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nt.quantity concept – The Riemann zeta duty and differential operators

I’ve revisited an ancient put up of mine–Dirac’s Delta Functions and Riemann’s Jump Function J(x) for the Primes–dealing with Riemann’s “jump” or “staircase” duty (aka, Π(x)) that has unit steps for every prime alongside the horizontal axis and smaller steps of measurement $1/n$ for an $n$-th energy of a primary. This duty is derived as an integral of the inverse Mellin remodel of $log(zeta(1-s))$. The inverse Mellin remodel can too breathe realized as a differential operator appearing on a delta duty:

$$log[zeta(1+xD_x)] delta(x-1)=sum_p sum_{n>0}frac{1}{n} delta(x-p^n),$$

the place the sum is over the primes $p$ and $D_x=d/dx.$

Another occasion during which values of the Riemann zeta emerge in a differential operator is offered within the MO-Q “Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus”

$$R_x = -log(x)-Psi(1+xD_x) = -log(x) +gamma + sum_{n=1}^{infty } (-1)^nzeta (n+1)(xD_x)^n,$$

the place $gamma$ is the Euler-Mascheroni ceaseless and $Psi$ the digamma duty. $R_x$ is an infinitesimal generator for a category of fractional calculus operators; i.e.,

$$ e^{beta R_x} frac{x^{alpha}}{alpha!}= D^beta frac{x^{alpha}}{alpha!}= frac{x^{alpha-beta}}{(alpha-beta)!}.$$

In addition, with a the change of variable $x=e^z$, it turns into the elevating operator for an Appell sequence of polynomials $p_n(z)$ associated to gamma lessons as proven within the MO-Q, which too has the generator

$$frac{1}{D_z!}z^n =expleft [-gamma D_z -sum_{k=2}^{infty } frac{zeta (k)D_z^k}{k} right ]z^n = p_n(z).$$

Another instance of the happening of the zeta duty, disguised because the Bernoulli numbers $b_n$ is a generator of the Bernoulli polynomials $B_n(x)$, which might breathe associated to differentiation: umbrally,

$$T ; x^n = frac{D_x}{e^{D_x} -1} x^n = exp(b.D_x)x^n= (b.+x)^n = B_n(x).$$

[Edit Jan 11, 2021: This is the Todd operator, whose relation to combinatorics, volumes of polytopes, summation/trace formulas, and the calculus in general is indicated with references in answers and comments to this MO-Q and this one. Also used by Hirzebruch to construct the Todd (characteristic) class.]Values for the zeta duty on the traverse integers, the Bernoullis once more, too pop up in elevating operators $R$ for a number of Appell Sheffer polynomial sequences $p_n(x)$ outlined by

$$R p_n(x) = p_{n+1}(x).$$

They too answer an evolution equation

$$frac{d}{dt}exp[tp.(x)]= R exp[tp.(x)],$$

the place

$$exp[tp.(x)]=e^{tp.(0)}e^{xt}$$

is the umbral illustration of the e.g.f. of the Appell sequence. Umbrally, $(p.(x))^n=p_n(x).$

- The elevating op for the reversed countenance polynomials of the simplices normalized by an integer,

$$p_n(x)=frac{(x+1)^{n+1}-x^{n+1}}{n+1}$$

(OEIS A074909, a truncated Pascal triangle),

$R = x – exp[-tfrac{b_{n+1}}{n+1}D] = x – exp[zeta(-n)D].$

- The elevating op for the Bernoulli polynomials

$R = x + exp[-tfrac{b_{n+1}}{n+1}D] = x + exp[zeta(-n)D].$

(These polynomials and those above are an inverse pair beneath umbral composition, which suggests their scowl triangular coefficient matrices are too an inverse pair. This property holds for Appell sequences whose elevating ops disagree solely by the one badge as above.)

- The elevating op for the integral, normalized Euler polynomials A081733

$$R = x – frac{2}{e^{-2D}+1}$$

with

$$frac{2}{e^{2t}+1} = 2 sum_{n g.e. 0} eta(-n) (-2t)^n/n!,$$

the place $eta(s)$ is the Dirichlet eta duty, and

$$2(-2)^n eta(-n) = (-1)^n [2^{n+1}-4^{n+1}] zeta(-n) = [2^{n+1}-4^{n+1}] tfrac{b_{n+1}}{n+1}.$$

- Its umbral compositional inverse is A119468 with

$$R = x + frac{2}{e^{-2D}+1}.$$

This generates a clique of polynomials which when multiplied by 2 provides basically the countenance polynomials of the hypercubes A038207, which is the sq. of the Pascal triangle.

- The elevating op for the Swiss-knife polynomials A119879, from which the Bernoulli, Gennochi, Euler, tangent, and Springer numbers can breathe computed.

$$R = x + tanh(D).$$

$tanh(x)$ is the e.g.f. of the zag numbers A000182,

$$zag(n) = 2^{2n} (2^{2n} – 1) frac{|b_{2n}|}{2n}=2^{2n} (2^{2} – 1)|zeta(-2n+1)|.$$

Note that the matrix of coefficients is a signed, masked Pascal triangle.

- Its umbral inverse is A119467, the identical masked Pascal triangle unsigned, with

$$R = x – tanh(D).$$

- (added 1/11/21) Dragovich presents operators of the figure

$$zeta(asquare+b),$$

the place $sq.= partial^2_t + triangledown^2$ is the D-dimensional d’Alembert operator, in “From p-Adic to Zeta Strings” and “p-Adic Mathematical Physics: the First 50 Years” together with Khrenikov, Kozyrev, Volovich, and Zelenov.

**Question**: **In what different differential operators does the Riemann zeta duty toy an vital function?**

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