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 $L(1,\chi )$, 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.