  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".
