ACTA ARITHMETICA

ACTA ARITHMETICA * (200*) Approximate formulae for L(1, ?), II by Olivier Ramare (Lille) 1. Introduction and results. Upper bounds of |L(1, ?)| are mainly useful in number theory to study class numbers of algebraic extensions. In [1]–[3] Louboutin establishes bounds for |L(1, ?)| that take into account the behavior of ? at small primes. His method uses special representations of L(1, ?) and does not extend to odd characters. For instance in [2] he uses L(1, ?) = 2∑n ∑ l≤n ?(l)/(n(n + 1)(n + 2)) which comes from an integra-tion by parts; such a formula fails in the odd case. But the effect of this integration by parts is in fact similar to the introduction of a smoothing, something we did in [5], the only difficulty being to handle properly the Fourier transform of functions behaving like 1/t near ∞. This method gives good numerical results in a uniform way. In this note we improve on the results given in [2] and [3] and extend them to the odd character case. Let us mention that we take this opportunity to correct several typos occurring in [5].

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