Modular forms and the Riemann Hypothesis

Question

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 $\zeta(s)$?

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@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.

Answer 1

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.

Answer 2

I know two statements about modular forms that are Riemann Hypothesis-ish.

First, note that the constant term of the level-one non-holomorphic Eisenstein series $E_s$ is $y^s+c(s)y^{-s}$, and that the poles of $c(s)$ are the same as the poles of $E_s$. We can directly calculate that $c(s)={\Lambda(s)\over\Lambda(1+s)}$ (this depends on your precise normalization of the Eisenstein series), where $\Lambda$ is the completed zeta function. We can actually say something about the location of the poles of $E_s$ (using the spectral theory of automorphic forms). Unfortunately, we only know how to control poles for ${\rm Re}(s)\ge 0$. This does give an alternate proof of the nonvanishing of $\zeta(s)$ at the edge of the critical strip (from the lack of poles of ${\Lambda(it)\over\Lambda(1+it)}$), but it doesn't seem possible to go further to the left (though it does generalize to other $L$-functions appearing as the constant term of cuspidal-data Eisenstein series).

Second, the values of modular forms at certain (Heegner) points in the upper-half plane can be related to zeta functions. For example, $E_s(i)={\Lambda_{{\mathbb Q}(i)}(s)\over \Lambda_{\mathbb Q}(2s)}$. The general statement is simple to express adelically. Take a quadratic extension $k_1$ of $k$, and let $H$ denote $k_1^\times$ as a $k$-group and $E_s$ the standard level-one Eisenstein series on $G=GL_2(k)$. Take a character $\chi$ on $Z_{\mathbb A}H_k\backslash H_{\mathbb A}$ then $$\int_{Z_{\mathbb A}H_k\backslash H_{\mathbb A}}E_s(h)\chi(h)\ dh={\Lambda_{k_1}(s,\chi)\over \Lambda_k(2s)}$$
where $Z$ denotes the center of $G$, and we have normalized the measure on the quotient space to be 1. Note that since $H$ is a non-split torus in $G$, the quotient is compact, so the integral is finite. In fact, the integrand is invariant (on the right) under a compact open subgroup $K$ of $H_{\mathbb A}$, so the integral is actually over the double coset space $Z_{\mathbb A}H_k\backslash H_{\mathbb A}/K$, which is actually a finite group.
In order to get the Riemann zeta function in the numerator on the right-hand-side, you would need to integrate over a split torus, which is precisely the Mellin transform, and you would have convergence issues. Note that if it did converge, the Mellin transform of $E_s$ would be $$\int_{Z_{\mathbb A}M_k\backslash M_{\mathbb A}} E_s(a)|a|^v\ da={\Lambda(v+s)\Lambda(v+1-s)\over\Lambda(2s)}$$

The second idea is more commonly discussed in the context of subconvexity problems for general $L$-functions. (See Iwaniec's Spectral Methods of Automorphic Forms, especially Chp 13.) A class of subconvexity results is the Lindelof Hypothesis, which is one of the stronger implications of the Riemann Hypothesis.

Answer 3

No.

Let's remember what the basic properties of a modular form are. A modular form is either a section of some high tensor power of a line bundle over $\Gamma \backslash \mathfrak{H}$ for some discrete cofinite $\Gamma \subset \mathrm{SL}_2(\mathbb{R})$ (holomorphic modular forms) or an element of $L^2(\Gamma \backslash \mathfrak{H})$ (Maass forms). A modular form gives rise to a Dirichlet series via Mellin transform, as you say. Now, if $\Gamma$ is a congruence subgroup and the modular form in question is an eigenform of all the Hecke operators, then it is expected that its corresponding L-function satisfies a Riemann hypothesis. However, this property is very sensitive: if $f$ and $g$ are modular eigenforms, then the modular form $f+10^{-10}g$ (say) will not be an eigenform, but it will look a lot like $f$. In particular, its Dirichlet series will look a lot like the Dirichlet series for $f$, but its RH will certainly be destroyed. There is no good criterion solely in terms of the modular form...

However, people expect that any dirichlet series with an Euler product and a functional equation (and some other mild properties) will satisfy RH. You can look up the "Selberg class" of Dirichlet series for more information on this. Of course, modular eigenforms are a primary source of Dirichlet series with Euler product...

Answer 4

The RH for an L-series L(s) is equivalent to an assertion about the locations of the poles of the logarithmic derivative (log L(s))' of L(s). In this way one can relate square-root savings in estimates for log-weighted partial sums of the traces of the Hecke operators of a given form to the RH for its L-series -- for the weight 1/2 theta series you've mentioned (whose Mellin transform is zeta(2s)), this corresponds to an improved error term in the prime number theorem. As David Hansen points out above, modular forms live in linear spaces but RH is not linearly robust, so any answer to your question will have to take some additional structure of those spaces (e.g. the action of the Hecke algebra) into account.

As for your second question, there are statements about families of modular forms that imply RH for zeta; see for instance MR0633666 (and papers that reference it).