A generalisation of an identity of Lehmer

Sanoli Gun¹ ; Ekata Saha² ; Sneh Bala Sinha³

¹ Institute of Mathematical Sciences C.I.T. Campus, Taramani Chennai, 600 113, India
² The Institute of Mathematical Sciences C.I.T. Campus, Taramani Chennai, 600 113, India
³ Harish-Chandra Research Institute, Jhunsi 211019 Allahabad, India

Acta Arithmetica, Tome 173 (2016) no. 2, pp. 121-131

Résumé

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).

Détail

DOI : 10.4064/aa8087-2-2016

Keywords: prove identity involving generalised euler briggs constants euler constant linear forms logarithms algebraic numbers generalises gives alternative proof identity lehmer further identity facilitates investigation conjectural transcendental nature generalised euler briggs constants earlier investigations similar type present authors involved interplay between additive multiplicative characters turn rendered inevitable careful analysis multiplicatively independent units suitable cyclotomic fields generalised lehmer identity derived here avoids leading natural transparent proofs earlier results allows prove stronger result see corollary nbsp