In this paper, we shall prove in Theorem 1 that Gauss’ famous closed formula for the values of the digamma function at rational arguments is equivalent to the well-known finite expression for the $L(1,\chi)$ , which in turn gives rise to the finite expression for the class number of quadratic fields. We shall also prove several equivalent expressions for the arithmetic function $N(q)$ introduced by Lehmer and reveal the relationships among them.
Publié le : 2008-01-15
Classification:
Gauss formula for the digamma function,
Dirichlet class number formula,
Hurwitz zeta-function,
Lehmer’s arithmetic function,
orthogonality of characters,
11R29,
33B15,
11R11
@article{1206367961,
author = {HASHIMOTO, Masahiro and KANEMITSU, Shigeru and TODA, Masayuki},
title = {On Gauss' formula for $\psi$ and finite expressions for the $L$ -series at 1},
journal = {J. Math. Soc. Japan},
volume = {60},
number = {1},
year = {2008},
pages = { 219-236},
language = {en},
url = {http://dml.mathdoc.fr/item/1206367961}
}
HASHIMOTO, Masahiro; KANEMITSU, Shigeru; TODA, Masayuki. On Gauss’ formula for $\psi$ and finite expressions for the $L$ -series at 1. J. Math. Soc. Japan, Tome 60 (2008) no. 1, pp. 219-236. http://gdmltest.u-ga.fr/item/1206367961/