Date: Jan 21, 2013 2:32 AM
Author: David Bernier
Subject: first zeta zero
Earlier, I wrote that PARI/gp seems pretty fast among free software

to compute Riemann zeta to high precision.

I posted to my blog what I think is the first zeta zero to

5000 decimal digits. TBD = "To Be Determined",

"There Be Dragons", and so on.

post URL is:

http://meditationatae.wordpress.com/2013/01/21/the-first-nontrivial-zero-of-the-riemann-zeta-function-to-5000d-accuracy/

The formula used was Euler-MacLaurin summation,

formula (1) of Section 6.4 of H. M. Edwards' book

"Riemann's Zeta Function".

What's not obvious is choosing N, essentially the number of terms

in the exponential/trigonometric series, and nu, the number of

even-index Bernoulli numbers to include. With faster Bernoulli

number algorithms, for 20,000 digits one can afford to include

hundreds or thousands of Bernoulli numbers, when 's', the

zeta argument, is about 1/2 + 14i . A large nu, or more even-indexed

Bernoulli numbers, means a smaller N is required, so a shorter

exponential/trigonometric sum.

There is also Peter Borwein's method of computing the

Dirichlet eta function:

http://en.wikipedia.org/wiki/Dirichlet_eta_function

I don't know how PARI/gp implements zeta(.) ....

This seems to compare favourably with Odlyzko's tables,

referenced below.

The sequence of digits: 77277554420656532052405 that

terminates the first zero in Odlyzko's web page

http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros2

occurs in the second block of 1000 in one my posts,

as: 77277554420656532052405180145559695717667493079382 etc.

David Bernier