Computation of Riemann Zeta function

  1. Euler-Maclaurin type formula with Bernoulli numbers
  2. One can accelerate the convergence of the Dirichlet series by acceleration methods to compute zeta function for small imaginary parts. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.9455&rep=rep1&type=pdf

  3. Poisson summation to get Riemann Siegel formula. Another way to see this is approximate functional equation. So now we are reduced to computing exponential sums e^{it\log (n)} . Using Taylor approximation of log to reduce to exponential sums for polynomials.

Theorem (Hiary, 2008) For fixed j, the sum

\displaystyle \frac{1}{K} \sum_{k=0}^{K-1} k^{\prime} \exp \left(2 \pi i a k+2 \pi i b k^{2}\right)

can be computed in O\left((\log K)^{2}\right) arithmetic operations.

3. The Odlyzko-Schonhage algorithm :If you want to compute the function at multiple values, one can use interpolation using FFT. The Fourier transform can be computed and then FFT gives a fast way to compute the zeta function for a large range of values. http://www.dtc.umn.edu/~odlyzko/doc/arch/fast.zeta.eval.pdf

Click to access annals-v174-n2-p04-p.pdf

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