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## Acta Arithmetica

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

### Tom 166 / 2014

Acta Arithmetica 166 (2014), 189-200 MSC: Primary 11M06; Secondary 11M26. DOI: 10.4064/aa166-2-4

#### Streszczenie

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.

#### Autorzy

• Yu. MatiyasevichSteklov Institute of Mathematics
St. Petersburg Department
(POMI RAN)
27, Fontanka
St. Petersburg, 191023, Russia
e-mail
• F. SaidakDepartment of Mathematics
University of North Carolina
Greensboro, NC 27402, U.S.A.
e-mail
• P. ZvengrowskiDepartment of Mathematics and Statistics
University of Calgary 