What is known about the transcendence of zeroes of Riemann zeta? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:06:04Z http://mathoverflow.net/feeds/question/32324 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/32324/what-is-known-about-the-transcendence-of-zeroes-of-riemann-zeta What is known about the transcendence of zeroes of Riemann zeta? Nick Salter 2010-07-17T23:11:29Z 2010-07-18T01:50:14Z <p>I was wondering if there are any well-known results or hunches about whether the non-trivial zeroes of Riemann-zeta (or zeta/L-functions in general) are algebraic or not. </p> http://mathoverflow.net/questions/32324/what-is-known-about-the-transcendence-of-zeroes-of-riemann-zeta/32327#32327 Answer by David Hansen for What is known about the transcendence of zeroes of Riemann zeta? David Hansen 2010-07-17T23:23:26Z 2010-07-17T23:23:26Z <p>Every non-trivial zero of every L-function, besides possible zeros at $s=1/2$, is conjectured to be of the form $s=1/2+i\gamma$ with $\gamma$ real (GRH) and transcendental. I learned this from (for example) the Rubinstein-Sarnak paper on Chebyshev biases, but they were not the first to enunciate it.</p> http://mathoverflow.net/questions/32324/what-is-known-about-the-transcendence-of-zeroes-of-riemann-zeta/32335#32335 Answer by Micah Milinovich for What is known about the transcendence of zeroes of Riemann zeta? Micah Milinovich 2010-07-18T01:50:14Z 2010-07-18T01:50:14Z <p>There is a paper by A. E. Ingham, "On two conjectures in the theory of numbers", Amer. J. Math. 64 (1942), 313-319, where he shows that if the ordinates of the non-trivial zeros of the Riemann zeta-function are linearly independent over $\mathbb{Q}$ then Merten's conjecture is false. This is, of course, weaker than the Rubinstein-Sarnak conjecture, but related and much earlier.</p>