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Leonhard Euler studied the function that is now known as the Riemann zeta function. I have not found the notation $\zeta$ in any of the works of any mathematicians prior to Bernhard Riemann's paper On the Number of Primes Less Than a Given Quantity dated November, 1859. Did Euler know the zeta function as the zeta function or did he use a different notation for it?

I am seeking a reference for the first use of the notation $\zeta$ for a function.

What I am getting at is this:

"Unfortunately, overshadowed by the complex version of the zeta function subsequently developed and used by Bernard Riemann, Euler's original real zeta function seems to have dropped out of sight in popular expositions of mathematics of late. With the hope of similarly inspiring another generation of future mathematicians, this month's column tries to rekindle interest in Euler's original and spectacular eighteenth century theorem." - Keith Devlin https://www.maa.org/external_archive/devlin/devlin_10_01.html

See, there are two zeta functions. So, which one do we denote by $\zeta$ and how do we denote the other?

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  • $\begingroup$ For any function, or for the Riemann zeta function? $\endgroup$ – Ben McKay Apr 22 '16 at 16:17
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    $\begingroup$ this was answered here: mathoverflow.net/questions/227209/the-zeta-word ---- Euler called it $P$, the name $\zeta$ is due to Riemann (although, of course, the letter $\zeta$ for another quantity may well have been used before) $\endgroup$ – Carlo Beenakker Apr 22 '16 at 16:28
  • $\begingroup$ Weierstrass used $\zeta$ to denote something completely different, and this is a standard notation nowadays. $\endgroup$ – Alexandre Eremenko Apr 22 '16 at 20:43
  • $\begingroup$ I'm not sure all the answers have been given, as many people wonder how did Dirichlet perceive $\zeta(s)$ and $L(s,\chi)$ ? it seems obvious that he knew quite a lot, and some historical facts on that part of the story would be nice. $\endgroup$ – reuns Apr 23 '16 at 4:11
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Georg Schuppener's "Geschichte der Zeta-Funktion von Oresme bis Poisson" (Deutsche Hochschulschriften v. 533) is the reference for the history of $\zeta(s)$ before Riemann. Although he uses the $\zeta$ notation throughout, it is never in direct quotations, which instead are all of the form, for example $$ "1+1/4+1/9+\ldots \text{etc.}" $$ instead of $\zeta(2)$. You should check yourself (my German is not very good) but I think there's no $\zeta$ before Riemann.

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It was Riemann that used it as zeta function. Before Riemann, Dirichlet and Euler had researched this infinite sum, but they just considered it as a function of real variable.

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  • $\begingroup$ are we sure Dirichlet did not consider $s \in \mathbb{C}$ ? I mean it seems difficult to prove the Dirichlet theorem on arithmetic progressions without thinking to functions of complex numbers $\endgroup$ – reuns Apr 23 '16 at 4:13

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