Speaker: Ade Irma Suriajaya

Venue: 203 Wentian Building, Weihai Campus

Time: May 8th, 15:00-15:50

Title: Zeros of the derivatives of the Riemann zeta function and Dirichlet L-functions

Abstract: Speiser in 1935 showed that the Riemann hypothesis is equivalent to the first derivative of the Riemann zeta function having no zeros on the left-half of the critical strip. This result shows that the distribution of zeros of the Riemann zeta function is related to that of its derivatives. The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates under the truth of the Riemann hypothesis. This result is further improved by Ge. In the first half of this talk, we introduce these results and generalize the result of Akatsuka to higher-order derivatives of the Riemann zeta function.

 Analogous to the case of the Riemann zeta function, the number of zeros and many other properties of zeros of the derivatives of Dirichlet L-functions associated with primitive Dirichlet characters were studied by Yildirim. In the second-half of this talk, we improve some results shown by Yildirim for the first derivative and show some new results. We also introduce two improved estimates on the distribution of zeros obtained under the truth of the generalized Riemann hypothesis. We also extend the result of Ge to these Dirichlet L-functions when the associated modulo is not small. Finally, we introduce an equivalence condition analogous to that of Speiser’s for the generalized Riemann hypothesis, stated in terms of the distribution of zeros of the first derivative of Dirichlet L-functions associated with primitive Dirichlet characters.