Playing with primes distribution
- * * * * * A quick introduction * * * * *
Assume Riemann Hypothesis is true.
It's known  that
pi0(x) = R(x) - sum(R(x^r), zeta(r)=0),
where R(x) is Riemann's function, sum(li(x^(1/n))*mu(n)/n, n=1..inf); pi0(x) = lim((pi(x+eps)+pi(x-eps))/2, eps-->0).
It's also known that R(x) - pi0(x) is noisy function with "mean" noise amplitude proportional to sqrt(x)/logx.
Delta(u) = (pi0(e^u) - R(e^u))*u*e^(-u/2),
R_(u) = sum(R(e^(-2m*u)), m=1..inf)*u*e^(-u/2),
R_main(u) = 2*sum(cos(Imag(r)*u-arg(r))/|r|, zeta(r)=0, Real(r)>0),
R_rem(u) = Delta(u) + R_(u) + R_main(u),
C = sum(1/|r|^2, zeta(r)=0, Real(r)>0) ~ 0.02309571,
D = integral(R_rem(u)du, 0, +inf).
R_(u) = (1-(u/pi)*arctan(pi/u))*e^(-u/2),
integral(R_(u)du, 0, +inf) = 1,
R_main(0) = C.
* * * * * Playing * * * * *
It's seen that Delta, R_main and R_rem are noisy functions, but "mean" noise amplitude for R_rem tends to 0 as u tends to +inf, so R_main becomes very close to Delta. BTW, heuristically (assuming Grand Simplicity Hypothesis) |R_main(u)| exceeds 2*sqrt((C+eps)*logu) only finite number of times for eps>0, and, hence, |Delta(u)| does the same.
Mathematically integral D looks very difficult to compute, but we can estimate it with the help of computer. Rough estimation gives about 4.31.
As for integral(R_main(u)du, 0, t), this functon oscillates around 2*C and has no limit for t --> +inf, but 2*C may be considered as "mean" value. Is R_main, the sum of cosines, "absolutely noisy"? Or we can extract some smooth function f(u) with integral(f(u)du, 0, +inf) = 2*C?
Thus, we see that integral(Delta(u)du, 0, t) oscillates around D-1-2*C ~ 3.26.
It can be seen on a plot that the smooth part of R_rem(u) definitely isn't equal to 0. Another question is to extract this smooth part g(u) such as integral(g(u)du, 0, +inf) = D.
So we see that R(x) isn't the best estimator for pi(x), i.e. Delta(u) isn't "absolutely noisy", g(u) - f(u) - R_(u) being it's smooth component. It's interesting that g(u) looks much more simple (on a plot) than Delta's smooth part. E.g., g(u) seems to be positive, and it's derivative - negative for all u>0.
These noisy functions are very interesting to investigate, and I can't believe there's still no book/article concerning their surprising properties. Let's define f(x,eps) = sqrt(x*(loglogx)^(1+eps))/logx. Maybe we it's possible to obtain pi0(x)-R(x) = O(f(x,eps)) for eps>1 and Omega(f(x,eps)) for eps<1 from RH?
Great thanks for comments,
Andrey V. Kulsha,
Belarussian state university,
[Non-text portions of this message have been removed]
- I mailed about that project some time ago. Now they have seen the light :
8/5: NEW CLIENT (v0.9.2) -- MAJOR PERFORMANCE INCREASE!
This is still not v1 but it is based on Woltman's prp
which is significantly faster than GMP.
(from their web page : http://sb.pns.net/)