Saturday, 15 October 2016

sequences and series - How to evaluate a zero of the Riemann zeta function?

Here is a super naive question from a physicist:




Given the zeros of the Riemann zeta function,
ζ(s)=n=1ns,
how do I actually evaluate them?





On this web page I found a list of zeros. Well I guess if the values, call one azero, are just given in decimal expansion, then I'd have to run a program and see how it approaches zero
ζ(azero)=n=1nazero=11azero+12azero+13azero+0.;(a)>1



But are there also analytical solutions, let's call one bzero, which I can plug in and see ζ(bzero)=0 exactly? Specifically the non-trivial ones with imaginary part are of interest. What I find curious is that these series, with each term being a real number multiplied by some niIm(bzero), then also kind of resemble Fourer series.

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