UPPER BOUNDS FOR |L(1, χ)|

Given a non-principal Dirichlet character χ (mod q), an important problem in number theory is to obtain good estimates for the size of L(1, χ). The best bounds known give that q−ǫ ≪ǫ |L(1, χ)| ≪ log q, while assuming the Generalized Riemann Hypothesis, J.E. Littlewood showed that 1/ log log q ≪ |L(1, χ)| ≪ log log q. Littlewood’s result reflects the true range of the size of |L(1, χ)| as it is known that there exist characters χ± for which L(1, χ+) ≍ log log q and L(1, χ−) ≍ 1/ log log q. In this paper we focus on sharpening the upper bounds known for |L(1, χ)|; in particular, we wish to determine constants c (as small as possible) for which the bound |L(1, χ)| ≤ (c+o(1)) log q holds. To set this in context, observe that if X is such that ∑