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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.

%I A353055 #56 Dec 26 2024 20:29:02
%S A353055 2,4,7,10,19,47,58,73,109,113,1109,1123,1129,1307,1321,1327,1418,1419,
%T A353055 1420,1421,1422,5379,5380,7449,7450,10343,11676,11761,11762,11763,
%U A353055 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
%N 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.
%C A353055 Two possibilities:
%C A353055 (1) this sequence is finite;
%C A353055 (2) this sequence is infinite.
%C A353055 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).
%C A353055 In case (2) there exists a real constant J such that lim_{x->oo} f(x) = J.
%C A353055 Then for every positive integer x, abs((R(x) - pi(x))/x^J) <= 1.
%C A353055 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:
%C A353055 0.27835121240340474... <= J.
%C A353055 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).
%C A353055 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.
%H A353055 Lowell Schoenfeld, <a href="https://doi.org/10.1090/S0025-5718-1976-0457374-X">Sharper Bounds for the Chebyshev Functions theta(x) and psi(x). II</a>, Mathematics of Computation Vol. 30, No. 134 (Apr., 1976), pp. 337-360.
%e A353055 x      f(x)                comment
%e A353055 1     -infinity            a(1)
%e A353055 2     -0.8862754573970588  a(2)
%e A353055 3     -4.883591467412115   removed because f(3) < f(2)
%e A353055 4     -0.614424415865155   a(3)
%e A353055 5     -1.0695141714266385  removed because f(5) < f(4)
%e A353055 ...    ...
%t A353055 gg = {1}; imax = -1000; Do[
%t A353055 kk = Log[Abs[PrimePi[x] - RiemannR[x]]]/Log[x];
%t A353055 If[kk > imax, AppendTo[gg, x]; imax = kk], {x, 2, 20000}]; gg
%Y A353055 Cf. A057793, A057794, A215663, A216709, A226473.
%K A353055 nonn
%O A353055 1,1
%A A353055 _Artur Jasinski_, Apr 20 2022