Ramanujan's formula for the logarithmic derivative of the gamma function
Abstract
We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in Ramanujan's notebooks. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant, gamma.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.math/0505125
 arXiv:
 arXiv:math/0505125
 Bibcode:
 2005math......5125B
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Number Theory;
 33B15 (Primary);
 11Y60 (Secondary)
 EPrint:
 AMSTeX