# In general is it even possible to evalutate the values of the zeta function?

So after much confusion on my last question (still unanswered) and reading the comments, I thiiiink I came to realization that: in general you can not just plug some numbers into some neat functions and get values for the zeta function? Incase there is any confusion by that I mean: take a random arbitrary point on the complex plane s, is it easy/possible to compute z(s)? In other words I'm now under the impression that in general they are guessing at the values by computing large partial sums. Is this true? If so what kind of confidence do they have in those guesses? What about the infamous zeros? (do they know the zeros are zeros? Trying to read further I came across the Basel problem apparently it took them the better part of a century to figure out zeta of two, before zeta was even extended to complex numbers etc...

• Let me turn this question around for an easier function. How would you compute the sine of an arbitrary complex number? For instance, how do I know $\sin(1+\mathrm{i}\sqrt{3}) = 2.452{\dots} + \mathrm{i} 1.479{\dots}$? What sort of argument would suffice to accept that equality? Oct 11, 2019 at 7:03

The Riemann-Siegel formula was developed by Riemann and published by Siegel. It expresses the Riemann zeta function as a sum of three parts, two are finite sums (meaning only finitely many terms are added) and the third is usually called "the error term". The method allows the user to select nonnegative integer parameters, call them $$M$$ and $$N$$ (as in the link), and then add up the two sums to get an approximation to the value of $$\zeta$$. (Perhaps we could call this $$\zeta_{M,N}$$ to keep track of the parameters, but I'm not going to do that.) Then one uses other methods to bound the error term. The error term depends on $$M$$ and $$N$$ and as those parameters get larger, the error term gets smaller. If you need to know $$\zeta(z)$$ to high precision, just keep increasing $$M$$ and $$N$$ until you reach your precision goal (i.e., until your estimated upper bound on the value of the error term is small enough to meet your precision target). See the link for references for bounding the error term and the reasonable starting values for $$M$$ and $$N$$, $$\sqrt{2 \pi \mathrm{Im}(z)}$$.

Odlyzko discusses the method used in 1988 to determine the locations of several zeroes in section 3 (starting on document p. 296, PDF page 24) of the cited paper. This paper discribes locating the $$10^{12}$$th through $$10^{12}+100\,000$$th zeroes to within $$10^{-8}$$ to see if certain statistical properties of the locations of the zeroes continued for zeroes this far up the critical line.

Similar algorithms are described in Odlyzko and Schonhage for computing "'medium accuracy' values". These compute $$\zeta(s+\mathrm{i}t)$$ with error proportional to $$t^{-c}$$, where $$c$$ is a parameter which, once selected, controls certain aspects of the computation. When $$-1 < -c < 0$$, this gives small errors for $$t$$ near $$0$$. When $$-c < -1$$, this gives small errors for $$t$$ away from zero -- the larger $$|t|$$, the smaller the error.

This is not the end of the story. There is still work to find more efficient ways to evaluate $$\zeta$$ to high precision.

 Odlyzko, A.M., "On the Distribution of Spacings Between Zeros of the Zeta Function", Mathematics of Computation, vol. 48, no.177, January 1987, pp. 273-308. (The link is to a copy of the paper through the AMS.)

 Odlyzko, A.M. and A. Schonhage, "Fast Algorithms for Multiple Evaluations of the Riemann Zeta Function", Transactions of the AMS, vol. 309, no. 2, October 1988, pp. 797-809. (The link is to a copy of the paper through the AMS.)

 Borwein, J.M., D.M. Bradley, and R.E. Crandall, "Computational strategies for the Riemann zeta function", Journal of Computational and Applied Mathematics, vol. 121, issues 1-2, September 2000, pp. 247-296

The exact values are known for certain rational exponents, see https://en.wikipedia.org/wiki/Riemann_zeta_function#Specific_values.

At other points, numerical algorithms can be used and the values obtained are not "guesses": rigorous error analysis can be made to guarantee the number of exact digits (which can in theory be unlimited).

As regards the zeroes, numerical exploration has been made (to huge extents) probably with the hope of observing "something", but the eventual proof of Riemann's hypothesis (if there's ever one) will be done with rigourous analytical arguments, not relying on particular values.

• Also the main point is that the functional equation means $\pi^{-s/2}\Gamma(s/2)\zeta(s)$ is real on $\Re(s)=1/2$ thus it has a zero at every sign change, in there we are sure there is a non-trivial zero of real part exactly 1/2, only its imaginary part needs to be approximated Oct 12, 2019 at 1:16