Download or view RiemannZeta.frink in plain text format
// Formula for calculating the Riemann Zeta function.
//
// This follows the very efficient algorithm set out by P. Borwein in
// "An Efficient Algorithm for the Riemann Zeta Function", Jan. 20, 1995.
//
// http://eprints.cecm.sfu.ca/archive/00000107/
//
// or
// https://vdoc.pub/download/an-efficient-algorithm-for-the-riemann-zeta-function-6p64dg2ligs0
//
// This implements Algorithm 2 of the paper.
//
// As noted by Borwein, "These algorithms do not compete with the
// Riemann-Siegel formula for computations concerning zeros on the critical
// line (Im[s] = 1/2) where multiple low precision evaluations are required."
//
// This means that it'll work around the critical line, but there are known
// faster algorithms if you just need low precision and only work around the
// critical line.
//
// This is the prototype of the (not-yet-implemented) Riemann Zeta function
// in Frink.
// Calculate the value of the Riemann Zeta function at s
// n is the approximate number of digits of accuracy. This automatically
// sets n to a minimum of 30 because the algorithm does not converge accurately
// with fewer terms.
RiemannZeta[s, n=30] :=
{
if n<30
n = 30
rnn = RiemannD[n]
sum = 0
for k = 0 to n-1
sum = sum + (-1)^k (rnn@k - rnn@n)/(k+1)^s
return sum * (-1/(rnn@n * (1 - 2^(1-s))))
}
// Calculate an array of values for d_0 ... d_n
// n is the approximate number of digits of precision in the result.
// This array should be stored and re-used across calculations.
RiemannD[n] :=
{
ret = new array[n+1]
sum = 0
for i = 0 to n
{
sum = sum + ((n+i-1)! 4^i)/((n-i)! (2i)!)
ret@i = n * sum
}
return ret
}
Download or view RiemannZeta.frink in plain text format
This is a program written in the programming language Frink.
For more information, view the Frink
Documentation or see More Sample Frink Programs.
Alan Eliasen was born 20217 days, 15 hours, 45 minutes ago.