EXTREME VALUES OF arg L ( 1 , χ )

Values of Dirichlet L-functions at s = 1 have always attracted great attention due to their central role in number theory. In particular the prime number theorem for arithmetic progressions relies on the non-vanishing of L(1, χ) for any non-principal character χ (mod q). Moreover improving the existing lower bounds for |L(1, χ)| (the famous Siegel’s bound |L(1, χ)| ≫ǫ q) would imply many important consequences. Various authors have studied the distribution of these values (one can refer to [5] for a history of the subject including results for other families of L-functions). In [3] A. Granville and K. Soundararajan studied the distribution of extreme values of |L(1, χ)|, showing that the proportion of characters χ (mod q) for which |L(1, χ)| > eτ equals