Small values of $| L^\prime/L(1,χ) |$

Abstract

In this paper, we investigate the quantity m q := min χ χ 0 |L /L(1, χ)|, as q → ∞ over the primes, where L(s, χ) is the Dirichlet L-function attached to a non trivial Dirichlet character modulo q. Our main result shows that m q log log q/ log q. We also compute m q for every odd prime q up to 10 7 . As a consequence we numerically verified that for every odd prime q, 3 ≤ q ≤ 10 7 , we have c 1 /q < m q < 5/ √ q, with c 1 = 21/200. In particular, this shows that L (1, χ) 0 for every non trivial Dirichlet character χ mod q where 3 ≤ q ≤ 10 7 is prime, answering a question of Gun, Murty and Rath in this range. We also provide some statistics and scatter plots regarding the m q -values, see Section 6. The programs used and the computational results described here are available at the following web address: http://www.math.unipd.it/~languasc/smallvalues.html.