A generalisation of an identity of Lehmer

Abstract

We prove an identity involving generalised Euler–Briggs constants, Euler’s constant, and linear forms in logarithms of algebraic numbers. This generalises and gives an alternative proof of an identity of Lehmer (1975). Further, this identity facilitates the investigation of the (conjectural) transcendental nature of generalised Euler–Briggs constants. Earlier investigations of similar type by the present authors involved the interplay between additive and multiplicative characters. This in turn rendered inevitable a careful analysis of multiplicatively independent units in suitable cyclotomic fields. The generalised Lehmer identity derived here avoids this, leading to natural and transparent proofs of earlier results. It also allows us to prove a stronger result (see Corollary 2).