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My question is about connectedness of the Riemann zeta function and the theory of Bergman spaces. Because I'm working in this area of Bergman spaces, and I had Analytic number theory as a subject on my graduate studies, I would like to know more about applications of the theory of Bergman spaces to Analytic number theory. By Bergman space I mean a space of all holomorphic functions on some domain that are $L^{p}$ there, i.e. the theory of Bergman spaces from the book of Kehe Zhu. Is there any equivalent of Riemann hypothesis in the theory of Bergman spaces? I would like to know more about the intersection between the Analytic number theory and the theory of Bergman spaces(Kehe Zhu).

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  • $\begingroup$ Do you have any reason to believe that such a connection exists? $\endgroup$ – Qiaochu Yuan Jan 4 '14 at 20:46
  • $\begingroup$ There is a lot of common tools used in both areas. For example: Jensen's formula, distributions of zeros etc...In this paper journals.tubitak.gov.tr/math/issues/mat-01-25-4/… authors used Hardy spaces. I would like to see something like this with Bergman spaces. $\endgroup$ – Alem Jan 5 '14 at 5:25
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I am not aware of such an "Riemann" conjecture in the theory of Bergman spaces. However, a result which lies perhaps "in the intersection" of both areas is Voronin's Universality Theorem, about the universality of zeta-functions, i.e., the property of zeta-functions (and Dirichlet $L$-functions) to approximate arbitrary non-vanishing holomorphic functions arbitrarily well. The proof uses theory of Bergman spaces, see S.M. Voronin, "Theorem on the Universality of the Riemann Zeta Function." Izv. Akad. Nauk SSSR, Ser. Matem. 39, (1975) pp.475-486.

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I think you should be interested in the following criterion for the Riemann Hypothesis: The Riemann Hypothesis is true if and only if there exists a sequence of Dirichlet polynomials $A_N(s) := \sum_{n \leq N} a(n) n^{-s}$ such that $$ \lim_{N \rightarrow \infty} \int_{-\infty}^{\infty} \frac{|1 - A_N(1/2 + it)\zeta(1/2 + it)|^2}{|1/2 + it|^2} dt \rightarrow 0 $$ Surely the function $(1-A_N(s)\zeta(s)/s$ is $L^2$ for example for $\Re s$ in a small vicinity of $1/2$. For references you should check the Neyman-Beurling criterion for the Riemann Hypothesis, and papers by authors such as Baez-Duarte, Balazard, Saias, Landreau, Roton, Burnol, etc. These authors do not approach the Riemann Hypothesis through the theory of Bergmann spaces but rather through the theory of Hardy spaces.

It is quite possible that the above statement, with $L^p$ norms in place of $L^2$ norms, corresponds to non-vanishing in a half-plane of the form $1/p$ for example. This however needs to be checked.

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