How was the importance of the zeta function discovered? - MathOverflow

How was the importance of the zeta function discovered?

2011-03-15T07:11:27Z

This question is similar to http://mathoverflow.net/questions/1880/why-do-zeta-functions-contain-so-much-information , but is distinct. If the answers to that question answer this one also, I don't understand why.

The question is this: with the benefit of hindsight, the zeta function had become the basis of a great body of theory, leading to generalizations of CFT, and the powerful Langlands conjectures. But what made the 19th century mathematicians stumble on something so big? After all $\sum \frac{1}{n^s}$ is just one of many possible functions one can define that have to do with prime numbers. How and why did was the a priori fancifully defined function recognized as being of fundamental importance?

Answer by Franz Lemmermeyer for How was the importance of the zeta function discovered?

2011-03-15T08:48:57Z

It was a classical problem going back to Mengoli to find a closed expression for the sum of inverse squares. This was solved by Euler, who saw more generally how to evaluate $\zeta(2k)$ at the positive even integers. Later, Euler "computed" the values of $\zeta(s)$ at negative integers as well and conjectured the functional equation of the zeta function. Euler also saw the connection with prime numbers and used the Euler factorization for estimating the number of primes up to $x$.

Most of Euler's results were made rigorous by Dirichlet (his proof of the infinitude of primes in arithmetic progression was built on Euler's results) and Riemann (who interpreted $\zeta(s)$ as a function on the complex plane, proved the functional equation, and indicated how the number of primes is connected with zeroes of the zeta function). There are many more names that should be mentioned (Kummer, Dedekind, Mertens, Landau, ...).

In any case, it was Euler who stumbled upon the zeta function more or less by accident, and he already recognized its importance.

Answer by Sasha for How was the importance of the zeta function discovered?

2011-03-15T11:59:53Z

Here is a text which I remember I found nice:

http://www.dpmms.cam.ac.uk/~wtg10/zetafunction.ps

Answer by Ramin for How was the importance of the zeta function discovered?

2011-03-15T14:39:54Z

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.