On the summation of some divergent series of Euler and the zeta functions (Q1924555)

The paper can be seen as a supplement to G. Faber's survey on infinite series in Euler Opera omnia Ser. I Vol. 16.2. -- Attention is paid to work of the Swiss Hermann Kinkelin (1832-1913) who developed in 1858 (one year previous to Riemann's paper) properties of the zeta function, and also of the `Hurwitz' zeta function, which extended contemporary work of Malmsten, Schlömilch and Clausen. Kinkelin derived an equivalent to Riemann's functional equation, and proved an essentially equivalent conjecture of Euler. -- By a variant of his and Maclaurin's formula, Euler had attempted to evaluate \(\Sigma (-1)^{k-1} k^{-s}\) for \(s=-1/2\) and \(s=3/2\). The values, up to 6 decimals, were corrected by Faber. By a summation method due to Charles Hutton (1737-1823) the evaluation is extended here to 10 decimals. Hutton's method is briefly discussed.