Multiplicative Hecke operators and their applications

In this paper, we define the multiplicative Hecke operators $T\left(n\right)$ for any positive integer on the integral weight meromorphic modular forms for ${\Gamma }_0\left(N\right)$. We then show that they have properties similar to those of additive Hecke operators. Moreover, we prove that multiplicative Hecke eigenforms with integer Fourier coefficients are eta quotients, and vice versa. In addition, we prove that the Borcherds product and logarithmic derivative are Hecke equivariant with the multiplicative Hecke operators and the Hecke operators on the half-integral weight harmonic weak Maass forms and weight 2 meromorphic modular forms.