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first zeta zero
Posted:
Jan 21, 2013 2:32 AM


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/thefirstnontrivialzerooftheriemannzetafunctionto5000daccuracy/
The formula used was EulerMacLaurin 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 evenindex 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 evenindexed 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



