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|Title:||Improvements in the Small Sieve Estimate of Selberg by Iteration|
|Author(s):||Rawsthorne, Daniel Andrew|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||In the notation of H. Halberstam and H.-E. Richert, the small sieve estimate of Selberg is
(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
where (tau) = log (xi)('2)/log z. Repeated iteration of (*) with the Buchstab identity produces improved upper (and lower) bounds for ( ; ,z). Sieve results of the form (*) are produced for each intermediate function produced by iteration, and also for the limit functions approached.
The limit functions are solutions of a pair of difference-differential equations, and Laplace transform methods are used in an effort to define the functions analytically. Two curves are produced such that the coordinates of one of their intersections are the two parameters that need to be found. Computer calculations indicate that there is in fact a unique intersection, but this is not proved; and by assuming the intersection is unique, approximate values of the parameters are given for various dimensions.
In addition to the above work, it is also shown that the L in the 0-term of (*) is unnecessary; and the iteration is studied when one begins with a general "upper bound function" (defined in the paper).
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980.
|Date Available in IDEALS:||2014-12-14|