On the size of $L(1,\chi)$ and S. Chowla's hypothesis implying that $L(1,\chi)>0$ for $s>0$ and for real characters $\chi$

Tom 130 / 2013

S. Louboutin Colloquium Mathematicum 130 (2013), 79-90 MSC: Primary 11M20. DOI: 10.4064/cm130-1-8


We give explicit constants $\kappa$ such that if $\chi$ is a real non-principal Dirichlet character for which $L(1,\chi ) \le\kappa$, then Chowla's hypothesis is not satisfied and we cannot use Chowla's method for proving that $L(s,\chi )>0$ for $s>0$. These constants are larger than the previous ones $\kappa =1-\log 2=0.306\ldots$ and $\kappa =0.367\ldots$ we obtained elsewhere.


  • S. LouboutinInstitut de Mathématiques de Luminy
    UMR 6206
    163, avenue de Luminy
    Case 907
    13288 Marseille Cedex 9, France

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