## Bagchi’s Theorem

Bagchi’s Theorem is a functional version of earlier results of Bohr and Jessen related to the statistical properties of the Riemann zeta function on a vertical line between the critical line and the region of absolute convergence. It seems that it is not as well-known as it could, partly because Bagchi proved it in his thesis, and did not publish a paper with this result (his only related paper explicitly states that he removed the probabilistic language that a referee did not like). It seems therefore useful to describe the result. I will then sketch the proof I gave last semester

Consider an open disc $D$ contained in the region $1/2<\mathrm{Re}(s)< 1$ (other compact regions may be considered, for instance an open rectangle). For any real number $t$, we can look at the function $\zeta_t\colon s\mapsto \zeta(s+it)$ on $D$. This is a holomorphic function on $D$, continuous on the closed disc $\bar{D}$. What kind of functions arise this way? Bagchi proved the following (this is essentially Theorem 3.4.11 in his thesis):

Theorem. Let $H$ denote the Banach space of holomorphic functions on $D$ which are continuous on the closed disc. For $T>0$, define a probability measure $\mu_T$ on $H$ to be the law of the random variable $t\mapsto \zeta_t$, where $t$ is uniformly distributed on $[-T,T]$. Then $\mu_T$ converges in law, as $T\to +\infty$, to the random holomorphic function
$Z(s)=\prod_{p}(1-X_pp^{-s})^{-1}$,
where $(X_p)$ is a sequence of independent random variables indexed by primes, all uniformly distributed on the unit circle.

This is relatively easy to motivate: if we could use the Euler product
$\zeta(s+it)=\prod_p (1-p^{-s-it})^{-1}$
in $D$, then we would be led to an attempt to understand the probabilistic behavior of the sequence $(p^{-it})_p$, viewed as a random variable on $[-T,T]$ with values in the infinite product $\widehat{U}$ of copies of the unit circle indexed by primes. This is a compact topological group, and the easy answer (using the Weyl criterion) is simply that this sequence converges to the Haar measure on $\widehat{U}$. In other words, the random sequence $(p^{-it})$ converges in law to a sequence $(X_p)$ of independent, uniform, random variables on the unit circle. Then it is natural to expect that $Z_t$ should converge to the random function $Z(s)$, which is obtained formally by replacing $(p^{-it})$ by its limit $(X_p)$.

Bagchi’s proof is somewhat intricate, in comparison with this heuristic justification, especially if one notices that if $D$ is replaced by a compact region in the domain of absolute convergence, then the same idea applies, and is a completely rigorous proof (one need only observe that the assignment of an Euler product
$\prod_p (1-x_pp^{-s})^{-it}$
to a sequence $(x_p)$ of complex numbers of modulus one is a continuous operation in the region of absolute convergence.)

The proof I give in my script tries to remain closer to the basic intuition, and is indeed less involved (it avoids both a use of the pointwise ergodic theorem that Bagchi required and any use of tightness or weak-compactness). It makes it easy to see exactly what arithmetic ingredients are needed, beyond the convergence in law of $(p^{-it})_p$ to the Haar measure on $\widehat{U}$. Roughly speaking, it goes as follows:

1. One checks that the random Euler product $Z(s)$ does exist (as an $H$-valued random variable), and that it has the Dirichlet series expansion
$Z(s)=\sum_{n\geq 1} X_nn^{-s}$
converging for $\mathrm{Re}(s)> 1/2$ almost surely, where $(X_n)_{n\geq 1}$ is defined as the totally multiplicative extension of $(X_p).$ This is done as Bagchi did using fairly standard probability theory and elementary facts about Dirichlet series.
2. One shows that $Z(s)$ has polynomial growth on vertical lines for $\mathrm{Re}(s)> 1/2$. This is again mostly elementary probability with a bit of Dirichlet series theory.
3. Consider next smoothed partial sums of $Z(s)$, of the type
$Z^{(N)}(s)=\sum_{n\geq 1}X_n\varphi(n/N)n^{-s},$
where $\varphi$ is a compactly supported test function with $\varphi(0)=1$. Using again standard techniques (including Cauchy’s formula for holomorphic functions), one proves that
$\mathbf{E}(\sup_{s\in D}|Z(s)-Z^{(N)}(s)|)\ll N^{-\delta}$
for some $\delta>0$.
4. One next shows that the smoothed partial sums of the zeta function
$\zeta^{(N)}(s)=\sum_{n\geq 1}\varphi(n/N)n^{-s}$
satisfy
$\mathbf{E}_T(\sup_{s\in D}|\zeta(s+it)-\zeta^{(N)}(s+it)|)\ll N^{-\delta}+NT^{-1}$
(the second term arises because of the pole), where $\mathbf{E}_T(\cdot)$ denotes the expectation with respect to the uniform measure on $[-T,T]$. This step is also in Bagchi’s proof, and is essentially the only place where a specific property of the Riemann zeta function is needed: one requires the boundedness on average of $\zeta(s)$ in vertical strips to the right of the critical line. The standard proof of this uses the Cauchy inequality and the mean-value property
$\frac{1}{2T}\int_{-T}^T|\zeta(\sigma+it)|^2dt\to \zeta(2\sigma)$
for any fixed $\sigma$ with $\sigma> 1/2$. It is here that the bottleneck lies if one wishes to generalize Bagchi’s Theorem to any “reasonable” family of $L$-functions.
5. Finally, we just use the definition of convergence in law: for any continuous bounded function $f\colon H\to\mathbf{C}$, we should prove that
$\mathbf{E}_T(f(\zeta_T))\to \mathbf{E}(f(Z)),$
where $\zeta_T$ is the $H$-valued random variable giving the translates of $\zeta(s)$, and $Z$ is the random Dirichlet series. The minor tweak that is useful to notice (and that I wasn’t consciously aware of before) is that one may assume that $f$ is Lipschitz: there exists a constant $C$ such that
$|f(g_1)-f(g_2)|\leq C\sup_{s\in D}|g_1(s)-g_2(s)|$
(this is hidden in standard references — e.g., Billingsley’s — in the proof that one may assume that $f$ is uniformly continuous; the functions used to prove this are in fact Lipshitz…).

