#### Translator’s note

*This page is a translation into English of the following:*

Balazard, M., Saias, E., and Yor, M. “Notes sur la fonction \zeta de Riemann, 2.” *Advances in Mathematics* **143** (1999), 284–287.

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `94f6dce`

# Introduction

We denote by \sum_{\Re\rho>1/2} a sum over the possible zeros of \zeta(s) with real part greater than \frac12, where the zeros of multiplicity m are counted m times.
The goal of this note is the proof of the following result.

We have
\frac{1}{2\pi}\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s|^2}|\operatorname{d}\!s|
= \sum_{\Re\rho>1/2} \log\left|\frac{\rho}{1-\rho}\right|.
\tag{1}
In particular, the Riemann hypothesis is true if and only if
\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s|^2}|\operatorname{d}\!s| = 0.

*Proof*. This proof consists of two steps.

*First step.*
We start by stating some properties satisfied by a generic function f in the Hardy space H^p(\mathbf{D}), where \mathbf{D}=\{z\in\mathbb{C}:|z|<1\}, and p is a positive real number.
We denote by f^* the function defined almost everywhere on the trigonometric circle \partial\mathbf{D}=\{z\in\mathbb{C}:|z|=1\} by f^*(e^{i\theta})=\lim_{r\to1^-}f(re^{i\theta}).
We use the letter z to denote an element of the trigonometric disc \mathbf{D}, and write
s = s(z) = \frac12+\frac{1+z}{2(1-z)} = \frac{1}{1-z}.
This formula defines a conformal representation of the disc \mathbf{D} in the semi-plane \Re(s)>1/2.

By Jensen’s formula (see, for example, [4, Theorem 3.61]), we have, for f(0)\neq0 and r<1,
\frac{1}{2\pi}\int_{-\pi}^\pi \log|f(re^{i\theta})|\operatorname{d}\!\theta
= \log|f(0)| + \sum_{\substack{|\alpha|<r\\f(\alpha)=0}} \log\frac{r}{|a|}
\tag{2}
where, in the sum, the zeros of multiplicity m are counted m times.
Denote by
\exp\left\{
-\int_{-\pi}^\pi \frac{e^{i\theta}+z}{e^{i\theta}-z}\operatorname{d}\!\mu(\theta)
\right\}
the singular interior factor of f.
As r tends to 1, Equation (2) becomes (cf. [2])
\frac{1}{2\pi}\int_{-\pi}^\pi \log|f(re^{i\theta})|\operatorname{d}\!\theta
= \log|f(0)| + \sum_{\substack{|\alpha|<1\\f(\alpha)=0}} \log\frac{1}{|a|} + \int_{-\pi}^\pi\operatorname{d}\!\mu(\theta).
\tag{3}
This formula is a consequence of the factorisation theorem for functions in H^p;
it is stated in [2] for p=1, but also holds for all positive values of p.

*Second step.*
Now consider the function
f(z) = (s-1)\zeta(s)
(where s=1/(1-z)).
The elementary properties of the Riemann \zeta function (see, for example, [5]) allow us to show that, on one hand, f belongs to the Hardy space H^{1/3}(\mathbf{D}), and, on the other hand, that the measure \mu associated to the singular interior factor of f is zero (for this latter point, it suffices to reuse the argument developed by Bercovici and Foias for the interior factor of the functions (\theta-\theta^s)\zeta(s)(s+1/2)/s, found in the proof of [1, Proposition 2.1]).
We can equally show that
\begin{aligned}
\int_{-\pi}^\pi \log|f^*(e^{i\theta})|\operatorname{d}\!\theta
&= \int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s|^2}|\operatorname{d}\!s|,
\\\log|f(0)|
&= 0,
\\\sum_{\substack{|\alpha|<1\\f(\alpha)=0}} \log\frac{1}{|\alpha|}
&= \sum_{\Re\rho>1/2} \log\left|\frac{\rho}{1-\rho}\right|.
\end{aligned}
With all this information, our result follows from Equation (3).

We finish with some remarks.
There are statements related to ours in the works [6,7] of Wang and Volchkov.
It is even possible that Jensen himself was aware of Equation (1) (the reader can consult the article [3] where Jensen informs Mittag–Leffler of his discovery of Equation (2)).
It seem interesting, however, to present things as we have done here, and this is for the following three reasons:

- Equation (1) is simpler than those that appear in [6,7];
- we show here that, to establish Equation (1), it is natural to place ourselves in the framework of Hardy spaces;
- the form of the integral in Equation (1) allows us to interpret this result via Brownian motion, as we show below.

Denote by Z=X+iY the planar Brownian motion from 0 (or from 1), and by Z_{T_{1/2}}=\frac12+iY_{T_{1/2}} its first point of impact on the critical line \Re s=1/2, where T_{1/2}:=\inf\{t:X_t=1/2\}.
We know that Y_{T_{1/2}} follows a Cauchy law with parameter 1/2.
In other words, the law of Y_{T_{1/2}} has density 1/2\pi(1/4+t^2).
Thus the second part of the theorem can be stated in the following manner: the Riemann hypothesis is true if and only if
\mathbb{E}[\log\vert\zeta(Z_{T_{1/2}})] = 0.

## Thanks

We thank Luis Báez-Duarte, Michel Delasneri, Catherine Donati, Laurent Habsieger, Aleksandar Ivić, and Alain Plagne for useful conversations.