# Additive and multiplicative convolution deeply related in modular forms

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 $$1$$.

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