Now pick some parameter $N>0$, and write
$|\mathbf{E}_T(f(\zeta_T))-\mathbf{E}(f(Z))|\leq A_1+A_2+A_3$,
where
$A_1=|\mathbf{E}_T(f(\zeta_T))\to \mathbf{E}_T(f(\zeta_T^{(N)}))|\leq C\ \mathbf{E}_T(\sup_{s\in D}|\zeta(s+it)-\zeta^{(N)}(s+it)|),$
$A_2=|\mathbf{E}_T(f(\zeta_T^{(N)}))\to \mathbf{E}(f(Z^{(N)}))|,$
$A_3=|\mathbf{E}(f(Z^{(N)}))\to \mathbf{E}(f(Z))|\leq C\ \mathbf{E}(\sup_{s\in D}|Z(s)-Z^{(N)}(s)|).$
Fix $\varepsilon>0$. For some fixed $N=N_0$ big enough, $A_3$ is less than $\varepsilon$ by Step 3, and $A_1$ is at most $\varepsilon+N_0T^{-1}$. For this fixed $N_0$, $A_2$ tends to $0$ as $T$ tends to infinity because of the convergence in law of $(p^{-it})$ to $(X_p)$ — the sum defining the truncations are finite, so there is no convergence issue. So for all $T$ large enough, we will get
$|\mathbf{E}_T(f(\zeta_T))\to \mathbf{E}(f(Z))|\leq 4\varepsilon.$

## Number Theory Days 2016

As usual, with Spring comes the annual Number Theory Days of EPFL and ETHZ, this time in Zürich during the week-end of April 15 and 16. The website is online, and the poster should be ready very soon (I will update the post when it is…)

The meeting is organized by the Forschungsinstitut für Mathematik, and (again as usual!) there is a certain amount of funding for local expenses made available by FIM for young researchers (graduate students and postdocs). Please register on the FIM web page before March 21 if you are interested!

## Probabilistic number theory

During the fall semester, besides the Linear Algebra course for incoming Mathematics and Physics students, I was teaching a small course on Probabilistic Number Theory, or more precisely on a few aspects of probabilistic number theory that I find especially enjoyable.

Roughly speaking, I concentrated on a small number of examples of statements of convergence in law of sequences of arithmetically-defined probability measures: (1) the Erdös-Kac Theorem; (2) the Bohr-Jessen-Bagchi distribution theorems for the Riemann zeta function on the right of the critical line ; (3) Selberg’s Theorem on the normal behavior of the Riemann zeta function on the critical line ; (4) my own work with W. Sawin on Kloosterman paths. My goal in each case was to do with as little arithmetic, and as much probability, as possible. The reason for this is that I wanted to get and give as clear a picture as possible of which arithmetic facts are involved in each case.

This led to some strange and maybe pedantic-looking discussions at the beginning. But overall I think this was a useful point of view. I think it went especially well in the discussion of Bagchi’s Theorem, which concerns the “functional” limit theorem for vertical translates of the Riemann zeta function in the critical strip, but to the right of the critical line. The proof I gave is, I think, simpler and better motivated than those I have seen (all sources I know follow Bagchi very closely). In the section on Selberg’s Theorem, I used the recent proof found by Radziwill and Soundararajan, which is very elegant (although I didn’t have much time to cover full details during the class).

I have started writing notes for the course, which can be found here. Note that, as usual, the notes are potentially full of mistakes! But those parts which are written are fairly complete, both from the arithmetic and probabilistic point of view.

## Ventotene

I just came back from a week on the Italian island of Ventotene, where I participated in a very nice conference on “Manifolds and groups”. This is of course not my usual topic, and besides giving a minicourse on expanders and coverings (with a focus on geometric applications of expanders having to do with coverings…), I learnt a lot of interesting things. I particularly enjoyed the other two minicourses, by T. Gelander on invariant random subgroups and by R. Sauer on Lücks’s Approximation Theorem. (For both of these, as well as for a number of other talks, the slides are available on the conference web page; my own — handwritten — notes on my course should appear there soon, after I scan them).

Being an island, Ventotene is reached by boat. One of the interesting things about the trip to Ventotene was to observe how birds

would follow us until a very definite point, and then suddenly disappear.

Also, the instructions to launch a lifeboat are rather daunting…

The organizers had scheduled a lot of free time during the conference besides the scientific programme. I was therefore able to take a few pictures, such as some of the local lizards,

and some of the wonderful local cats.

The end of the week also coincided with the beginning of the ten-day long celebration of the island’s patron saint, Santa Candida. Among the festivities were the evening launches

(over a week) of huge hot-air balloons (“Mongolfieri”), with some fireworks

(note the amusing effects of my camera’s fireworks scene setting without tripod…). Some Galois-theoretic persiflage was also notable…

Update The scans of my lectures can be found here.