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.

  • $\begingroup$ See Landau's Handbuch der Lehre von der Verteilung der Primzahlen $\endgroup$ – 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.

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    $\begingroup$ 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. $\endgroup$ – KConrad Feb 4 '12 at 5:26
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    $\begingroup$ Well, Riemann used the letter $s$, but it's wrong that he used $\sigma$. $\endgroup$ – KConrad Feb 4 '12 at 5:27
  • $\begingroup$ Sorry for the confusion. I have edited the answer. $\endgroup$ – timur Feb 4 '12 at 5:36
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    $\begingroup$ @timur: maybe you read that above! $\endgroup$ – José Hdz. Stgo. Feb 4 '12 at 5:40
  • $\begingroup$ No, it was explicitly stated that Landau seems to be the first one to use $\sigma$. $\endgroup$ – timur Feb 4 '12 at 17:46

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