A trio of Bernoulli relations, their implications for the Ramanujan polynomials and the zeta constants

TL;DR

This paper explores deep connections between Bernoulli relations, Ramanujan polynomials, and zeta constants, revealing new identities and implications for the Riemann Hypothesis and transcendence of zeta at odd integers.

Contribution

It introduces a generalized Ramanujan polynomial derived from Bernoulli relations and explores their implications for zeta functions and related conjectures.

Findings

New Bernoulli relations for zeta functions

Generalized Ramanujan polynomial properties

Insights into the Riemann Hypothesis and zeta transcendence

Abstract

We study the interplay between recurrences for zeta related functions at integer values, `Minor Corner Lattice' Toeplitz determinants and integer composition based sums. Our investigations touch on functional identities due to Ramanujan and Grosswald, the transcendence of the zeta function at odd integer values, the Li Criterion for the Riemann Hypothesis and pseudo-characteristic polynomials for zeta related functions. We begin with a recent result for \zeta(2s) and some seemingly new Bernoulli relations, which we use to obtain a generalised Ramanujan polynomial and properties thereof.