arXiv subjects

Pawan Singh Mehta

Publications and source records attributed to Pawan Singh Mehta.

4 recordsLinked to original sources

arXiv2026

A note on the multiple zeta functions and their variants at identical arguments

In this article, we study the multiple zeta functions (MZF) and some of its variants at identical arguments. Using the harmonic product, these functions can be expressed as polynomials in the Riemann zeta function. Firstly, we note that an explicit description of the coefficients of these polynomials can be given in terms of the values of the complete Bell polynomials. This for instance, easily leads to the complete description of the singularities of these functions. But more importantly, this enables us to establish a functional relation between MZF, Multiple $t$-functions (M$t$F) and their star variants at identical arguments.

math.NT↗

arXiv2026

Laurent type expansion of multiple polylogarithms at integer points

In this article, we study the local behaviour of the multiple polylogarithm functions at integer points, in the $s$-aspect. This is done by writing a Laurent type expansion at integer points, involving certain power series and rational functions. The coefficients of these power series are the regularised values of the multiple polylogarithm functions at certain related integer points.

math.NT↗

arXiv2026

Ramaswami Type translation formulae for the polylogarithm functions

In 1934, Ramaswami proved a number of curious translation formulae satisfied by the Riemann zeta function. Such translation formulae, in turn give the meromorphic extension of the Riemann zeta function. In 1954, Apostol extended those identities to establish a family of such similar translation formulae. In this article, we establish many such Ramaswami and Apostol type translation formulae for the Dirichlet series defining the polylogarithm functions. This extended set up has many interesting applications, for example, it allows us to also find some (seemingly new) recurrence relations between the Bernoulli numbers, and use them to deduce some congruence properties of the tangent numbers.

math.NT↗

arXiv2025

Multiple polylogarithms, a regularisation process and an admissible open domain of convergence

In this article, we study the analytic properties of the multiple polylogarithms in the $s$-aspect. Although the domain of absolute convergence of the series defining the multiple polylogarithms is well-known, the study towards a larger open domain of (conditional) convergence has been limited, particularly when the depth is $\ge 2$. Here, we exhibit a larger open domain of (conditional) convergence for this series by writing certain translation formulas satisfied by them. The series moreover defines a holomorphic function in this open set. We then introduce a regularisation process for the multiple polylogarithms, extending an earlier work of the second author. This regularisation process requires a generalisation of the Euler-Boole summation formula that we derive in the appendix of this article. The regularisation process leads to a larger open domain, where the series (conditionally) converges at integer points. The holomorphicity at such points is a more delicate question and this regularisation process is to be used to study the local behaviour of the multiple polylogarithms around such points.

math.NT↗