Andre Weil has an article called "Prehistory of the zeta function" (reviewed by Jutila on mathscinet). I read this article many years ago, but this is basically what I remember of its content. Apparently the divergence of the harmonic series was known in 1650. Euler computed the special values at even integers and derived a some kind of a functional equation. He also proved the Euler product formula and gave a proof of the infinitude of prime numbers using the Euler product. Dirichlet defined general L functions that now bear his name but only for real s>1. Riemann extended the definition of the zeta function to all complex values and proved the functional equation. According to Weil there were other people who had proved functional equations for functions that were closely related to the zeta function (namely, Malmstén, Schlömilch and Clausen from the review), but perhaps Riemann's contribution is the singular paper that established the importance of the zeta function as an importance object to study. Weil believes that Riemann was influenced by his discussion with Eisenstein.

Andre Weil has an article called "Prehistory of the zeta function" (reviewed by Jutila on mathscinet). I read this article many years ago, but this is basically what I remember of its content. Apparently the divergence of the harmonic series was known in 1650. Euler computed the special values at even integers and derived some kind of a functional equation. He also proved the Euler product formula and gave a proof of the infinitude of prime numbers using the Euler product. Dirichlet defined general L functions that now bear his name but only for real s>1. Riemann extended the definition of the zeta function to all complex values and proved the functional equation. According to Weil there were other people who had proved functional equations for functions that were closely related to the zeta function (namely, Malmstén, Schlömilch and Clausen from the review), but perhaps Riemann's contribution is the singular paper that established the importance of the zeta function as an important object to study. Weil believes that Riemann was influenced by his discussion with Eisenstein.