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On Gauss’ formula for \(\psi\) and finite expressions for the \(L\)-series at 1. (English) Zbl 1211.11097

The paper under review is motivated by the work of D. H. Lehmer [“Euler constants for arithmetic progressions”, Acta Arith. 27, 125–142 (1975; Zbl 0302.12003)], where he reproves a remarkable expression for \(\psi(p/q)=\frac{d}{dx}\log(\Gamma(x))|_{x=p/q}\) with \(1\leq p<q\), due to Gauss. Authors prove that Gauss’ formula for \(\psi(p/q)\) is equivalent to well-known finite expressions for \(L(1,\chi)\), and meanwhile, they give some expressions for the Lehmer’s number-theoretic function \(N(q)\) defined by \(\log N(q)=-q\sum_{d|q}(\mu(d)\log d)/d\).

MSC:

11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
11R29 Class numbers, class groups, discriminants
33B15 Gamma, beta and polygamma functions
11R11 Quadratic extensions

Citations:

Zbl 0302.12003
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Full Text: DOI

References:

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