# Equivalent forms of the Grand Riemann Hypothesis

I have long been curious about equivalent forms of the Riemann hypothesis for automorphic L-functions.

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.

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.

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.

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.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.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 consequences of the Riemann hypothesis for automorphic L-functions, again, especially those with qualitative interpretations.

• Would GRH be a Generalized Riemann Hypotesis? You might want your question easier to read by expanding that in the title. – Ilya Nikokoshev Oct 27 '09 at 20:36