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.