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Michael Milgram

Publications and source records attributed to Michael Milgram.

At least 19 recordsLinked to original sources
arXiv2026

Five Parameter Hypergeometric 3F2(1) when One or more Parameters are Integers or Separated by Integers: Derivations, Review, Exotics and More

This work was intended to be all about, and only about, hypergeometric 3F2(1). The initial goal was to revisit many identities from the literature that have been derived over the years and show that they can be obtained in a simpler way armed, with only a minimum of elementary identities. That goal has been achieved as a (patient) reader will discover. In another sense, this work is a partial review of the last half-century's worth of progress in the evaluation of a particular set of 3F2(1), in particular those cases where at least one parameter is an integer or two or more parameters are separated by an integer. The result is a collection of very general identities (or techniques) that an analyst seeking to evaluate a particular 3F2(1), might want to consider as a starting point. That is the secondary goal. Along the way however, the temptation arose to investigate at least one of the unanswered questions that others have raised. This led to a few digressions, and some possibly new results. The reader is invited to follow where curiosity led me to depart from a straightforward review of the state-of-the-art.

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arXiv2024

On a Generalized Moment Integral containing Riemann's Zeta Function: Analysis and Experiment

Here, we study both analytically and numerically, an integral $Z(\sigma,r)$ related to the mean value of a generalized moment of Riemann's zeta function. Analytically, we predict finite, but discontinuous values and verify the prediction numerically, employing a modified form of Ces\`aro summation. Further, it is proven and verified numerically that for certain values of $\sigma$, the derivative function $Z^{\prime}(\sigma,n)$ equates to one generalized tine of the Dirac comb function without recourse to the use of limits, test functions or distributions. A surprising outcome of the numerical study arises from the observation that the proper integral form of the derivative function is quasi-periodic, which in turn suggests a periodicity of the integrand. This possibility is also explored and it is found experimentally that zeta function values offset (shifted) over certain segments of the imaginary complex number line are moderately auto-correlated.

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arXiv2024

An Extension of Glasser's Master Theorem and a Collection of Improper Integrals Many of Which Involve Riemann's Zeta Function

Glasser's Master Theorem arXiv:1308.6361v2 is essentially a restatement of Cauchy's integral Theorem reduced to a specialized form. Here we extend that theorem by introducing two new parameters, but still retain a simple form. Because of wide interest in entities involving Riemann's zeta function, the focus is on the evaluation of improper integrals with almost arbitrary integrands involving that function, but we also consider some other instances, the purpose being to demonstrate the power of the extended theorem. This is achieved by the presentation of a large number of examples that illustrate the ubiquity of the range of possibilities. One simple outcome of the study is the use of the extended theorem to show how it is possible to evaluate an integral when series or other representations of an integrand function do not converge.

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arXiv2023

On the Use of the Mellin Transform to Generate Families of Power, Hyperpower, Lambert and Dirichlet Type Series and Some Consequences

This note is concerned with series of the forms $\sum f(a^n)$ and $\sum f(n^{-a})$ where f(a) possesses a Mellin transform and $a > 1$ or $a<0$ respectively. Integral representations are derived and used to transform these series in several ways yielding a selection of interesting integral evaluations involving Riemann's function $\zeta(s)$, limits and series representations containing hyperpowers. A number of examples of such sums are provided, each of which is investigated for possible new structure. In one case, we obtain a generalization of Riemann's classic relationship among the Zeta, Gamma and Jacobi Theta functions.

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arXiv2023

A Curious Trigonometric Infinite Product in Context

By treating the multiple argument identity of the logarithm of the Gamma function as a functional equation, we obtain a curious infinite product representation of the $sinc$ function in terms of the cotangent function. This result is believed to be new. It is then shown how to convert the infinite product to a finite product, which turns out to be a simple telescoping of the double angle $sin$ function. In general, this result unifies known infinite product identities involving various trigonometric functions when the product term index appears as an exponent. In one unusual case, what appears to be a straightforward limit, suggests a counterexample to Weierstrass' factor theorem. A resolution is offered. An Appendix presents the general solution to a simple functional equation. This work is motivated by its educational interest.

