Convergence of Riemann's Product representation of Xi

During his investigation of zeta Riemann defined the $$xi$$ function as $$xi(s):= Gamma(frac{s}{2})(s-1)pi^{-s/2}zeta(s)$$ which is an entire function that is invariant under the substitution $$s to 1-s$$. Moreover $$xi$$ shares its zeros with Riemann zeta function $$zeta$$.

Riemann wanted to write $$xi(s)$$ in the form $$xi(0)prod_{rho}(1-frac{s}{rho})$$ where the product is taken over all zeros of the zeta function. How does one prove the convergence of this product?

MathOverflow Asked by Mustafa Said on February 7, 2021

You need to group the complex conjugates pairs of non-trivial zeros together

$$2sum_{rho} log(1-frac{s}{rho})= 2sum_{Im(rho)le 2|s|} log(1-frac{s}{rho})+ 2sum_{Im(rho)> 2|s|}log(1-frac{s}{rho})+log(1-frac{s}{overline{rho}})$$ $$=2sum_{Im(rho)le 2|s|} log(1-frac{s}{rho})+ 2sum_{Im(rho)> 2|s|}O(frac{|s|+|s^2|}{Im(rho)^2})tag{1}$$ Due to the density of zeros theorem $$# {rho, |Im(rho)| $$(1)$$ converges locally uniformly, with a rate of convergence similar to $$sum_{n>2|s|log |s|} frac{(|s|+|s^2|) log^2 n}{n^2 }$$.

Answered by reuns on February 7, 2021

As a comment says, this is "standard" textbook material nowadays, but I'm guessing you want a more historical perspective from Riemann's point of view.

The general theory of the Hadamard product (for entire functions of order 1) obviously wasn't available, but one really only needs linear exponential factors, so Riemann essentially did this ad hoc.

The relevant part is in the middle of page 139 of .

The translation is given as

If one denotes by $$alpha$$ all the roots of the equation $$xi(alpha)$$ = 0, one can express $$log xi(t)$$ as $$sum_alpha log(1-t^2/alpha^2)+logxi(0)$$ for, since the density of the roots of the quantity $$t$$ grows with $$t$$ only as $$log t/2pi$$, it follows that this expression converges [the important point]... thus it differs from $$logxi(t)$$ by a function of $$t^2$$... This difference is consequently a constant, whose value can be determined through setting $$t = 0$$.

So in other words, I think a fair answer to your question is that Riemann instead considers $$xi(s)=xi(0)prod_rho (1-s^2/rho^2),$$ which softens the analytic difficulties compared to $$xi(s)=prod_rho (1-s/rho)$$.

Answered by user171793 on February 7, 2021

Related Questions

Is there a source in which Demazure’s function $p$ defined in SGA3, exp. XXI, is calculated?

0  Asked on December 15, 2021 by inkspot

Potential p-norm on tuples of operators

1  Asked on December 15, 2021 by chris-ramsey

Understanding a quip from Gian-Carlo Rota

5  Asked on December 13, 2021 by william-stagner

What is Chemlambda? In which ways could it be interesting for a mathematician?

1  Asked on December 13, 2021

Degree inequality of a polynomial map distinguishing hyperplanes

1  Asked on December 13, 2021

The inconsistency of Graham Arithmetics plus $forall n, n < g_{64}$

2  Asked on December 13, 2021 by mirco-a-mannucci

Subgraph induced by negative cycles detected by Bellman-Ford algorithm

0  Asked on December 13, 2021

How to obtain matrix from summation inverse equation

1  Asked on December 13, 2021 by jdoe2

Is $arcsin(1/4) / pi$ irrational?

3  Asked on December 13, 2021 by ikp

A mutliplication for distributive lattices via rowmotion

0  Asked on December 13, 2021

Tensor product of unit and co-unit in a closed compact category

1  Asked on December 13, 2021 by andi-bauer

Finite subgroups of $SL_2(mathbb{C})$ arising as a semi-direct product

1  Asked on December 13, 2021 by user45397

Uniqueness of solutions of Young differential equations

0  Asked on December 13, 2021

Differentiation under the integral sign for a $L^1$-valued function (shape derivative)

0  Asked on December 13, 2021

Digraphs with exactly one Eulerian tour

2  Asked on December 13, 2021

Exactness of completed tensor product of nuclear spaces

1  Asked on December 11, 2021

An integral with respect to the Haar measure on a unitary group

1  Asked on December 11, 2021

Best known upper bound for Dedekind zeta function on line $sigma=1$ in the $t$ aspect

0  Asked on December 11, 2021

Regularity with respect to the Lebesgue measure through dimensions

0  Asked on December 11, 2021 by titouan-vayer

Reference request on Gentzen’s proof of the consistency of PA

1  Asked on December 11, 2021