Sayed Barez

math.NT — Number Theory

Ramanujan's prime counting inequality holds for all x ≥ 38,358,837,683: ranges from the zeros, a spectral barrier, and trade-offs

Contributed by Sayed Barez

Ramanujan proved that $\pi(x)^2 < \frac{ex}{\log x}\pi(x/e)$ for all sufficiently large $x$. Let $N_R$ be the least integer such that this holds for every real $x \ge N_R$. Dudek and Platt showed that the Riemann hypothesis implies $N_R = 38{,}358{,}837{,}683$. We prove that $N_R = 38{,}358{,}837{,}683$. To our knowledge, this is the first determination of $N_R$ without assuming the Riemann hypothesis. The proof uses the zero-free half-plane $\mathrm{Re}s > 7/8$ of a recent OpenAI preprint. Its main statement for $\zeta$ is formalized in Lean 4 and has been rebuilt independently; its written proof has not been peer reviewed. The deduction is our Theorem B: if $\zeta(s) \ne 0$ for $\mathrm{Re}s > 0.945092$, then $N_R = 38{,}358{,}837{,}683$. The proof of Theorem B uses only refereed inputs, together with Bhattacharjee's correction of a theorem of Büthe. Relying only on the peer-reviewed literature, and on preprints where stated, we also prove the following by rigorous computer-assisted interval arithmetic. (A) The inequality holds for $38{,}358{,}837{,}683 \le x \le e^{753.98}$. The previous range ended at $e^{103}$. (C) It holds for $x \ge e^{3035.01}$. This replaces Axler's threshold $e^{3158.442}$, whose proof applies a lemma of Dudek and Platt in a case where that lemma's proof is defective. (B′, D) Either of the following also suffices: a zero-free region $\beta \le 1 - 1/(1.7373\log|\gamma|)$ above height $3\cdot10^{12}$, or RH verified to height $4.1602\cdot10^{14}$. We tabulate the trade-off between the two. (E) Arguments that use only non-negativity, analytic continuation and the poles up to height $T$ cannot go beyond $e^{754.40}$ when $T = 3\cdot10^{12}$. So (A) is optimal among such arguments to within 0.41 in $\log x$. Building on Bhattacharjee's discovery that the constant $E_2$ in Büthe's Theorem 1 lacks a factor $\log(3c)$, we show that the affected rows of the Broadbent–Kadiri–Lumley–Ng–Wilk table inherit the error, by 1.3–2.2% after re-optimization. We also trace the consequences in the literature.