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arXiv2021

Determining the Indeterminate: On the Evaluation of Integrals that connect Riemann's, Hurwitz' and Dirichlet's Zeta, Eta and Beta functions

By applying the inverse Mellin transform to some simple closed form identities, a number of relationships are established that connect integrals containing Riemann's and Hurwitz' zeta functions ($\zeta(s)$ and $\zeta(s,a)$) and their alternating equivalents $\eta(s)$ and $\eta(s,a)$. In particular, special cases involving improper integrals containing $\zeta(\sigma+it)$ and few other functions in the integrand are identified. Many of these integrals that do not appear in the literature, can be, and were, verified numerically. In one limit, the use of analytic continuation generates a family of improper integrals containing only the real and imaginary parts of $\zeta(\sigma+it)$ with and without simple trigonometric factors; the associated closed form contains an (unclassified) entity that has many of the attributes of an essential singularity. Consequently, this means that the associated integrals are indeterminate (i.e. non-single valued), so a new symbol is introduced to label the indeterminism. Much of this paper examines this singularity from several angles in order to resolve the associated ambiguities, before eventually showing how it blends into the classical study of functions of real and complex variables in an unusual manner. This is done by establishing a self-consistent way to remove the singularity and thereby evaluate new members of a family of integrals of general interest that contain $\zeta(s,a)$ and $\eta(s,a)$. Some implications are proposed.

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arXiv2021

Some Additions to a Family of Sums and Integrals related to Hurwitz' Zeta Function(s), Euler polynomials and Euler Numbers

Integrals involving the kernel function $sech (πx)$ over a semi-infinite range are of general interest in the study of Riemann's function $ζ(s)$ and Hurwitz' function $ζ(s,a)$. Such integrals that include the $arctan$ and $log$ functions in the integrand are evaluated here in terms of $ζ(s,a)$, thereby adding some new members to a known family of related integrals. A claimed connection between $ζ(s)$ of odd integer argument and such integrals is verified.

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arXiv2020

A Series Representation for Riemann's Zeta Function and some Interesting Identities that Follow

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $η(s)$, and hence Riemann's function $ζ(s)$, is obtained in terms of the Exponential Integral function $E_{s}(iκ)$ of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions $ζ^{\pm}(s)$ and $η^{\pm}(s)$ are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.

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arXiv2019

An Integral Equation for Riemann's Zeta Function and its Approximate Solution

Two identities extracted from the literature are coupled to obtain an integral equation for Riemann's $ξ(s)$ function, and thus $ζ(s)$ indirectly. The equation has a number of simple properties from which useful derivations flow, the most notable of which relates $ζ(s)$ anywhere in the critical strip to its values on a line anywhere else in the complex plane. From this, I obtain both an analytic expression for $ζ(σ+it)$ everywhere inside the asymptotic ($t\rightarrow\infty)$ critical strip, and an approximate solution, within the confines of which the Riemann Hypothesis is shown to be true. The approximate solution predicts a simple, but strong correlation between the real and imaginary components of $ζ(σ+it)$ for different values of $σ$ and equal values of $t$; this is illustrated in a number of Figures.

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arXiv2018

Variations on a Hypergeometric Theme

The question was asked: Is it possible to express the function \begin{equation} \tag{1.1} h(a)\equiv\,{_4F_3}(a,a,a,a;2a,a+1,a+1;1) \label{question} \end{equation} in closed form? After considerable analysis, the answer appears to be "no", but during the attempt to answer this question, a number of interesting (and unexpected) related results were obtained, either as specialized transformations, or as closed-form expressions for several related functions. The purpose of this paper is to record and review both the methods attempted and the related identities obtained (specifically new $_4F_3(1)$, $_5F_6(1)$ and (generalized Euler) sums containing digamma functions) - the former for their educational merit, since they appear to be not-very-well-known, the latter because they do not appear to exist in the literature.

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arXiv2017

On Some Sums of Digamma and Polygamma functions - Version (2017) and Review

This paper is an enhanced version of a more than decade-older paper with a similar title. Many formulae involving both finite and infinite sums of digamma and polygamma functions up to quadratic order, few of which appear in standard reference works or the literature, but which periodically arise in applications, are collected, reviewed, listed and developed to the point that a knowledgeable reader could devise a formal proof. Several errors in the literature are corrected

