Origin of the notation s=\sigma+it in analytic number theory I was wondering if the standard notation of denoting a complex variable by "$s$" had an interesting origin, or if it dates back to Riemann or Weierstrass. Almost every book in analytic number theory seems to uses that alphabet with 
$ s = \sigma + i t$ 
denoting its real and imaginary parts. 
I shall be happy if anyone could enlighten me about it. I tried searching MO for relevant questions but couldn't find it. 
 A: In skimming through Narkiewicz "The Development of Prime Number Theory", one sees a reference on p. 155 (footnote 38) to a R. Lipschitz, who in Crelle volume 54 in 1857 "studied the series $\sum_{n=1}^\infty\exp(nui)n^{-\sigma}$ for real values of $\sigma$."  I checked the reference, which is here (there are two papers by Lipschitz in this volume of Crelle, and it's the second one that's relevant); Lipschitz was indeed using $\sigma$.  
Lipschitz is referred to several times in this section of Narkiewicz for later work on functional equations of various $L$-functions.
A: To expand on KConrad's comment: Edmund Landau's 1909 book Handbuch der Lehre von der Verteilung der Primzahlen certainly uses $\sigma = \Re s$, see the footnote on page 30 at Google books
It reads in English "I understand $\sigma = \Re(s)$ as the real part of the complex number $s = \sigma + ti$, (and) $t = \Im(s)$ as the coefficient of $i$ in the purely imaginary part.
A: Riemann uses the notation $s=\frac12+it$ for $s$ on the critical line, but I cannot find any appearance of $\sigma$ in his paper. On the contrary, the notation $a+bi$ appears often there.
However, the following sentence occurs in the first paragraph of Ivic's book:

Riemann wrote $s=\sigma+it$ ($\sigma,t$ real) for the complex variable $s$, and this tradition still persists, although some authors prefer the more logical notation $s=\sigma+i\tau$.

I cannot remember where, but I vaguely recall reading that the tradition was initiated by Landau's book.
