Monday, 2 September 2013

number theory - Reformulation of riemann zeta




Does this extend to C?



ζ(x)=01txdt, where for 0t<1 we say that t=1.


Answer



For (x)>1, this integral converges. Further, this integral converges to exactly the regular Riemann zeta function. Thus an analytic continuation of the regular Riemann zeta is an analytic continuation of this function.



[Given the clarification]:



Instead of converging to the regular Riemann zeta function, this zeta function is exactly the regular Riemann zeta function +1 (for the bit between 0 and 1), and thus we still get our continuation from the regular Riemann zeta function.



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