On dd-dimensional cycles and the vanishing of simplicial homology

In this paper we introduce the notion of a $d$-dimensional cycle which is a homological generalization of the idea of a graph cycle to higher dimensions. We examine both the combinatorial and homological properties of this structure and use these results to describe the relationship between the combinatorial structure of a simplicial complex and its simplicial homology. In particular, we show that over any field of characteristic 2 the existence of non-zero $d$-dimensional homology corresponds exactly to the presence of a $d$-dimensional cycle in the simplicial complex. We also show that $d$-dimensional cycles which are orientable give rise to non-zero simplicical homology over any field.Comment: Substantially revised and expande

A new derivation of Hermite's integral for the Hurwitz zeta function

We obtain another proof of Hermite's integral for the Hurwitz zeta function

A Ramanujan enigma involving the first Stieltjes constant

We provide a rigorous formulation of Entry 17(v) in Ramanujan's Notebooks and show how this relates to the first Stieltjes constant. A new proposition 4.5 is included to show the close relationship with some analysis presented by Candelpergher in 2017

Some possible approaches to the Riemann Hypothesis via the Li/Keiper constants

In this paper we consider some possible approaches to the proof of the Riemann Hypothesis using the Li criterion

A formula connecting the Bernoulli numbers with the Stieltjes constants

We show that the formula recently derived by Coffey for the Stieltjes constants in terms of the Bernoulli numbers is mathematically equivalent to the much earlier representation derived by Briggs and Chowla

Euler-Hurwitz series and non-linear Euler sums

In this paper we derive two expressions for the Hurwitz zeta function involving the complete Bell polynomials in the restricted case where q is a positive integer greater than 1. The arguments of the complete Bell polynomials comprise the generalised harmonic number functions. These in turn give rise to Euler-Hurwitz series which may then be used to determine identities for combinations of both linear and non-linear Euler sums

A summation by Gencev

Gencev has recently reported a closed form summation for an infinite series involving the harmonic numbers and the central binomial numbers. This note indicates a possible approach to the proof involving the dilogarithm function.Comment: Contains typographical correction

On the generalized Bernoulli numbers

We derive an expression for the generalized Bernoulli numbers in terms of the Bernoulli numbers involving the (exponential) complete Bell polynomials

Elementary evaluations of some Euler sums

This short note contains elementary evaluations of some Euler sums

Determination of the Stieltjes constants at rational arguments

The Stieltjes constants have attracted considerable attention in recent years and a number of authors, including the present one, have considered various ways in which these constants may be evaluated. The primary purpose of this paper is to belatedly highlight the fact that Deninger actually ascertained the first generalised Stieltjes constant at rational arguments as long ago as 1984 and that all of the higher constants (at rational arguments) were determined in principle by Chakraborty, Kanemitsu and Kuzumaki in 2009. Equivalent results were obtained by Musser in his 2011 thesis. The authors of the above papers simply referred to the constants as the Laurent coefficients which explains why various electronic searches conducted by this author for Stieltjes constants did not readily highlight these particular sources. In this paper the author has employed a slightly different argument to obtain a simpler expression for the results originally derived by Chakraborty et al. in 2009.Comment: Sign errors corrected in (6.6) and (6.9