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Thursday 31 Oct 2019NT Seminar: Distribution of Inverses and Moments of Dirichlet $L$-functions

Michael Yiasemides - University of Exeter

H103 14:30-16:30

 Let $p$ be a prime and $A_1 , A_2 \subseteq \mathbb{F}_p^*$ be intervals. We define $f(A_1 A_2 , 1) :=\lvert \{(a_1 , a_2 ) \in A_1 \times A_2 : a_1 a_2 \equiv 1 (\modulus p) \} \rvert$. That is, $f(A_1 A_2 , 1)$ is the number of elements in $A_1$ whose inverses lie in $A_2$. We can generalize this to $f(A_1 , \ldots , A_n , r) :=\lvert \{(a_1 , \ldots , a_n ) \in A_1 \times \ldots \times A_n : a_1 \ldots a_n \equiv r (\modulus p) \} \rvert$, for any intervals $A_1 , \ldots , A_n \subseteq \mathbb{F}_p^*$ and $r \in \mathbb{F}_p^*$. I will explain some results on the distribution of $f$, demonstrate the relationship to moments of Dirichlet $L$-functions, and state some problems for future research.

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