Horizontal monotonicity of the modulus of the zeta function, $L$-functions, and related functions

Yu. Matiyasevich ¹   ; F. Saidak ²   ; P. Zvengrowski ³

¹ Steklov Institute of Mathematics Russian Academy of Sciences St. Petersburg Department (POMI RAN) 27, Fontanka St. Petersburg, 191023, Russia
² Department of Mathematics University of North Carolina Greensboro, NC 27402, U.S.A.
³ Department of Mathematics and Statistics University of Calgary Calgary, Alberta, Canada T2N 1N4

Acta Arithmetica, Tome 166 (2014) no. 2, pp. 189-200

Résumé

As usual, let $s = \sigma + it$. For any fixed value of $t$ with $|t| \geq 8$ and for $\sigma 0$, we show that $|\zeta(s)|$ is strictly decreasing in $\sigma$, with the same result also holding for the related functions $\xi$ of Riemann and $\eta$ of Euler. The following inequality related to the monotonicity of all three functions is proved: $$ \Re\biggl(\frac {\eta'(s)}{\eta(s)} \biggr) \Re\biggl(\frac {\zeta'(s)}{\zeta(s)}\biggr) \Re\biggl(\frac {\xi'(s)}{\xi(s)} \biggr). $$ It is also shown that extending the above monotonicity result for $|\zeta(s)|$, $|\xi(s)|,$ or $|\eta(s)| $ from $\sigma 0$ to $\sigma 1/2$ is equivalent to the Riemann hypothesis. Similar monotonicity results will be established for all Dirichlet $L$-functions $L(s,\chi)$, where $\chi$ is any primitive Dirichlet character, as well as the corresponding $\xi(s,\chi)$ functions, together with the relation of this to the generalized Riemann hypothesis. Finally, these results will be interpreted in terms of the degree $1$ elements of the Selberg class.

Détail

DOI : 10.4064/aa166-2-4

Keywords: usual sigma fixed value geq sigma zeta strictly decreasing sigma result holding related functions riemann eta euler following inequality related monotonicity three functions proved biggl frac eta eta biggr biggl frac zeta zeta biggr biggl frac biggr shown extending above monotonicity result zeta eta sigma sigma equivalent riemann hypothesis similar monotonicity results established dirichlet l functions chi where chi primitive dirichlet character corresponding chi functions together relation generalized riemann hypothesis finally these results interpreted terms degree elements selberg class