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
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}
    \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,
    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.

Jacques Ménard, author of Nicolas Bourbaki

My punning title about James Maynard must have given me somewhere the undeserved reputation of a Borges specialist, since I’ve just received a curious reworking of the story of Pierre Ménard.

The email address from which it came ( jlb@limbo.ow ) is probably not genuine, so I wonder who the author could be (the final note “Translated, from the Spanish, by H.A.H” is of course suggestive, but one would then like to see the original Spanish…)

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!

Is MathOverflow insane?

Since my post contra MathOverflow, already five years ago, I’ve continued watching the site and enjoying many of its mathematical discussions, and seeing myself evolve a bit concerning some of my critical opinions. However, I read today with amazement the discussion that evolved from a question of Richard Stanley on the topic of gravitational waves. I applaud the question, the answer and (among the voices of reason) the comments of Lucia.

The negative comments embody the perfect distillation of the perverse puritanical hair-splitting competition known as “Is this question a good fit for MO?” (to be read in a slightly hysterical voice) that is now what I find most annoying on the site. This is not what mathematics (not even “research” mathematics, that seems to replace here the “pure” mathematics illusion of yesteryears) is about for me. I must confess to finding particularly annoying that some of the most vocal critics (e.g., the pseudonymous “quid”) seem to be people with little actual mathematical contributions and too much time to spend and to write for ever and ever on the finer points of etiquette of a web site as if it were some platonic object to protect from all interlopers.

What would Arnold think of this discussion, where “mathematicians” throw away much (he would say “most”) of the whole history, motivation and insights of their science? Would a question of Kolmogorov on what the brain looks like as graph have passed through the fourches caudines of Signor Quid?