I was wondering about classical notations in number theory. I will not ask here about special functions in general but about the more ubiquitous number theory functions. That which made me wonder originally is $\zeta$: is it really Riemann's $\zeta$? I guess so, but then why did he choose that? Thinking about it, it is interesting that number theory functions seem to have had sticky names, rather varied too. Weierstrass's $\wp$, Dirichlet's $L$, Jacobi's $\theta$,... Subsequent writers have not come up with widely-used variations. I can guess a few reasons why the original notations stuck. The inventors were aware of previous research and chose noncolliding names coherent with what came before. There were few mathematicians in the 19th century and they knew relatively well each other and each other's work. Later mathematicians were then quite rightly very respectful of the traditions set by those illustrious pioneers, especially because the whole was rather coherent and for this reason any attemp at changing part of it would probably clash with some other piece of the "nomencla(pic)ture". Research in number theory is also probably among the most self sufficient of mathematics. Of course there are many connections to other subjects but number theorists can live "within" number theory and outside workers will "borrow" respectfully to the majestuous monument.


So the standard questions would be:


inventor? reason(s)? historical alternatives?


The notations I think about now:


$\zeta$, $\wp$ $\theta$, $\sigma$ (Weierstrass), $G$ (Eisenstein), $\tau$ (Ramanujan), $\eta$ (Dedekind), $s$ for complex numbers, $\tau$ for upper half-plane points, $q$ for $e^{2\pi i\tau}$.


For Eisenstein series I can guess he thought about $F$ but that was too common and $G$ was not used for groups then and anyway general groups are not used close to $G$. Now with general Lie groups and automorphic forms $G$ has morphed into $E$.


Feel free to add to these lists. Any comment is welcome. Thanks.


EDIT: I see my question does not satisfy moderators. If I may: I think notation reflects many deep facts about mathematical concepts and intuitions. I also think that the notations I mention are far from trivial to understand -but I probably overestimate my capacities and they are actually obvious to everybody else but me. Anyway I would really appreciate any indication, like what Carlo replied.


2ND EDIT: I consider that Carlo replied. Thanks to him I convinced myself $\zeta$ stands for "zahl...". It is quite surprising that nowhere I've read this plausible explanation for the notation. I looked at Edwards book, where he translates Riemann and could not find any comment on that. Yet reading the masters as he advises is important only to understand them best, and notation is one of the choices they make that readers should understand. Finally I would like to tell the moderators sorry that I was upset in my 1st EDIT, and also that I think they should probably be more open to questions outside the strict rules set up to now. I think, like Thurston, that mathematics is not foremost about precise statements and proofs, but about understanding. And my belief is that understanding mathematics involves some psychology, sociology, and history among other sciences. All this I would hope to find their place on mathoverflow, to efficiently contribute to the furthering of knowledge in mathematics proper.



The $\zeta$-word
Well, Riemann himself says "I denote this function by $\zeta(s)$" ("*Die Function [...] bezeichne ich durch $\zeta(s)$*"), so I would think the choice of which letter to use for this function was his.


![](https://ilorentz.org/beenakker/MO/Riemann_zeta.jpg)


first page of Riemann's [Über die Anzahl der Primzahlen unter einer gegebenen Grösse](https://en.wikipedia.org/wiki/File:Ueber_die_Anzahl_der_Primzahlen_unter_einer_gegebenen_Gr%C3%B6sse.pdf) (1859).




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The OP also asks: Why did Gauss choose the letter $\zeta$? For integer $s$, Euler had called this product $P$, which seems natural. 


![](https://ilorentz.org/beenakker/MO/Euler_zeta.jpg)


page from [Introductio in analysin infinitorum](https://archive.org/stream/introductioanaly00eule#page/222/mode/2up) (1748)


According to this [source](https://web.archive.org/web/20161013014422/http://www.math.uconn.edu/~kconrad/math5121f15/handouts/dirichletseries.pdf), the use of $s$ as the variable in a series of powers of primes $p$ goes back to Dirichlet, who took $s$ to be real and positive. Riemann emphasized the importance of letting $s$ be complex. Why he chose the letter $\zeta$ for the function of $s$ is not documented, but I can guess that it would make sense to use a different character set for the function and its argument. (We still do this routinely today.) Upper case Greek letters ($\Pi$ or $\Sigma$) were already committed, so the choice for lower case Greek seems natural.


