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Euler constants for arithmetical progressions. (English) Zbl 0302.12003

Author’s introduction: Euler’s constant \(\gamma\) is defined by \[ \gamma=\lim_{n\to\infty}\left\{1+\frac 12+\frac 13+\dots+\frac 1m-\log m\right\}= .5772156649. \] In this paper we study the properties of the corresponding limit \(\gamma(r,k)\) obtained by considering the sum of the reciprocals of the terms of the arithmetic progression \(r, r+k, r+2k, \dots\) \((0<r\leq k)\). In §2, \(\gamma(r,k)\) is defined precisely and shown to exist. In §3 we show that \(\gamma(r,k)\) differs from \(\gamma/k\) by a linear combination of logarithms of cyclotomic integers in the field of \(k\)th roots of unity. From this a formula for \(\gamma(r,k)\) involving only real numbers is deduced and specialized for certain small \(k\). In §4 a study is made of the \(\varphi(k)\) primitive \(\gamma(r,k)\) in which \(r\) and \(k\) are coprime. In §5 the connection is made between \(\gamma(r,k)\) and the logarithmic derivative \(\psi\) of the Gamma function at the point \(r/k\). The results of §3 are now seen to give a really elementary proof of Gauss’ theorem on \(\psi(z)\) for rational \(z\). In §6 we make applications of \(\gamma(r,k)\) to certain infinite series. In particular, we develop the connection between \(\gamma(r,k)\) and the class number of the quadratic fields \(\mathbb Q(\sqrt{\pm k})\). The final §7 contains some comments on the numerical evaluation of \(\gamma(r,k)\).

MSC:

11Y60 Evaluation of number-theoretic constants
11B25 Arithmetic progressions
11R11 Quadratic extensions
11R18 Cyclotomic extensions
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