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arXiv2016

Further Exploration of Riemann's Functional Equation

A previous exploration of the Riemann functional equation that focussed on the critical line, is extended over the complex plane. Significant results include a simpler derivation of the fundamental equation developed previously, and its generalization from the critical line to the complex plane. A simpler statement of the relationship that exists between the real and imaginary components of $ζ(s)$ and $ζ^{\prime}(s)$ on opposing sides of the critical line is developed, reducing to a simpler statement of the same result on the critical line. An analytic expression is obtained for the sum of the arguments of $ζ(s)$ on opposite sides of the critical line, reducing to the analytic expression for $arg(ζ(1/2+iρ))$ first obtained in the previous work. Relationships are obtained between various combinations of $|ζ(s)|$ and $|ζ^{\prime}(s)|$, particularly on the critical line, and it is demonstrated that $arg(ζ(1/2+iρ))$ and $arg(ζ^{\prime}(1/2+iρ))$ uniquely define $|ζ(1/2+iρ)|$. A comment is made about the utility of such results as they might apply to putative proofs of Riemann's Hypothesis (RH).

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arXiv2016

Exploring Riemann's Functional Equation

An equivalent, but variant form of the Riemann functional equation is explored, and several discoveries are made. Properties of the Riemann zeta function $ζ(s)$ from which a necessary and sufficient condition for the existence of zeros in the critical strip are deduced. This in turn, by an indirect route, eventually produces a simple, solvable, differential equation for $arg(ζ(s))$ on the critical line $s=1/2+iρ$, the consequences of which are explored, and the "LogZeta" function is introduced. A singular linear transform between the real and imaginary components of $ζ$ and $ζ^\prime$ on the critical line is derived, and an implicit relationship for locating a zero ($ρ=ρ_0$) on the critical line is found between the arguments of $ζ(1/2+iρ)$ and $ζ^{\prime}(1/2+iρ)$. Notably, the Volchkov criterion, a Riemann Hypothesis (RH) equivalent is analytically evaluated and verified to be half equivalent to RH, but RH is not proven. Numerical results are presented, some of which lead to the identification of {\it anomalous zeros}, whose existence in turn suggests that well-established, traditional derivations such as the Volchkov criterion and counting theorems require re-examination. It is proven that the derivative $ζ^{\prime}(1/2+iρ)$ will never vanish on the perforated critical line ($ρ\neqρ_0$). Traditional asymptotic and counting results are obtained in an untraditional manner, yielding insight into the nature of $ζ(1/2+iρ)$ as well as very accurate asymptotic estimates for distribution bounds and the density of zeros on the critical line.

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arXiv2015

Uncovering functional relationships at zeros with special reference to Riemann's Zeta Function

A Master equation has been previously obtained which allows the analytic integration of a fairly large family of functions provided that they possess simple properties. Here, the properties of this Master equation are explored, by extending its applicability to a general range of an independent parameter. Examples are given for various values of the parameter using Riemann's Zeta function as a template to demonstrate the utility of the equation. The template is then extended to the derivation of various sum rules among the zeros of the Zeta function as an example of how similar rules can be obtained for other functions.

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arXiv2014

Master Theorems for a Family of Integrals

A family of general Master theorems for analytic integration over the real (or imaginary) axis with various reciprocal hyperbolic (trig) kernels ($\sinh and/or \cosh$) with varying arguments is developed. Several examples involving closed-form integrations which do not appear to exist in the standard tables are given in detail and general results are listed in an Appendix. As well, it is shown how to convert the special case of an infinite series involving ratios of Gamma functions into a finite series.

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arXiv2010

On Hypergeometrics 3F2(1) - A Review

By systematically applying ten well-known and inequivalent two-part relations between hypergeometric sums 3F2(...|1) to the published database of all such sums, 62 new sums are obtained. The existing literature is summarized, and many purportedly novel results extracted from that literature are shown to be special cases of these new sums. The general problem of finding elements contiguous to Watson's, Dixon's and Whipple's theorems is reduced to a simple algorithm suitable for machine computation. Several errors in the literature are corrected or noted. The present paper both summarizes and extends a previous work on this subject.

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arXiv2006

On Hypergeometric 3F2(1)

By systematically applying ten inequivalent two-part relations between hypergeometric sums 3F2(1) to the published database of all such sums, 66 new sums are obtained. Many results extracted from the literature are shown to be special cases of these new sums. In particular, the general problem of finding elements contiguous to Watson's, Dixon's and Whipple's theorem is reduced to a simple algorithm suitable for machine computation. Several errors in the literature are corrected or noted.

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