Atle Selberg

Positive proportion for linear combinations of L-functions [UPDATED]

Selberg's celebrated positive proportion result concerning zeros of linear combinations of L-functions was first obtained early in 1998 at age 80, following some preliminary work that he did during his stay in Uppsala the preceding Fall.

The first two files below are scans of the transparencies from two of his lectures on this topic: Hong Kong (May 1998) and MSRI (June 1999). The latter had the title "Linear combinations of L-functions and zeros on the critical line". The former was a supplement to his prime number theory lectures there. (File 3 is simply the typed version of the Hong Kong talk.)

This topic is very closely connected with items #9 and 44 in Selberg's list of publications. See also items 10 and 17 (p. 196).

In response to several questions raised since files 1--3 were uploaded in June 2012, five additional variants have now been appended. The most important of these is file #4, a scan of a revised version of the 1999 MSRI lecture that was found at home in Selberg's study. The file contains some additional assertions, as well as brief indications about possible improvements in the 1999 version. (Those in the audience at MSRI may remember Selberg's impromptu remarks there about several of these matters. File 4 should probably be viewed as a bit tentative, since no further details concerning the revisions could be located among Selberg's papers, either in his office or at home.)

One of the improvements that Selberg mentions pertains to obtaining (c/n)TlogT when Lambda = 1/2. Files 6 and 8 (below) also cite this result, albeit briefly.

Files 5--7 are scans of some pages found on the windowsill near Selberg's desk. File 5 is quite similar to the MSRI talk and is likely part of the preparation for that lecture. The pages scanned in file 6 have a certain resemblance to both the MSRI lecture and its revision; unfortunately, they stop at page 12. The natural conjecture is that they are part of the preparations for a lecture subsequent to MSRI. File 7 is a scan of a xerox copy of the transparencies that Selberg used in his 1998 talk at Uppsala; it is interesting to compare file 7 with file 5.

The close similarity of file 8 to Selberg's Hong Kong lecture (i.e., file 1) suggests that the former may be an earlier draft of what he ultimately decided to present there. (The discrepancy with file 1, page 15, fits with a comment that Selberg made to D.H. in July 1998 that the case alpha = 1 encountered a technical snag, and that his current-best estimate had coefficient c/(n log n).) The transparencies scanned in file 8 were found on a shelf near Selberg's desk.