Prev: consensus on how much Missing Mass, NOVA's tv show Chapter 4, Missing Mass #226 Atom Totality
Next: product of hessian matrix and gradient
From: mjc on 20 Jul 2010 10:39
See this paper: "Prime Number Races" by Andrew Granville and Greg
Martin in American Mathematical Monthly
vol 113 (2006) pages 1-33, url
http://www.dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf

Among many fascinating results, they prove (actually provide a
reference to a proof) "there “usually” seem to be more primes up to x
of the form qn + b than of the form qn + a if a is a square modulo q
and b is not. Indeed, under our two assumptions (that is, the
Generalized Riemann Hypothesis and the linear independence of the
relevant γ s), Rubinstein and Sarnak proved that this is true: the
logarithmic measure of the set of x for which there are more primes of
the form qn + b up to x than of the form qn + a is strictly greater
than 1/2, although always less than 1. In other words, any nonsquare
is ahead of any square more than half the time, though not 100% of the
time."

The reference also proves "We can ask the same question when a and b
are either both squares modulo q or both nonsquares modulo q. In this
case, under the same assumptions, Rubinstein and Sarnak demonstrated
that
#{primes qn + a ≤ x} > #{primes qn + b ≤ x} exactly half the time."

Here is the reference:
M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experiment. Math. 3
(1994) 173–197.

 | 
Pages: 1
Prev: consensus on how much Missing Mass, NOVA's tv show Chapter 4, Missing Mass #226 Atom Totality
Next: product of hessian matrix and gradient