{"id":4762,"date":"2016-04-09T12:36:52","date_gmt":"2016-04-09T10:36:52","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/?p=4762"},"modified":"2024-02-20T13:40:32","modified_gmt":"2024-02-20T11:40:32","slug":"bagchis-theorem","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2016\/04\/09\/bagchis-theorem\/","title":{"rendered":"Bagchi’s Theorem"},"content":{"rendered":"

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<\/a>…<\/p>\n

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


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

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

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

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 $latex (p^{-it})_p$ to the Haar measure on $latex \\widehat{U}$. Roughly speaking, it goes as follows:<\/p>\n

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

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

    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 … Continue reading Bagchi’s Theorem<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[157,2],"tags":[],"class_list":["post-4762","post","type-post","status-publish","format-standard","hentry","category-eth","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/4762","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=4762"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/4762\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=4762"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=4762"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=4762"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}