In Apostol's "Introduction to Analytic Number Theory", there is a definition for the Riemann Zeta Function for $0<s<1$.
Is there a justification for this formula?
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2analytic continuation, obviously – terrace Apr 13 '20 at 22:36
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1Yes; you can use Riemann-Stieltjes integration for the summation and obtain some cancellation. What is left will converge. – Clayton Apr 13 '20 at 22:39