Abstract
This paper deals with the matter of the non-vanishing of Dirichlet L-functions at the central point for all primitive characters χ. More precisely, S. Chowla conjectured that \(L(\frac{1}{2},\chi)\not =0\), but this remains still unproved. We first give an efficient algorithm to compute the order n χ of zero of L(s,χ) at \(s=\frac{1}{2}\). This enables us to efficiently compute n χ for L-functions with very large conductor near 1016. Then, we prove that \(L(\frac{1}{2},\chi)\not =0\) for all real characters χ of modulus less than 1010. Finally we give some estimates for n χ and the lowest zero of L(s,χ) on the critical line in terms of the conductor q.
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Omar, S. (2008). Non-vanishing of Dirichlet L-functions at the Central Point. In: van der Poorten, A.J., Stein, A. (eds) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol 5011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79456-1_30
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DOI: https://doi.org/10.1007/978-3-540-79456-1_30
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