I am reading some material which makes use of Landau(-Bachmann) notation for the asymptotic behavior of some error.

I started to wonder when this notation (especially the little $o$ on) has been first introduced and checked the Wikipedia source on this which states:

Earliest Uses of Symbols of Number Theory, 22. September 2006: (Memento vom 19. Oktober 2007 im Internet Archive) According to Wladyslaw Narkiewicz in The Development of Prime Number Theory:“The symbols O(·) and o(·) are usually called the Landau symbols. This name is only partially correct, since it seems that the first of them appeared first in the second volume of P. Bachmann’s treatise on number theory (Bachmann, 1894). In any case Landau (1909a, p. 883) states that he had seen it for the first time in Bachmann's book. The symbol o(·) appears first in Landau (1909a).”

I traced back the book and the page (where the reference is supposed to be found) using the quoted link, it's called

Handbuch der Lehre von der Verteilung der Primzahlen, Landau (p. 883)

but unfortunately, the book Handbuch der Lehre von der Verteilung der Primzahlen simply has not that many pages. It's a long shot, but: Does anyone know the precise page where I could find it?

  • $\begingroup$ I dont know if I understood correctly... You are asking to other people to search some information in a book that you have because you are so lazy to search for yourself? $\endgroup$ – Masacroso Feb 17 '17 at 3:21
  • $\begingroup$ @Masacroso First: I don't have the book, just the link to the scanned one. Second: I am not even sure, whether it's the correct one, since the page numbers don't add up. Third: It would take like forever to search the whole book so I was curious if someone would know just the precise page or reference. I wouldn't call it laziness myself but do as you please... $\endgroup$ – user190080 Feb 17 '17 at 3:28
  • $\begingroup$ There's a new beta SE for history of science and mathematics! This post might be well suited there as well: hsm.stackexchange.com $\endgroup$ – Stella Biderman Feb 17 '17 at 5:21
  • $\begingroup$ @StellaBiderman thanks for the tip, I posted a follow up question on First use of litte $o_p$ (little $o$ in probability) notation?. $\endgroup$ – user190080 Feb 22 '17 at 3:54

I downloaded the Landau book and searched it.

The first occurrence of "big-oh" appears to be on page 31 with a discussion of Riemann's estimate of the number of zeros of the zeta function.

The first occurrences of "little-oh" appear to be on pages 69 and 70 in the form $\pi(x) = o(x)$. The definition is on page 70.

(Added after OP's comment - I somehow overlooked these)

The definition of big-oh is on page 59 and the definition of little-oh is on page 61.

  • 1
    $\begingroup$ Hey, thanks for your effort! I actually found out that one could download the book myself - and guess what, I did it. I found the first discussion of $o$-notation to be on p.61 (I mean the printed page number), introduced as a definition w/o reference to anything, so this really might be the first introduction (at least in the book for sure), not sure if it's possible to trace it back any further in history! If you like you could add this to your answer and then I'll go ahead and accept it. $\endgroup$ – user190080 Feb 17 '17 at 4:53
  • $\begingroup$ Interesting that the definition on big-oh on page 59 comes after a number of uses (and its own definition) starting on page 31. $\endgroup$ – marty cohen Feb 17 '17 at 5:14
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    $\begingroup$ The first part of the book, where he already uses big $\mathcal O$ notation, is devoted to the historical development, so maybe something like a historical abstract, w/o definitions at all (as it seems). When he introduces the actual definitions, this is where the new part begins I guess. $\endgroup$ – user190080 Feb 17 '17 at 5:29

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