nt.number theory - Modular forms and the Riemann Hypothesis

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 $c(s)={Lambda(s)overLambda(1+s)}$ (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 ${Lambda(it)overLambda(1+it)}$), 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 $Z_{mathbb A}H_kbackslash H_{mathbb A}$ then
$$int_{Z_{mathbb A}H_kbackslash H_{mathbb A}}E_s(h)chi(h) dh={Lambda_{k_1}(s,chi)over Lambda_k(2s)}$$
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 $Z_{mathbb A}H_kbackslash H_{mathbb A}/K$, 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
$$int_{Z_{mathbb A}M_kbackslash M_{mathbb A}} E_s(a)|a|^v da={Lambda(v+s)Lambda(v+1-s)overLambda(2s)}$$

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.

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