On some historical aspects of Riemann zeta function, 3

In this third note, we deepen, from either an historical and histori-ographical standpoint, the arguments treated in the previous preprint hal.archives-ouvertes.fr/hal-00907136- version 1. Roughly, the history of entire function theory starts with the theorems of factorization of a certain class of complex functions, later called entire transcen-dental functions by Weierstrass (see (Loria 1950, Chapter XLIV, Section 741) and (Klein 1979, Chapter VI)), which made their explicit appearance around the mid-1800s, within the wider realm of complex function theory which had its paroxysmal moment just in the 19th century. But, if one wished to identify, with a more precision, that chapter of complex function theory which was the crucible of such a theory, then the history would lead to elliptic function theory and related factorization theorems for doubly periodic elliptic functions, these latter being meant as a generalization of trigonometric functions. Following (Stillwell 2002, Chapter 12, Section 12.6), the early idea which was as at the basis of the origin of elliptic functions as obtained by inversion of elliptic in-tegrals, is due to Gauss, Abel and Jacobi. Gauss has such an idea in the late 1790s but didn't publish it; Abel had the idea in 1823 and published it in 1827, independently of Gauss. Jacobi's independence instead is not quite so clear. He seems to have been approaching the idea of inversion in 1827, but he was only spurred by the appearance of Abel's paper. At any rate, his ideas sub-sequently developed at an explosive rate, up until he published the rst and major book on elliptic functions, the celebrated Fundamenta Nova Theoriæ