## 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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