A353055 Successive records of function f(x) = log(abs(pi(x) - R(x)))/log(x) where pi(x) is the number of primes <= x and R(x) is Riemann's prime counting function.

2, 4, 7, 10, 19, 47, 58, 73, 109, 113, 1109, 1123, 1129, 1307, 1321, 1327, 1418, 1419, 1420, 1421, 1422, 5379, 5380, 7449, 7450, 10343, 11676, 11761, 11762, 11763, 11764, 11765, 11766, 11767, 11768, 11769, 11770, 11771, 11772, 11773, 11774, 11775, 11776, 19360, 19361, 19362, 19363, 19364, 19365, 19366, 19367, 19368, 19369, 19370, 19371, 19372

Offset: 1

Author

Artur Jasinski, Apr 20 2022

Keywords

• nonn

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.

Examples

			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)
...    ...
		

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.

Crossrefs

Cf. A057793, A057794, A215663, A216709, A226473.

Programs

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