A Diorama of the Digamma

(Under construction)

This series is divergent, so we may be able to do something with it. — Heaviside

The divergent series for the pole of the Riemann zeta function is \zeta(1) = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + ... . Lets’s use Mellin transform interpolation (essentially the master’s (Ramanujan) master formula) to interpolate the harmonic numbers H_n = \frac{1}{1}+ \frac{1}{2}+\frac{1}{3}+ \cdots + \frac{1}{n}, the partial sums of the divergent series, in the hope that we can glean some global numerics of the Riemann zeta. The digamma function and its various avatars will naturally spring forth.

But first these excerpts:

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The Riemann Zeta and the Calculus

(Under construction: Reprising investigations over several years.)

By virtue of the relation between the values of the Riemann zeta function at the negative integers, \zeta(-n<1), and the Bernoulli numbers and between the Bernoulli polynomials and the partial sums of the powers of the natural numbers and derivatives of analytic functions, the Riemann zeta can be related to the integration and differentiation of analytic functions.

Through the relation between the values of the Riemann zeta function at the positive natural numbers greater than one, \zeta(n>1), and a series expansion of the digamma function and between a digamma differential operator and the infinigen (infinitesimal generator) of a fractional calculus, the Riemann zeta can be related to the fractional calculus-the calculus of fractional integral and differential operators acting on real functions analytic on the positive real axis. .

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Jumpin’ Riemann !_ !__ !___ ! !_____ ! Mangoldt–da mon–got it !___!_!

The magic of Mangoldt summoning Riemann’s miraculous miniscules-the nontrivial zeros.

In response to observations initiated by Matt McIrvin of a sum of exponentials of the imaginary part of the non-trivial zeroes of the Riemann zeta function, assuming the Riemann hypothesis is true, as presented on a stream through Mathstackexchange (MSE), Mathoverflow (MO), and the n-Category Cafe. One thread is the MO-Q Quasicrystals and the Riemann Hypothesis posed by John Baez.

The main actors are the Riemann zeta function \zeta(s), the Landau Xi function \xi_L(s) (aka, the Riemann Xi function with the two poles removed), the von Mangoldt function \Lambda(n), the Chebyshev function \psi(x) (aka, the von Mangoldt summatory function), and the Riemann jump function J(x) (aka, the Riemann prime number counting function) with Mellin, Heaviside, and Dirac directing, with a cameo by Fourier.

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Quotes Post:

Ongoing collection of quotes attributed (mod my lapses in memory) to mathematicians and physicists or about their work:

d’Alembert:

Go forward, faith will follow! 

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In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms

In the earlier post Compositional Inverse Operators and Sheffer Sequences, I constructed relations among a generic power series, call it f(x)=x \cdot H(-x), or ordinary generating function (o.g.f.), its compositional inverse f^{(-1)}(x)= [xH(-x)]^{(-1)} and four sets of Sheffer polynomial sequences–two Appell sequences p_n(x) and q_n(x) and two binomial Sheffer sequences u_n(x) and v_n(x) intimately related by

e^{tp.(x)}= \frac{t}{tH(-t)} e^{xt},

e^{tq.(x)}= \frac{[tH(-t)]^{(-1)}}{t} e^{xt},

e^{tu.(x)}= e^{x \cdot tH(-t)},

e^{tv.(x)}= e^{x \cdot [tH(-t)]^{(-1)}}.

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Squaring Triangles

This post illustrates what Feynman praised as a beautiful facet of mathematics–abstraction from the concrete–as well as the fascinating synergy at one of its crossroads–that of algebra and enumerative geometry.

One day last fall in a class, several curious 12-th graders marvelled at the relationship I showed them between the Pascal triangle (OEIS A007318) and the enumerative geometry of triangles and squares and their n-dimensional extensions/abstractions the hypertriangles (HTs) and hypersquares (HSs) (or, equivalently, the tetrahedrons and hypertetrahedrons, and the cubes and hypercubes). By looking at certain physico-geometric ways of generating the n-dimensional extensions, we can relate simple algebraic manipulations–multiplication of polynomials and a matrix by itself–to counting the components of these geometric constructs, enumerated by their face-polynomials.

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Commutators, matrices and an identity of Copeland

The arXiv “Commutators, matrices and an identity of Copeland” by Darij Grinberg proves and extends an identity I proposed for a matrix computation of the partition polynomials generated by iterated multiplication of a tangent vector(g(x)D)^n,where D = d/dx and g(x) is a function or formal series.

Some background and refs are given in the body and comments of the MathOverflow question “Expansions of iterated, or nested, derivatives, or vectors–conjectured matrix computation.” And another proof was added on Oct. 14, 2019.

The exponentiation and resultant partition polynomials are central to unveiling the relationships among compositional inversion via series; the differential geometry of vector fields; solutions of evolution equations, including the inviscid Burgers’ equation and the soliton solution of the KdV equation; the enumeratuve combinatorics of analytic rooted Cayley trees, Dyck paths, the associahedra, dissections of polygons, noncrossing partitions, phylogenetic trees, among other combinatorial geometric constructs; algebraic geometry and certain moduli spaces; generalized permutahedra, Hopf algebra of monoids, and optimization (see Aguiar and Ardila ref) ; free cumulants and their associated moments in free probability; enumerative combinatorics of iterated convolutions associated to the Hirzebruch criterion for the Todd class; Koszul duality and quadratic operads; and pre-Lie algebras and Butcher series (also this post).

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