Values of symmetric polynomials and a truncated analogue of the Riemann zeta function

TL;DR

This paper characterizes symmetric functions satisfying certain prime modulus congruences related to Bernoulli numbers, contingent on a conjecture, and explores a truncated analogue of the Riemann zeta function and its relations.

Contribution

It provides a conditional classification of symmetric functions fulfilling specific prime-related congruences and investigates a new truncated zeta function analogue and its relations.

Findings

Identifies symmetric functions satisfying congruences modulo p^n

Introduces a truncated analogue of the Riemann zeta function

Analyzes relations among values of the truncated zeta function

Abstract

For each positive integer n, we determine the set of symmetric functions f for which the congruence f(p/1,p/2,...,p/(p-1)) \equiv 0 mod p^n holds for all sufficiently large primes p. Our determination is conditional on a conjecture regarding the modulo p independence of Bernoulli numbers. In a recent work the author introduced a new truncated analogue of the multiple zeta function and investigated a class of relations among values of this function at positive integers. The question answered in the present work is equivalent to the determination of the relations satisfied by values of the corresponding analogue of the ordinary Riemann zeta function.