# On the value distribution of two Dirichlet L-functions over sums of two zeros of the Riemann zeta-function

## Abstract

Let $\rho=&space;\beta&space;+\imath&space;\dot{\gamma&space;}$ and $\rho{}'&space;=&space;\beta&space;{}'+\imath\dot{}&space;{y}'$denote the non-trivial zeros of the Riemann zeta-function $\zeta&space;\left&space;(&space;s\right&space;)$ with $\gamma&space;,{\gamma&space;}'$ and $\sigma&space;\epsilon&space;\left&space;(&space;1/2,1&space;\right&space;)$. Under the assumption of the Riemann Hypothesis, we show that for a positive proportion of $1/2+\dot{\imath&space;}\left&space;(&space;\gamma&space;+{\gamma&space;}'&space;\right&space;)$, the values of the Dirichlet $L$-function  $L\left&space;(&space;\sigma&space;+\dot{\imath&space;}&space;\left&space;(&space;\gamma&space;+{\gamma&space;}'&space;\right&space;),\chi&space;_{1}\right&space;)$ and $L\left&space;(&space;\sigma&space;+\dot{\imath&space;}\left&space;(&space;\gamma&space;+{\gamma&space;}'&space;\right&space;),\chi&space;_{2}&space;\right&space;)$ associated with the primitive characters $\chi&space;_{1}$ and $\chi&space;_{2}$ to different prime moduli are linearly independent over $\mathbb{R}$.

## Article Details

How to Cite
1.
Maugmai N, Srichan T. On the value distribution of two Dirichlet L-functions over sums of two zeros of the Riemann zeta-function. Prog Appl Sci Tech. [Internet]. 2020 Jun. 18 [cited 2024 Aug. 11];10(1):136-47. Available from: https://ph02.tci-thaijo.org/index.php/past/article/view/242914
Section
Mathematics and Applied Statistics

## References

Fujii A. An additive theory of the zeros of the Riemann zeta function. Rikkyo Daigaku sugaku zasshi. 1996. 45 : 49-116.

Fujii A. On the zeros of Dirichlet L-functions. V. Acta Arith. 1976. 28: 395-403.

Laaksonen N. and Petridis Y. N. On the value distribution of two Dirichlet L-functions. Functiones et Approximatio Commentarii Mathematici. 2018. 58 : 43-68.

Lavrik A.F. An approximate functional equation for Dirichlet L-functions. Trans. Moscow Math. Soc. 1859. 18 : 101-115.

Riemann B. Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Ges. Math. Werke und Wissenschaftlicher Nachlaß. 1968. 2 : 145-155.

Steuding J. Value-distribution of L-functions. Springer, Berlin. 2007.

Titchmarsh E.C., Titchmarsh E.C.T. and Heath-Brown D.R. The theory of the Riemann zeta- function. Oxford University Press. 1986.