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Mark W. Coffey

Publications and source records attributed to Mark W. Coffey.

52 records · Page 3Linked to original sources
arXiv2009

The Stieltjes constants, their relation to the eta_j coefficients, and representation of the Hurwitz zeta function

The Stieltjes constants gamma_k(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about its only pole at s=1. We present the relation of gamma_k(1) to the eta_j coefficients that appear in the Laurent expansion of the logarithmic derivative of the Riemann zeta function about its pole at s=1. We obtain novel integral representations of the Stieltjes constants and new decompositions such as S_2(n) = S_gamma(n) + S_Lambda(n) for the crucial oscillatory subsum of the Li criterion for the Riemann hypothesis. The sum S_γ(n) is O(n) and we present various integral representations for it. We present novel series representations of S_2(n). We additionally present a rapidly convergent expression for γ_k= γ_k(1) and a variety of results pertinent to a parameterized representation of the Riemann and Hurwitz zeta functions.

math-ph↗
arXiv2009

Series of zeta values, the Stieltjes constants, and a sum S_γ(n)

We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function zeta(s,a) about s=1. Additionally we obtain series and integral representations of a sum S_γ(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum S_γ(n)+n is an important subsum in application of the Li criterion for the Riemann hypothesis.

math-ph↗
arXiv2008

On representations and differences of Stieltjes coefficients, and other relations

The Stieltjes coefficients $γ_k(a)$ arise in the expansion of the Hurwitz zeta function $ζ(s,a)$ about its single simple pole at $s=1$ and are of fundamental and long-standing importance in analytic number theory and other disciplines. We present an array of exact results for the Stieltjes coefficients, including series representations and summatory relations. Other integral representations provide the difference of Stieltjes coefficients at rational arguments. The presentation serves to link a variety of topics in analysis and special function and special number theory, including logarithmic series, integrals, and the derivatives of the Hurwitz zeta and Dirichlet $L$-functions at special points. The results have a wide range of application, both theoretical and computational.

math-ph↗
arXiv2008

On harmonic binomial series

We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and calculations of theoretical physics, as well as other applications.

math-ph↗
arXiv2008

Alternative evaluation of a ln tan integral arising in quantum field theory

A certain dilogarithmic integral I_7 turns up in a number of contexts including Feynman diagram calculations, volumes of tetrahedra in hyperbolic geometry, knot theory, and conjectured relations in analytic number theory. We provide an alternative explicit evaluation of a parameterized family of integrals containing this particular case. By invoking the Bloch-Wigner form of the dilogarithm function, we produce an equivalent result, giving a third evaluation of I_7. We also alternatively formulate some conjectures which we pose in terms of values of the specific Clausen function Cl_2.

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arXiv2008

On a three-dimensional symmetric Ising tetrahedron, and contributions to the theory of the dilogarithm and Clausen functions

Perturbative quantum field theory for the Ising model at the three-loop level yields a tetrahedral Feynman diagram C(a,b) with masses a and b and four other lines with unit mass. The completely symmetric tetrahedron C^Tet \equiv C(1,1) has been of interest from many points of view, with several representations and conjectures having been given in the literature. We prove a conjectured exponentially fast convergent sum for C(1,1), as well as a previously empirical relation for C(1,1) as a remarkable difference of Clausen function values. Our presentation includes Propositions extending the theory of the dilogarithm Li_2 and Clausen Cl_2 functions, as well as their relation to other special functions of mathematical physics. The results strengthen connections between Feynman diagram integrals, volumes in hyperbolic space, number theory, and special functions and numbers, specifically including dilogarithms, Clausen function values, and harmonic numbers.

math-ph↗
arXiv2008

Evaluation of a ln tan integral arising in quantum field theory

We analytically evaluate a dilogarithmic integral that is prototypical of volumes of ideal tetrahedra in hyperbolic geometry. We additionally obtain new representations of the Clausen function Cl_2 and the Catalan constant G=Cl_2(π/2), as well as new relations between sine and Clausen function values.

math-ph↗
arXiv2006

Special functions and the Mellin transforms of Laguerre and Hermite functions

We present explicit expressions for the Mellin transforms of Laguerre and Hermite functions in terms of a variety of special functions. We show that many of the properties of the resulting functions, including functional equations and reciprocity laws, are direct consequences of transformation formulae of hypergeometric functions. Interest in these results is reinforced by the fact that polynomial or other factors of the Mellin transforms have zeros only on the critical line Re s = 1/2. We additionally present a simple-zero Proposition for the Mellin transform of the wavefunction of the D-dimensional hydrogenic atom. These results are of interest to several areas including quantum mechanics and analytic number theory.

math-ph↗
arXiv2006

Theta and Riemann xi function representations from harmonic oscillator eigensolutions

From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical zeta function. A key result provides a basis for generalizing the important Riemann-Siegel integral formula.

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arXiv2006

On the coefficients of the Baez-Duarte criterion for the Riemann hypothesis and their extensions

We present analytic properties and extensions of the constants ck appearing in the Baez-Duarte criterion for the Riemann hypothesis. These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal of the Riemann zeta function. We present generalizations of this representation to the Hurwitz zeta and many other special functions. We relate the corresponding coefficients to other known constants including the Stieltjes constants and present summatory relations. In addition, we generalize the Maslanka hypergeometric-like representation for the zeta function in several ways.

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arXiv2006

On hypergeometric series reductions from integral representations, the Kampe de Feriet function, and elsewhere

Single variable hypergeometric functions pFq arise in connection with the power series solution of the Schrodinger equation or in the summation of perturbation expansions in quantum mechanics. For these applications, it is of interest to obtain analytic expressions, and we present the reduction of a number of cases of pFp and p+1F_p, mainly for p=2 and p=3. These and related series have additional applications in quantum and statistical physics and chemistry.

math-ph↗
arXiv2005

Polygamma theory, the Li/Keiper constants, and validity of the Riemann Hypothesis

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. We investigate a related set of constants c_n, n = 1,2,..., showing in detail that the leading behaviour (1/2) ln n of lambda_n/n is absent in c_n. Additional results are presented, including a novel explicit representation of c_n in terms of the Stieltjes constants gamma_j. We conjecture as to the large-n behaviour of c_n. Should this conjecture hold, validity of the Riemann hypothesis would follow.

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arXiv2005

On one dimensional digamma and polygamma series related to the evaluation of Feynman diagrams

We consider summations over digamma and polygamma functions, often with summands of the form (\pm 1)^nψ(n+p/q)/n^r and (\pm 1)^nψ^{(m)} (n+p/q)/n^r, where m, p, q, and r are positive integers. We develop novel general integral representations and present explicit examples. Special cases of the sums reduce to known linear Euler sums. The sums of interest find application in quantum field theory, including evaluation of Feynman amplitudes.

math-ph↗
arXiv2005

Toward verification of the Riemann hypothesis: Application of the Li criterion

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{λ_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we determine new bounds for relevant Riemann zeta function sums and the sequence itself. We find that the Riemann hypothesis holds if certain conjectured properties of a sequence η_j are valid. The constants η_j enter the Laurent expansion of the logarithmic derivative of the zeta function about s=1 and appear to have remarkable characteristics. {\em On our conjecture}, not only does the Riemann hypothesis follow, but an inequality governing the values λ_n and inequalities for the sums of reciprocal powers of the nontrivial zeros of the zeta function.

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