HeathBrown, D. R. (2001) The largest prime factor of . Proceedings of the London Mathematical Society (3), 82 . pp. 554596.

PDF
290kB 
Abstract
The largest prime factor of has been investigated by Hooley, who gave a conditional proof that it is infinitely often at least as large as , with a certain positive constant . It is trivial to obtain such a result with . One may think of Hooley's result as an approximation to the conjecture that is infinitely often prime. The condition required by Hooley, his R conjecture, gives a nontrivial bound for short RamanujanKloosterman sums. The present paper gives an unconditional proof that the largest prime factor of is infinitely often at least as large as , though with a much smaller constant than that obtained by Hooley. In order to do this we prove a nontrivial bound for short RamanujanKloosterman sums with smooth modulus. It is also necessary to modify the Chebychev method, as used by Hooley, so as to ensure that the sums that occur do indeed have a sufficiently smooth modulus.
Item Type:  Article 

Additional Information:  This is a preprint version. The original journal should be consulted for the final version. 
Uncontrolled Keywords:  Largest prime factor, Cubic polynomial, Hypothesis R$^{*}$, Unconditional 
Subjects:  H  N > Number theory 
Research Groups:  Number Theory Group 
ID Code:  165 
Deposited By:  Roger HeathBrown 
Deposited On:  27 Jan 2005 
Last Modified:  29 May 2015 18:17 
Repository Staff Only: item control page