Tokyo mini workshop, Zeta's afternoon
O Zeta, Zeta, wherefore art thou Zeta?
Friday, 23 January 2015
Room 123
Graduate School of Mathematical Sciences, The University of Tokyo
3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan
15:00--15:45 Takashi Nakamura (The University of Tokyo)
Universality and zeros of the derivatives of zeta functions
We show that the derivatives of any polynomial of zeta or L-functions with some conditions has infinitely many complex zeros off the critical line. By using this result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain
spectral zeta-functions and the Euler-Zagier multiple zeta-functions, and their derivatives have infinitely many complex zeros off the critical line. Moreover, we show the non-existence of the universality for the derivatives of the Riemann zeta function in the left half side of the critical strip.
16:00--16:50 Yoonbok Lee (Incheon National University)
Simple zeros of primitive Dirichlet L-functions and the asymptotic large sieve (Joint work with Chandee, Liu and Radziwill)
Assuming the Generalized Riemann Hypothesis (GRH), we show using the asymptotic large sieve that 91\% of the zeros of primitive Dirichlet
$L$-functions are simple. This improves on earlier work of \"{O}zl\"{u}k which gives a proportion of at most 86\%. We further compute $q$-analogue of the Pair Correlation Function $F(\alpha)$ averaged over all primitive Dirichlet $L$-functions in the range $|\alpha| < 2$. Previously such a result was available only when the average included all the characters $\chi$.
17:05--17:45 Takashi Nakamura (The University of Tokyo)
Real zeros of Hurwitz-Lerch zeta and Hurwitz-Lerch type of Euler-Zagier double zeta functions
Let $0 < a ¥le 1$, $s,z ¥in {¥mathbb{C}}$ and $0 < |z|¥le 1$. Then the Hurwitz-Lerch zeta function is defined by $¥Phi (s,a,z) := ¥sum_{n=0}^¥infty z^n(n+a)^{-s}$ when $¥sigma :=¥Re (s) >1$. We show that the Hurwitz zeta function $¥zeta (¥sigma,a) := ¥Phi (¥sigma,a,1)$ does not vanish for all $0 <¥sigma <1$ if and only if $a ¥ge 1/2$. Moreover, we prove that $¥Phi (¥sigma,a,z) ¥ne 0$ for all $0 <¥sigma <1$ and $0 < a ¥le 1$ when $z ¥ne 1$. Real zeros of Hurwitz-Lerch type of Euler-Zagier double zetafunctions are studied as well.