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 contained in the region (other compact regions may be considered, for instance an open rectangle). For any real number , we can look at the function on . This is a holomorphic function on , continuous on the closed disc . What kind of functions arise this way? Bagchi proved the following (this is essentially Theorem 3.4.11 in his thesis):
Theorem. Let denote the Banach space of holomorphic functions on which are continuous on the closed disc. For , define a probability measure on to be the law of the random variable , where is uniformly distributed on . Then converges in law, as , to the random holomorphic function
where 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
in , then we would be led to an attempt to understand the probabilistic behavior of the sequence , viewed as a random variable on with values in the infinite product 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 . In other words, the random sequence converges in law to a sequence of independent, uniform, random variables on the unit circle. Then it is natural to expect that should converge to the random function , which is obtained formally by replacing by its limit .
Bagchi’s proof is somewhat intricate, in comparison with this heuristic justification, especially if one notices that if 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
to a sequence 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 to the Haar measure on . Roughly speaking, it goes as follows:
- One checks that the random Euler product does exist (as an -valued random variable), and that it has the Dirichlet series expansion
converging for almost surely, where is defined as the totally multiplicative extension of This is done as Bagchi did using fairly standard probability theory and elementary facts about Dirichlet series.
- One shows that has polynomial growth on vertical lines for . This is again mostly elementary probability with a bit of Dirichlet series theory.
Consider next smoothed partial sums of , of the type
where is a compactly supported test function with . Using again standard techniques (including Cauchy’s formula for holomorphic functions), one proves that
for some .
- One next shows that the smoothed partial sums of the zeta function
(the second term arises because of the pole), where denotes the expectation with respect to the uniform measure on . 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 in vertical strips to the right of the critical line. The standard proof of this uses the Cauchy inequality and the mean-value property
for any fixed with . It is here that the bottleneck lies if one wishes to generalize Bagchi’s Theorem to any “reasonable” family of -functions.
- Finally, we just use the definition of convergence in law: for any continuous bounded function , we should prove that
where is the -valued random variable giving the translates of , and 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 is Lipschitz: there exists a constant such that
(this is hidden in standard references — e.g., Billingsley’s — in the proof that one may assume that is uniformly continuous; the functions used to prove this are in fact Lipshitz…).
Now pick some parameter , and write
Fix . For some fixed big enough, is less than by Step 3, and is at most . For this fixed , tends to as tends to infinity because of the convergence in law of to — the sum defining the truncations are finite, so there is no convergence issue. So for all large enough, we will get