```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 softwareto compute Riemann zeta to high precision.I posted to my blog what I think is the first zeta zero to5000 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 termsin the exponential/trigonometric series, and nu, the number ofeven-index Bernoulli numbers to include. With faster Bernoullinumber algorithms, for 20,000 digits one can afford to includehundreds or thousands of Bernoulli numbers, when 's', thezeta argument, is about 1/2 + 14i . A large nu, or more even-indexedBernoulli numbers, means a smaller N is required, so a shorterexponential/trigonometric sum.There is also Peter Borwein's method of computing theDirichlet eta function:http://en.wikipedia.org/wiki/Dirichlet_eta_functionI don't know how PARI/gp implements zeta(.) ....This seems to compare favourably with Odlyzko's tables,referenced below.The sequence of digits: 77277554420656532052405 thatterminates the first zero in Odlyzko's web pagehttp://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros2occurs in the second block of 1000 in one my posts,as: 77277554420656532052405180145559695717667493079382 etc.David Bernier
```