Linear independence of digamma function and a variant of a conjecture of Rohrlich
The first result of this paper deals with the digamma function \(\psi(z)=\Gamma'(z)/\Gamma(z)\). Let \(q\), \(r\) be two coprime integers \(>1\) and \(K\) be a number field over which both the \(q\)th and \(r\)th cyclotomic polynomials are irreducible. Then at least one of the two following sets of real numbers \[ \bigl\{ \psi(a/q)\; ;\; 1\leq a\leq q,\; (a,q)=1\bigr\} \] \[ \bigl\{ \psi(b/r)\; ;\; 1\leq b\leq r,\; (b,r)=1\bigr\} \] is linearly independent over \(K\). The other statements deal with the \(\mathbb Q\)-vector space (resp. the \(\overline{\mathbb Q}\)-vector space) \(V_\Gamma(q)\) (resp. \(\overline{V}_\Gamma(q)\)) spanned by the numbers \(\log\Gamma(a/q)\) with \( 1\leq a\leq q\), \((a,q)=1\). The authors conjecture that for \(q\in\mathbb Z_{>1}\), the dimension \(d(q)\) of \(V_\Gamma(q)\) is \(\varphi(q)\), where \(\varphi\) is Euler totient function. They notice that \(d(3)=\varphi(3)=2\), \(d(4)=\varphi(4)=2\), \(d(6)=\varphi(6)=2\), and \(d(5)\geq 3\). They prove that \(d(q)\geq \varphi(q)\) when \(q\) is a power of a prime. They also consider the dimension \(\overline{d}(q)\) of \(\overline{V}_\Gamma(q)\); when \(q\) and \(r\) are powers of distinct primes, then one at least of the two inequalities \(\overline{d}(q)\geq \varphi(q)/2\), \(\overline{d}(r)\geq \varphi(r)/2\) is true. Under Schanuel's Conjecture, they deduce the lower bound \(\overline{d}(q)\geq \varphi(q)/2\) when \(q\) is a prime power. This paper contains also a number of related remarks dealing with values of \(L\)--functions.
scientific article; zbMATH DE number 5569027
scientific article