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This question is similar to 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?

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  • $\begingroup$ +1 While I don't believe the related question of why the zeta function is so important has a good answer at this point, the 'how' question seems a reasonable and interesting one. I might have a few remarks to contribute, but I am looking forward to reading other answers first. $\endgroup$
    – Minhyong Kim
    Commented Mar 15, 2011 at 10:49
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    $\begingroup$ Do you not believe that many other functions were considered? Even just by Euler? Look at the size of his collected works! Mathematicians consider many more things than turn out to be beautiful, or interesting, or useful, and those that prove their worth stick around for us to learn about them. I like to think that given enough time, each useful idea would be discovered by someone. I think a more interesting question is WHEN an idea will be discovered. It seems that many ideas "have their time", the almost simultaneous invention of calculus by Newton and Leibnitz being the archetypical example $\endgroup$
    – Barry
    Commented Mar 15, 2011 at 12:19

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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.

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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.

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    $\begingroup$ The divergence of the harmonic series was proven by Nicole Oresme in Questiones super geometriam Euclidis, around 1350. See plato.stanford.edu/entries/nicole-oresme $\endgroup$
    – Stopple
    Commented Mar 15, 2011 at 15:18
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    $\begingroup$ The functional equation for Dirichlet's L-series for the quadratic character modulo 4 was discovered by Euler; before Landau made his work known, more general cases were covered by Malmsten et al. Precise references can be found in Landau's "Euler und die Funktionalgleichung der Riemannschen Zetafunktion", Bibl. Math. (3) 7 (1906), 69--79 as well as in Narkiewicz's book on the prime number theorem. $\endgroup$
    – Franz Lemmermeyer
    Commented Mar 15, 2011 at 17:21
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Here is a text which I remember I found nice:

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

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