20140805, 00:55  #12 
"Bob Silverman"
Nov 2003
North of Boston
2^{3}×937 Posts 

20140805, 14:10  #13  
Aug 2006
5,987 Posts 
Quote:
But it struck me while reading the theorem that even more important than the correction term in variance is the correction term in mean, and it is purely elementary to improve the error term in the average order of from O(1) to M + O(1/log x), where the error term comes both from the fraction of numbers below x which are prime and from Mertens' theorem. I know this isn't the same as the normal order, and I doubt something so sharp could be proved at that level, but this should improve the practical performance. 

20140812, 15:12  #14  
Aug 2006
5,987 Posts 
Quote:
Persi Diaconis, Frederick Mosteller, Hironari Onishi, Secondorder terms for the variances and covariances of the number of prime factors—Including the square free case, Journal of Number Theory 9:2 (May 1977), pp. 187202. 

20140813, 13:05  #15  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}×937 Posts 
Quote:
The authors surprise me. None of them, AFAIK, is a number theorist. OTOH, I got to take statistics from Mosteller. He was a terrific lecturer. 

20140813, 18:46  #16 
Aug 2006
5,987 Posts 

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