The eñe product over a commutative ring

Abstract: We define the e~ne product for the multiplicative group of polynomials and formal power series with coefficients on a commutative ring and unitary constant coefficient. This defines a commutative ring structure where multiplication is the additive structure and the e~ne product is the multiplicative one. For polynomials with complex coefficients, the e~ne product acts as a multiplicative convolution of their divisor. We study its algebraic properties, its relation to symmetric functions on an infinite number of variables, to tensor products, and Hecke operators. The exponential linearizes also the e~ne product. The e~ne product extends to rational functions and formal meromorphic functions. We also study the analytic properties over the complex numbers, and for entire functions. The e~ne product respects Hadamard-Weierstrass factorization and is related to the Hadamard product. The e~ne product plays a central role in predicting the phenomenon of the "statistics on Riemann zeros" for Riemann zeta function and general Dirichlet $L$-functions discovered by the author. It also gives reasons to believe in the Riemann Hypothesis as explained in the survey "Notes on the Riemann Hypothesis".