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I'm trying to find Rosser's original paper where he introduces his eponymous sieve. I've already found https://arxiv.org/pdf/math/0505521 (where the reference isn't given, but where it is indicated that Rosser has priority) and http://matwbn.icm.edu.pl/ksiazki/aa/aa36/aa36210.pdf (which is called "Rosser's sieve", but where there is no reference to his paper).

Any help would be very much appreciated.

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    $\begingroup$ I think the history of this is a bit complicated, and it might be Selberg's lectures on sieves that first discussed Rosser's work (possibly from manuscripts of Rosser?). Iwaniec had independently discovered this, and the title of his paper was influenced by Selberg's lecture notes. I don't believe the word Rosser appears anywhere in Iwaniec's paper, except in the title! $\endgroup$
    – Lucia
    Jul 2 at 19:36
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    $\begingroup$ The archived papers of Rosser at the University of Texas contain a document from 1955 entitled "sieve method". It says "Contact repository for retrieval." Might be worth a try... $\endgroup$ Jul 2 at 20:02
  • $\begingroup$ I contacted the repository and asked them to digitally publish the document online. $\endgroup$ Jul 2 at 22:21
  • $\begingroup$ They said they could provide a scan if I send an email to the appropriate office, which I'm about to do. I wonder whether they'll allow me to upload it myself; perhaps that would be too "socialist" for them. Well, if they don't want the credit, it's their own fault. $\endgroup$ Jul 28 at 21:07
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This is an addition to Lucia's remark. Check out in Selberg's collected papers entry #36: Sieve methods (Proc. Symp. Pure Math., 1971). There he writes:

Very important is also the unpublished work of Barkley Rosser who first settled the problem of the 1-residue sieve, and thus anticipated the work of Jurkat-Richert by about a decade. The part of §5 which deals with the Buchstab-Rosser sieve, refers to work done in the late 1950's after I had had the opportunity to see an unpublished manuscript by Rosser. [...] In the early 1950's Barkley Rosser devised a sieve procedure that essentially represents the limit of the Buchstab procedure. [...] Rosser's work was never published, but a paper about 10 years later by Jurkat and Richert [4] obtained essentially the same results for $k = 1$, by methods which combine Buchstab's ideas with estimations obtained by Brun's method and by a method mentioned in [6] and [7] for the upper bound. If one carefully analyzed their method, it would probably turn out that this is really the Buchstab-Rosser sieve again.

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