On Gauss’ formula for ψ and finite expressions for the L -series at 1

Abstract

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 Lð1;� Þ ,w hich 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 NðqÞ introduced by Lehmer and reveal the relationships among them.