Functiones et Approximatio Commentarii Mathematici

On the value distribution of two Dirichlet $\boldsymbol L$-functions

Niko Laaksonen and Yiannis N. Petridis

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Let $\rho$ denote the non-trivial zeros of the Riemann zeta function. We study the relative value distribution of $L(\rho+\sigma,\chi_{1})$ and $L(\rho+\sigma,\chi_{2})$, where $\sigma\in[0,1/2)$ is fixed and $\chi_{1}$, $\chi_{2}$ are two fixed Dirichlet characters to distinct prime moduli. For $\sigma>0$ we prove that a positive proportion of these pairs of values are linearly independent over $\mathbb{R}$, which implies that the arguments of the values are different. For $\sigma=0$ we show that, up to height $T$, the values are different for $cT$ of the Riemann zeros for some positive constant $c$.

Article information

Funct. Approx. Comment. Math., Volume 58, Number 1 (2018), 43-68.

First available in Project Euclid: 5 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses

Dirichlet $L$-function value-distribution


Laaksonen, Niko; Petridis, Yiannis N. On the value distribution of two Dirichlet $\boldsymbol L$-functions. Funct. Approx. Comment. Math. 58 (2018), no. 1, 43--68. doi:10.7169/facm/1640.

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