Item – Theses Canada

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OCLC number
272539412
Author
Knafo, Emmanuel Robert.
Title
Variance of distribution of almost primes in arithmetic progressions.
Degree
Ph. D. -- University of Toronto, 2006
Publisher
Ottawa : Library and Archives Canada = Bibliothèque et Archives Canada, [2007]
Description
8 microfiches
Notes
Includes bibliographical references.
Abstract
In counting primes up to 'x' in a given arithmetic progression, one resorts to the 'prime' counting function <display-math> <fd> <fl><g>y</g><fen lp="par">x;q,a<rp post="par"></fen>=<sum align="c"> <ll><stk><lyr>n<=x</lyr><lyr>na<hsp sp="0.265"><fen lp="par"> <rm>mod<hsp sp="0.212"></rm>q<rp post="par"></fen></lyr></stk> </ll></sum><g>L</g><fen lp="par">n<rp post="par"></fen></fl> </fd> </display-math>where [Lambda] is the usual von Mangoldt function. Analogously, to count those integers with no more than 'k' prime factors, one can use <display-math> <fd> <fl><g>y</g><inf>k</inf><fen lp="par">x;q,a<rp post="par"></fen> =<sum align="c"><ll><stk><lyr>n<=x</lyr><lyr>na<hsp sp="0.265"> <fen lp="par"><rm>mod<hsp sp="0.212"></rm>q<rp post="par"></fen> </lyr></stk></ll></sum><g>L</g><inf>k</inf><fen lp="par">n<rp post="par"></fen> </fl> </fd> </display-math>where [Lambda]'k' is the generalized von Mangoldt function defined by [Lambda]'k' = [mu] * log'k'. Friedlander and Goldston gave a lower bound of the correct order of magnitude for the mean square sum <display-math> <fd> <fl><sum align="c"><ll><stk><lyr>a<hsp sp="0.265"><hsp sp="0.265"> <fen lp="par"><rm>mod<hsp sp="0.212"></rm>q<rp post="par"></fen> </lyr><lyr><fen lp="par">a,q<rp post="par"></fen>=1</lyr></stk> </ll></sum><fen lp="par"><g>y</g><fen lp="par">x;q,a<rp post="par"></fen> -<fr><nu>x</nu><de><g>f</g><fen lp="par">q<rp post="par"></fen> </de></fr><rp post="par"></fen>2</fl> </fd> </display-math> for 'q' in the range <math> <f> <fr><nu>x</nu><de><fen lp="par"><rf>log</rf>x<rp post="par"></fen> A</de></fr></f> </math> <= 'q' <= 'x'. Later, Hooley extended this range to <math> <f> <fr><nu>x</nu><de><rf>exp</rf><fen lp="par">c<rad><rcd><rf>log</rf> x</rcd></rad><rp post="par"></fen></de></fr></f> </math> <= 'q' <= 'x'. We obtain, in the larger range, a lower bound of the correct order of magnitude and approaching the expected asymptotic 'exponentially fast' as 'k' approaches infinity.
ISBN
9780494158418
0494158417