In this series of seven papers, predominantly by means of elementary analysis, we establish a num... more In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
In this series of seven papers, predominantly by means of elementary analysis, we establish a num... more In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
New proofs of the duplication formulae for the gamma and the Barnes double gamma functions are de... more New proofs of the duplication formulae for the gamma and the Barnes double gamma functions are derived using the Hurwitz zeta function. Concise derivations of Gauss's multiplication theorem for the gamma function and a corresponding one for the double gamma function are also reported. This paper also refers to some connections with the Stieltjes constants.
In this paper we derive two expressions for the Hurwitz zeta function involving the complete Bell... more 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.
In this series of seven papers, predominantly by means of elementary analysis, we establish a num... more In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
This paper considers various integrals where the integrand includes the log gamma function (or it... more This paper considers various integrals where the integrand includes the log gamma function (or its derivative, the digamma function) multiplied by a trigonometric or hyperbolic function. Some apparently new integrals and series are evaluated.
In a rather straightforward manner, we develop the well-known formula for the Stirling numbers of... more In a rather straightforward manner, we develop the well-known formula for the Stirling numbers of the first kind in terms of the (exponential) complete Bell polynomials where the arguments include the generalised harmonic numbers. We also show how the (exponential) complete Bell polynomials feature in a number of other areas of mathematical interest.
A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants is derived ... more A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants is derived in this paper. In addition, we also report a formula for the Stieltjes constants in terms of the higher derivatives of the Riemann zeta function. A formula for the Stieltjes constants in terms of the (exponential) complete Bell polynomials containing the eta constants as the arguments is also derived.
We show that the formula recently derived by Coffey for the Stieltjes constants in terms of the B... more 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.
In this series of seven papers, predominantly by means of elementary analysis, we establish a num... more In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
In this paper we present some applications of the Stieltjes constants including, for example, new... more In this paper we present some applications of the Stieltjes constants including, for example, new derivations of Binet's formulae for the log gamma function and the evaluation of some integrals related to the Barnes multiple gamma functions.
In this paper we consider some possible approaches to the proof of the Riemann Hypothesis using t... more In this paper we consider some possible approaches to the proof of the Riemann Hypothesis using the Li criterion.
In this series of seven papers, predominantly by means of elementary analysis, we establish a num... more In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
In this series of seven papers, predominantly by means of elementary analysis, we establish a num... more In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
New proofs of the duplication formulae for the gamma and the Barnes double gamma functions are de... more New proofs of the duplication formulae for the gamma and the Barnes double gamma functions are derived using the Hurwitz zeta function. Concise derivations of Gauss's multiplication theorem for the gamma function and a corresponding one for the double gamma function are also reported. This paper also refers to some connections with the Stieltjes constants.
In this paper we derive two expressions for the Hurwitz zeta function involving the complete Bell... more 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.
In this series of seven papers, predominantly by means of elementary analysis, we establish a num... more In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
This paper considers various integrals where the integrand includes the log gamma function (or it... more This paper considers various integrals where the integrand includes the log gamma function (or its derivative, the digamma function) multiplied by a trigonometric or hyperbolic function. Some apparently new integrals and series are evaluated.
In a rather straightforward manner, we develop the well-known formula for the Stirling numbers of... more In a rather straightforward manner, we develop the well-known formula for the Stirling numbers of the first kind in terms of the (exponential) complete Bell polynomials where the arguments include the generalised harmonic numbers. We also show how the (exponential) complete Bell polynomials feature in a number of other areas of mathematical interest.
A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants is derived ... more A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants is derived in this paper. In addition, we also report a formula for the Stieltjes constants in terms of the higher derivatives of the Riemann zeta function. A formula for the Stieltjes constants in terms of the (exponential) complete Bell polynomials containing the eta constants as the arguments is also derived.
We show that the formula recently derived by Coffey for the Stieltjes constants in terms of the B... more 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.
In this series of seven papers, predominantly by means of elementary analysis, we establish a num... more In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods.
In this paper we present some applications of the Stieltjes constants including, for example, new... more In this paper we present some applications of the Stieltjes constants including, for example, new derivations of Binet's formulae for the log gamma function and the evaluation of some integrals related to the Barnes multiple gamma functions.
In this paper we consider some possible approaches to the proof of the Riemann Hypothesis using t... more In this paper we consider some possible approaches to the proof of the Riemann Hypothesis using the Li criterion.
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Papers by Donal Connon