From the fact spaces of modular forms are finite dimensional, from the decomposition in Hecke eigenforms, and from the duplication formula for $\Gamma(s)$ there are a lot of identities mixing additive convolution $\oplus$ and multiplicative convolution $\otimes$, the basic example being $1_{\mathbb{Z}^2} \oplus 1_{\mathbb{Z}^2} = 1 \otimes \chi_4$

That is in the algebra $M,\oplus$ generated by modular forms for $\Gamma_0(N)$ there will be a lot of identifies involving $\otimes$. How do the Hecke algebras and their representations come into play ? Is there a more precise formulation, using higher things like automorphic representations ?

Do you have some heuristics making visible what property of the primes tells that connexion between additive and multiplicative convolution ? (if the heuristic implies the RH it is good too)

Assuming the primes were wildly distributed, can we still expect such identifies ? The proof of additive-multiplicative identifies rely on the functional equation and Euler product of L-functions, so I'm asking if it is plausible the PNT and the RH still play a role in there. What about the inverse convolutions, the convolution with modular functions like $1/j(z)$, the additive convolution with $\mu(n)$ ?

There is another striking relation between additive and multiplicative convolution in the Riemann hypothesis

The RH holds iff $\text{Riesz}(x)=\sum_{k=0}^\infty \frac{(-1)^k}{\zeta(2+2k) } \frac{x^k}{k!} = O(x^{-3/4+\epsilon})$.

This is because $\text{Riesz}(x)= \sum_{n=1}^\infty \mu(n) n^{-2} e^{-x/n^2}$ so that for $\Re(s) \in (0,1-\sigma_0/2)$, $\int_0^\infty \text{Riesz}(x) x^{s-1}dx = \frac{\Gamma(s)}{\zeta(2-2s)}$. Then $\zeta(2+2k)$ are the odd coefficients of $\frac{x}{e^{2i\pi x}-1}$ ie. of the additive convolution inverse of $\frac{(2i\pi)^k}{k!}$. On the other hand, the RH is clearly about the partial sums of $\mu(n)$ the multiplicative convolution inverse of $1$.

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    $\begingroup$ Can you explain more precisely about these convolution identities? $\endgroup$ – Kimball Mar 16 '19 at 0:14

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