Why JSH's prime residue distribution axiom is incorrect
mjc
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.
MichaelW
> See this paper: "Prime Number Races" by Andrew Granville and Greg
> Martin in American Mathematical Monthly
>
> 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.
I recommend the article referred to in the original post. It is quite
accessible and a great overview about why some residues are
'preferred' after all.
As a bonus I note in the middle of page 31 the exact same "prime gap
equation" that JSH published in August 2006 and claimed to be new. The
article is dated Jan 2006.
Regards, Michael W.
Penny Hassett
Thank you frm me and Socrates. The google search
http://www.google.co.uk/#hl=en&source=hp&q=M.+Rubinstein+and+P.+Sarnak%2C+Chebyshev%E2%80%99s+bias&aq=f&aqi=&aql=&oq=&gs_rfai=&fp=7cee8c1057b5a0bd
finds the cited document at
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.5138&rep=rep1&type=pdf