Mellin convolution for generalized Hadamard product of functions/power series

(Rough draft . WordPress editor gone flaky. )

From representations of umbral substitution by differential operators, this note derives an inverse Mellin transform and its associated Mellin convolution giving the Hadamard product of two functions or power series.

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Jumpin’ Riemann!…..!..!.!.Mangoldt–da mon–got it!….!..!

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

(Originally published in Sept. 2019. Inadvertanly deleted in April)

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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Hacking Reality [Official Film]

Fun, well-done, elementary intro to E8 in physics (and dynamics and propaganda of the physics/math/Internet community). A variation on the motif Shadows of Simplicity.

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Witt-Lie algebra, Associated Groups, and the Right and Left Generalized Factorials

With D = d/dx, the Witt vectors -x^{m+1}D when exponentiated give rise to the action

\exp[-t x^{m+1}D] f(x) = f[h_m(x;t)] = f[x(1+mtx^m)^{-1/m}]

for m \geq -1. (Cf. the MathOverflow question Motivation of the Virasoro algebra.)

The function h_m(x;t) represents a Lie group under composition with respect to the parameter t. That is, with roots always chosen as positive and real for x in a suitably small neighborhood of the origin,

h_m(h_m(x;s);t) = h_m(x;s+t)

and the compositional inverse of h_m(x;t) is simply

h^{(-1)}_m(x;t) = h_m(x;-t).

The function h_m(x;t) is also an exponential generating function for the generalized left and right factorials (for positive and negative integral m) presented in OEIS A094638 (mod sign, index shifts, and an additional initial 1 in some cases).

For example,

h_2(x;t) = x(1+2tx^2)^{-1/2} = x - t x^3 + 3 \frac{t^2}{2!}x^5 - 15 \frac{t^3}{3!} x^7 + 105 \frac{t^4}{4!} x^9 + \cdots ,

and the sequence 1,-1,3,-15,105, ... . is signed A001147, which I will call the signed right double factorial with an additional initial 1 (cf. A094638). (The numerators and denominators of the reduced fractions are A098597 and A046161, apparently.)

The compositional inverse is

h_2(x;-t) = x(1-2tx^2)^{-1/2} = x + t x^3 + 3 \frac{t^2}{2!}x^5 + 15 \frac{t^3}{3!} x^7 + 105 \frac{t^4}{4!} x^9 + \cdots,

and the sequence 1,1,3,15,105, ... . is A001147.

We can say the augmented right double factorial as represented in this group is skew invariant under compositional inversion. It is also quasi-invariant under multiplicative inversion with

x/h_2(x;t) = (1 + 2tx^2)^{1/2} = 1 + t x^2 - \frac{t^2}{2!}x^4 + 3 \frac{t^3}{3!} x^6 - 15 \frac{t^4}{4!} x^8 + \cdots,

generating 1,1,-1,3,-15, … .

In contrast, for m = 3, we have

h_3(x;t) = x(1+3tx^3)^{-1/3} = x - t x^4 + 4 \frac{t^2}{2!}x^7 - 28 \frac{t^3}{3!} x^{10} + 280 \frac{t^4}{4!} x^{13} - \cdots,

generating 1,-1,4,-28,280, ...., signed A007559, the signed, right triple factorial augmented with an initial 1.

The shifted reciprocal gives

x/h_3(x;t) = (1 +3 t x^3)^{1/3} = 1 + t x^3 - 2 \frac{t^2}{2!}x^6 + 10 \frac{t^3}{3!} x^9 - 80 \frac{t^4}{4!} x^{12} + 880 \frac{t^5}{5!} x^{15} - \cdots ,

generating 1,1,-2,10,-80,880, .... , the signed left triple factorials A008544 augmented with an initial 1.

Note that the compositional inverse pairs can be related to dual families of trees and also the multiplicative inverse pairs, according to the OEIS entries. See also the MO-Q Combinatorial interpretation of series reversion coefficients and the Gessel link therein, which discusses both multiplicative and compositional inversion. The two types of inversions are also related in general to Hopf algebras/monoids and Koszul duality.

The compositional inverse of h_m(x;t) can be computed from the coefficients of x/h_m(x;t) using the compositional inversion formula A134264 related to free cumulants in free probability theory, non-crossing partitions, and Dyck paths, among other combinatorial constructs. This algorithm allows a transformation of right factorials into left factorials, as does A133437

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More on Formal Group Laws, Binomial Sheffer Sequences, and Linearization Coefficients

A formula for computing the structure, or linearization, constants for reducing products of pairs of polynomials of a binomial Sheffer sequence, p_n(t), is presented in terms of the umbral compositional inverses of the polynomials, \bar{p}_n(t). To say the pair are umbral inverses means

p_n[\bar{p}.(a.)] = (a.)^n = a_n = \bar{p}_n[p.(a.)],

which is equivalent to the lower-triangular coefficient matrices being an inverse pair.

This is used to give a simple computation for the first order structure constant of the pair, which determines the associated coefficients of the associated formal group law, defined by

FGL_f(x,y) = f[f^{((-1)}(x)+f^{(-1)}(y)].

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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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