Linear independence of digamma function and a variant of a conjecture of Rohrlich

Abstract

Let ψ(x) denote the digamma function. We study the linear independence of ψ(x) at rational arguments over algebraic number fields. We also formulate a variant of a conjecture of Rohrlich concerning linear independence of the log gamma function at rational arguments and report on some progress. We relate these conjectures to non-vanishing of certain L-series.

Special values of derivatives of $L$-series and generalized Stieltjes constants

“…In this paper, we utilize this link to throw light on the arithmetic nature of L ′ (1, f ) and certain Stieltjes constants. In particular, if p is an odd prime greater than 7, then we deduce the transcendence of at least (p − 7)/2 of the generalized Stieltjes constants, {γ1(a, p) : 1 ≤ a < p}, conditional on a conjecture of S. Gun, M. Ram Murty and P. Rath [8].…”

“…The nature of values of the gamma function at rational arguments and relations among them has been the subject of research for a long time. In light of this, a conjecture put forth by S. Gun, M. Ram Murty and P. Rath [8] will be useful towards a partial solution to our question. The conjecture is the following.…”

“…The motivation for studying these constants emanates from our desire to understand special values of L-series. More precisely, when f is an arithmetical function, with period q, the Dirichlet series L(s, f ) := ∞ n=1 f (n) n s , has been the focus of intense study (see for example the survey article by Tijdeman [17] as well as [4], [8] and [15]). However, these papers studied the special value L(1, f ) whenever it is defined.…”

“…has been the focus of intense study (see for example the survey article by Tijdeman [17] as well as [4], [8] and [15]). However, these papers studied the special value L(1, f ) whenever it is defined.…”

Explicit and asymptotic formulae for Vasyunin-cotangent sums

“…The geometry of ( ) is studied in [14,24]. In the next corollary we write log as a linear combination of ( ) for 1 − 1.…”