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Distribution of values of the Riemann zeta function on vertical arithmetic sequences


Hold Date 2019-05-13 15:30~2019-05-13 17:00

Place Seminar Room W1-C-615, West Zone 1, Ito campus, Kyushu University

Object person  

Speaker Ade Irma Suriajaya (Kyushu University)

Abstract :
C. R. Putnam (1954) showed that the set of all zeros of the Riemann zeta function $¥zeta(s)$ on the critical line $¥Re(s)=1/2$ cannot contain any arbitrarily long arithmetic progressions. We were interested in extending this result to any sets of values of $¥zeta(s)$ and to the whole critical strip $0<¥Re(s)<1$. Unfortunately, we could not obtain further extensions of Putnam's result on the critical line, but if we exclude this line in the critical strip, we could obtain a much more general result analogous to that of Putnam's. This result we obtained also reflects the irregular behavior of $¥zeta(s)$ in the critical strip.

The next natural question which comes into mind is how much the distribution of values of $¥zeta(s)$ on arithmetic progressions characterize that of $¥zeta(s)$ on the whole line. A. Reich (1982) shows that values of any general Dirichlet series $F(s) = ¥sum_{n¥geq1} a(n)n^{-s}$ on arithmetic progressions on vertical lines in the region of absolute convergence {¥it can} characterize values of $F(s)$ on the whole line if we take the closure of both sets of values. The question then comes down to whether $¥zeta(s)$ shares values on a vertical arithmetic progression. Surely we cannot expect this to be false without taking the closure, but on the region far to the right of the abscissa of absolute convergence, we could show that $¥zeta(s)$ does not share values on vertical arithmetic progressions. Using universality, we could also show similar results in the right half of the critical strip for a specific sequence of permutations of positive integers.

In this talk, I would like to introduce these results and provide key details of the proof. This is a joint work with Junghun Lee (formerly in Nagoya University, currently on military service), Athanasios Sourmelidis (University of Würzburg) and Jörn Steuding (University of Würzburg).