The Riemann zeta function and differential operators

I've revisited an old post of mine--Dirac's Delta Functions and Riemann's Jump Function J(x) for the Primes--dealing with Riemann's "jump" or "staircase" function (aka, Π(x)) that has unit steps for each prime along the horizontal axis and smaller steps of size $$1/n$$ for an $$n$$-th power of a prime. This function is derived as an integral of the inverse Mellin transform of $$\log(\zeta(1-s))$$. The inverse Mellin transform can also be realized as a differential operator acting on a delta function:

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

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

Another instance in which values of the Riemann zeta appear in a differential operator is presented in the MO-Q "Riemann zeta function at positive integers and an Appell sequence of polynomials related to fractional calculus"

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

where $$\gamma$$ is the Euler-Mascheroni constant and $$\Psi$$ the digamma function. $$R_x$$ is an infinitesimal generator for a class 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 becomes the raising operator for an Appell sequence of polynomials $$p_n(z)$$ related to gamma classes as shown in the MO-Q, which also has the generator

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

Another example of the ocurrence of the zeta function, disguised as the Bernoulli numbers $$b_n$$ is a generator of the Bernoulli polynomials $$B_n(x)$$, which can be related to differentiation: umbrally,

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

Values for the zeta function at the negative integers, the Bernoullis again, also pop up in raising operators $$R$$ for several Appell Sheffer polynomial sequences $$p_n(x)$$ defined by

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

They also satisfy an evolution equation

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

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

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

1) The raising op for the reversed face 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),

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

2) The raising op for the Bernoulli polynomials

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

(These polynomials and the ones above are an inverse pair under umbral composition, which means their lower triangular coefficient matrices are also an inverse pair. This property holds for Appell sequences whose raising ops differ only by the one sign as above.)

3) The raising op for the integral, normalized Euler polynomials A081733

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

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

where $$\eta(s)$$ is the Dirichlet eta function, and

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

4) Its umbral compositional inverse is A119468 with

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

This generates a set of polynomials which when multiplied by 2 gives essentially the face polynomials of the hypercubes A038207, which is the square of the Pascal triangle.

5) The raising op for the Swiss-knife polynomials A119879, from which the Bernoulli, Gennochi, Euler, tangent, and Springer numbers can be 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.

6) Its umbral inverse is A119467, the same masked Pascal triangle unsigned, with

$$R = x - \tanh(D).$$

Question: In what other differential operators does the Riemann zeta function play an important role?

• You meant $\exp(-D_x\log n) \delta(x) = \sum_{k=0}^\infty \frac{(-\log n)^k}{k!} \delta^{(k)}(x)=\delta(x-\log n)$ in the sense of analytic functionals, with a convergent Dirichlet series $F(s)=\sum_{n=1}^\infty a_n n^{-s}$ then $F(D_x)\delta = \sum_{n=1}^\infty a_n \exp(-D_x\log n)\delta$ converges in the sense of analytic functionals acting on bounded analytic functions and it is $= \sum_{n=1}^\infty a_n \delta(x-\log n)$ which is the inverse Laplace transform of $F(s)$ in the sense of distributions. – reuns Sep 17 '19 at 20:13
• I@reuns I mean exactly what I derive in my blog post that I linked to. The pdf file does present Dirichlet series as well, but they aren't central to the discussion per se. If there is a spot in the analysis you don't follow, please point it out. The Mellin transform and the diff op derivations are pretty straightforwardly based on manipulations of Euler's product formula. – Tom Copeland Sep 17 '19 at 22:59
• It is uncorrect – reuns Sep 17 '19 at 23:12