(Second edition by Chelsea, New York, 1953, p. 160, 933). ζ s − {\displaystyle \Re (s)\geq {\frac {1}{2}}} ∞ → ) n / ≥ 1 = ( 2 s ) 1 {\displaystyle \eta (2n)=(-1)^{n+1}{{B_{2n}\pi ^{2n}(2^{2n-1}-1)} \over {(2n)!}}.}. 0 , valid over the whole complex plane and also proven by Lindelöf. s , at One particularly simple, yet reasonable method is to apply Euler's transformation of alternating series, to obtain. n ) ℜ The first one follows from a change of variable of the integral representation of the Gamma function (Abel, 1823), giving a Mellin transform which can be expressed in different ways as a double integral (Sondow, 2005). 1 Since ( 1 n "This formula, remarquable by its simplicity, can be proven easily with the help of Cauchy's theorem, so important for the summation of series" wrote Jensen (1895). s n Oxford University Press. ) − {\displaystyle \zeta (s_{n})\,} 2 − 1 ≠ / {\displaystyle \Re (s)=1} This is valid for + I have tried to write a program to compute the Zeta function for complex arguments. Jahrhundert von Leonhard Euler, der sie im Rahmen des Basler Problems untersuchte. | = Indeed, suppose thisis the case; let . 1 < {\displaystyle 0

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