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.

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I am a professor of mathematics at ETH Zürich since 2008.