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