# 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.

• See Landau's Handbuch der Lehre von der Verteilung der Primzahlen – KConrad Feb 4 '12 at 5:27

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.

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.

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.

• This is wrong: look at the paper and you'll see he never mentions a variable for the real part of s. In fact the only time he does use a real part, it's 1/2. – KConrad Feb 4 '12 at 5:26
• Well, Riemann used the letter $s$, but it's wrong that he used $\sigma$. – KConrad Feb 4 '12 at 5:27
• Sorry for the confusion. I have edited the answer. – timur Feb 4 '12 at 5:36
• @timur: maybe you read that above! – José Hdz. Stgo. Feb 4 '12 at 5:40
• No, it was explicitly stated that Landau seems to be the first one to use $\sigma$. – timur Feb 4 '12 at 17:46