OFFSET
1,1
COMMENTS
Two possibilities:
(1) this sequence is finite;
(2) this sequence is infinite.
In case (1) there exists a maximal integer x_max such that J = f(x_max) = log(abs(pi(x_max) - R(x_max)))/log(x_max).
In case (2) there exists a real constant J such that lim_{x->oo} f(x) = J.
Then for every positive integer x, abs((R(x) - pi(x))/x^J) <= 1.
According to actual computations biggest x = 1090696 with log(-85020 + R(1090696))/log(1090696]) = 0.27835121240340474... and no more new terms up to x 3000000. Follow this:
0.27835121240340474... <= J.
J < 1/2 = limit((log(x) - 2*log((8*Pi)/log(x)))/(2*log(x)), x -> infinity) proof follow Lowell Schoenfeld 1976 proof on upper limit of Chebyshev function psi(x).
Constant J can be used to measure best proved upper limits of asymptotical behavior of function pi(x) when x->infinity. If J is smaller upper bound is better.
LINKS
Lowell Schoenfeld, Sharper Bounds for the Chebyshev Functions theta(x) and psi(x). II, Mathematics of Computation Vol. 30, No. 134 (Apr., 1976), pp. 337-360.
EXAMPLE
x f(x) comment
1 -infinity a(1)
2 -0.8862754573970588 a(2)
3 -4.883591467412115 removed because f(3) < f(2)
4 -0.614424415865155 a(3)
5 -1.0695141714266385 removed because f(5) < f(4)
... ...
MATHEMATICA
gg = {1}; imax = -1000; Do[
kk = Log[Abs[PrimePi[x] - RiemannR[x]]]/Log[x];
If[kk > imax, AppendTo[gg, x]; imax = kk], {x, 2, 20000}]; gg
CROSSREFS
KEYWORD
nonn
AUTHOR
Artur Jasinski, Apr 20 2022
STATUS
approved