Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function $\zeta(s)$ corresponds to a modular form $f$ (of weight 1/2). The functional equation of $\zeta(s)$ follows from the transform equation of $f$. So what is the property of $f$ that would be equivalent to the (conjectural) property that all the non-trivial zeros of $\zeta(s)$ lie on the critical line? Or perhaps is there any statement about some family of modular forms that would imply RH for $\zeta(s)$?
@Hansen and @Anonymous: your answers are appreciated. I want to know why people almost never discuss this question, so even the answer that the question is not a good one is appreciated, provided it also gives a reason, like you did.
As Emerton suggested, I want to know whether RH could be stated for eigenforms directly, instead of the L-functions. I'm no expert in this field, but it seems to me that analytic properties of modular forms are easier to understand (than those of L-functions), so why not expressing RH in the space of modular forms and working with them?
@Anonymous: do you know of any readily accessible source for statements about families of modular forms that imply RH for zeta? I don't have access to MathSciNet.
Best Answer
It can be hard to translate some properties between modular forms and L-functions. As far as I know, there is no simple property of a modular form that is equivalent to the Riemann Hypothesis for its corresponding L-function.
Here is a baby problem to think about. The exponential function and the gamma function form a Mellin pair. How would you detect the periodicity of $e^{ix}$ (to pick a well-known property off the top of my head) from its integral representation in terms of the gamma function?
In this paper, Conrey describes an approach of Iwaniec to RH using a family of elliptic curves. See page 12 for Iwaniec's method, as well as the conclusion with some comments on families.