixner-Pollaczek polynomials. We also obtain orthonormal expansions for eta and zeta restricted to the line $\{\Re(s)=1/2\}$. The coefficients of these expansions are given explicitly as linear combinations with rational coefficients of $\log(2),$ Euler's constant $\gamma$, and zeta values at positive integers.

math.NT↗

arXiv2018

Expansions of the Riemann Zeta function in the critical strip

We use expansions with functions related to some special functions such as Hermite or Laguerre to get some conjectural expansions of the Riemann Zeta function in the critical strip involving a set of polynomials which have their zeros on the line Re(s)=1/2..

math.NT↗

arXiv2009

Ramanujan Summation and the Exponential Generating Function $ \sum_{k=0}^{\infty}\frac{z^{k}}{k!}ζ^{\prime}(-k)$

In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from which follows a formula that connects the derivatives of the Riemann zeta - function at the negative integers to the Ramanujan summation of the divergent Euler sums $\sum_{n\ge 1} n^kH_n, k \ge 0$, where $H_n= H_n^{(1)}$. Further, we express our results on the Ramanujan summation in terms of the classical summation process called the Borel sum.

math.NT↗

arXiv2000

On a fate of hot quantum field theories

It is argued that for hot quantum fields, the necessary effective perturbation theories may be based on a resummation procedure which, contrarily to the zero temperature case, differs substantially from the one ordinarily in use. Important differences show up in the infrared sector of hot quantum field theories.

hep-ph↗