Explicit zero-free regions and a $\tau$-Li-type criterion
Abstract
$\tau$-Li coefficients describe if a function satisfies the Generalized Riemann Hypothesis or not. In this paper we prove that certain values of the $\tau$-Li coefficients lead to existence or non-existence of certain zeros. The first main result gives explicit numbers $N_1$ and $N_2$ such that if all real parts of the $\tau$-Li coefficients are non-negative for all indices between $N_1$ and $N_2$, then the function has non zeros outside a certain region. According to the second result, if some of the real parts of the $\tau$-Li coefficients are negative for some index $n$ between numbers $n_1$ and $n_2$, then there is at least one zero outside a certain region.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2018
- DOI:
- 10.48550/arXiv.1807.01506
- arXiv:
- arXiv:1807.01506
- Bibcode:
- 2018arXiv180701506P
- Keywords:
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- Mathematics - Number Theory