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 $\mu(n)$ ?