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Selberg's Central Limit Theorem

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Title: Selberg's Central Limit Theorem
Author(s): Mäkelä, Antti
Contributor: University of Helsinki, Faculty of Science
Degree program: Master 's Programme in Mathematics and Statistics
Specialisation: Mathematics and applied mathematics
Language: English
Acceptance year: 2023
This thesis follows a proof for Selberg’s Central Limit Theorem for log |ζ( 1/2 + it)|. The theorem states that the random variable ( 1/2 log log T )^(−1/2) log |ζ( 1/2 +it)| with T ≤ t ≤ 2T converges to N (0, 1) weakly as T → ∞. The proof we follow is by Kannan Soundararajan and Maxym Radziwill. The intention is to expand on the details that their original work leaves for the reader to fill in. Their proof is a four step approximation. The first step shifts the consideration right from the critical line Im(s) = 1/2. The second step is proving that a random variable based on a related Dirichlet polynomial converges weakly to N (0, 1). The third step ties another Dirichlet polynomial to the one from the previous step. The final step is to tie the Dirichlet polynomial from step 3 to the Riemann Zeta. One way to interpret Selberg’s Central Limit Theorem is that extreme ab- solute values of the Riemann Zeta become proportionally rarer when we look further on the critical line. The function does not linger long around its zeros and it does not stay close to its extreme values for long. Most of its values will have an absolute value close to √ (1/2 log log T) .
Keyword(s): Riemann Zeta Function Selberg's Central Limit Theorem

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