The impact of 𝜁(𝑠) complex zeros on 𝜋(𝑥) for 𝑥<10^{10¹³}

Abstract

An analysis of the local variations of the prime counting function π ( x ) \pi (x) due to the impact of the non-trivial, complex zeros ϱ k \varrho _k of ζ ( s ) \zeta (s) is provided for x &gt; 10 10 13 x&gt;10^{10^{13}} using up to 200 billion ζ ( s ) \zeta (s) complex zeros. A new bound for | l i ( x ) − π ( x ) | &gt; x 1 / 2 ( l o g l o g l o g x + e + 1 ) / e l o g x |\mathrm {li}(x)-\pi (x)|&gt;x^{1/2}(\mathrm {log}\,\mathrm {log}\,\mathrm {log}\,x+e+1)/e\, \mathrm {log}\,x is proposed consistent with the error growth rate in Littlewood’s proof that l i ( x ) − π ( x ) \mathrm {li}(x)-\pi (x) changes sign infinitely often. This bound is also consistent with all presently known cases where π ( x ) &gt; l i ( x ) \pi (x)&gt;\mathrm {li}(x) including many new examples listed. This implies that Littlewood’s constant K = 1 / e \mathrm {K}=1/e , the lower bound for Skewes’ number is 3.17 × 10 114 3.17\times 10^{114} and the positive constant c c in the Riemann Hypothesis equivalent | l i ( x ) − π ( x ) | &gt; c l o g ( x ) x 1 / 2 |\mathrm {li}(x)-\pi (x)|&gt;c\,\mathrm {log}(x)x^{1/2} is less than 3 × 10 − 27 3\times 10^{-27} .

Effective estimates for some functions defined over primes

“…Lehman [35] improved this last upper bound considerably by showing that exists a number x 0 with x 0 < 1.65 × 10 1165 such that π(x 0 ) > li(x 0 ). After some further improvements (see, for instance, te Riele [55], Bays and Hudson [7], Chao and Plymen [15], Saouter and Demichel [51], Stoll and Demichel [54] and Saouter, Trudgian, and Demichel [50]), the current best upper bound was found by Platt and Trudgian [42]. They proved that there exists a number x 0 with x 0 < e 727.951332668 such that π(x 0 ) > li(x 0 ).…”

“…This lower bound was improved in a series of papers. For details, see Rosser and Schoenfeld [49], Brent [9], Kotnik [30], Platt and Trudgian [42], and Stoll and Demichel [54]. For our further inverstigation, we use the following improvement.…”