Equivalent forms of the Grand Riemann Hypothesis - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:58:30Z http://mathoverflow.net/feeds/question/2826 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2826/equivalent-forms-of-the-grand-riemann-hypothesis Equivalent forms of the Grand Riemann Hypothesis Jonah Sinick 2009-10-27T15:19:22Z 2009-11-04T02:11:34Z <p>I have long been curious about equivalent forms of the Riemann hypothesis for automorphic L-functions.</p> <p>In the case of the ordinary Riemann hypothesis, one gets a very good error term for the prime number theorem, one has the formulation involving the Mobius mu function which is a result to the effect of the parity of prime factors in a square free number having a distribution related to that of flips of an unbiased coin, and one also has the reformulation in terms of Farey fractions.</p> <p>I know that for L-functions attached to Dirichlet characters, one gets a very good error term for the prime number theorem for primes in arithmetic progressions. Presumably if one focuses on Dedekind zeta functions and Hecke L-series one gets a very strong effective Chebotarev density theorem or something like that.</p> <p>But for L-functions attached to Hecke eigenforms for GL(2), or more abstract things like symmetric n-th power L-functions attached to automorphic forms or automorphic representations, it seems quite unclear to me what the significance of the Riemann hypothesis for these L-functions is. I think that I remember something about a zero free region to the left of the boundary of the critical strip being related to the Sato-Tate conjecture, so I have a vague impression that one might be able to get a good bound on the speed of convergence to the Sato-Tate distribution as an equivalent to the Riemann hypothesis for some of these L-functions.</p> <p>What are some interesting equivalents to the Riemann hypothesis for automorphic L-functions that you know? I'm particularly interested in statements that have qualitative interpretations.</p> <p>P.S. I've blurred the distinction between an equivalent of the Riemann hypothesis for a single L-function and equivalents to the Riemann hypothesis for a specified family of L-functions. I am interested in both things</p> <p>P.P.S. I am more interested in equivalents than in consequences of the Riemann hypotheses for these L-functions in so far as equivalents "capture the essence" of the statement in question to a greater extent than consequences do. Still, I would would welcome references to interesting <em>consequences</em> of the Riemann hypothesis for automorphic L-functions, again, especially those with qualitative interpretations.</p> http://mathoverflow.net/questions/2826/equivalent-forms-of-the-grand-riemann-hypothesis/4037#4037 Answer by David Hansen for Equivalent forms of the Grand Riemann Hypothesis David Hansen 2009-11-04T02:11:34Z 2009-11-04T02:11:34Z <p>Well, suppose pi is a cuspidal automorphic representation of GL(n)/Q. This has the structure of a tensor product, indexed by primes p, of representations pi_p of the groups GLn(Q<sub>p</sub>). The Satake isomorphism tells us that at almost all primes, each pi<sub>p</sub> is determined by a conjugacy class A(p) in GLn(C). In this language, the Riemann hypothesis for the L-function associated to pi says that the partial sums of tr(A(p)) over p &lt; X show "as much cancellation as possible," and are of size sqrt(X). But if n>1, we are dealing with very complicated objects, and the local components of these automorphic representations vary in some incomprehensible way...</p> <p>You are right, there are certainly special cases. If we knew GRH for L-functions associated to Artin representations then the Cebotarev density theorem would follow with an optimal error term. Likewise, GRH for all the symmetric powers of a fixed elliptic curve E implies (and is in fact equivalent to; see Mazur's BAMS article for a reference) the Sato-Tate conjecture for E with an optimal error term. But in general, reformulations like this simply don't exist.</p> <p>There are many interesting consequences of GRH for various families of automorphic L-functions. I recommend Iwaniec and Kowalski's book (Chapter 5), the paper "Low-lying zeros of families of L-functions" by Iwaniec-Luo-Sarnak, and Sarnak's article at <a href="http://www.claymath.org/millennium/Riemann_Hypothesis/Sarnak_RH.pdf" rel="nofollow">http://www.claymath.org/millennium/Riemann_Hypothesis/Sarnak_RH.pdf</a></p>