Oberseminar ZAHLENTHEORIE (online)

Dr. Masahiro Mine (Sophia University, Tokyo)

Probabilistic study of the Hurwitz zeta-function with algebraic parameter

Abstract. Cassels proved in 1961 that the Hurwitz zeta-function has infinitely many zeros for Re(s)>1 if the parameter is algebraic and irrational. A key step in his proof is to show at least 51% of log(n+alpha) are linearly independent over rational numbers. In this talk, I will introduce a certain random variable closely related to the value-distribution of the Hurwitz zeta-function with algebraic parameter. Then I explain how Cassels' method is applied in the study of the Hurwitz zeta-function with this random variable. The main result is that any complex number can be approximated by values of the Hurwitz zeta-function for arbitrary quadratic irrational parameters but with finite exceptions.

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