math.NT — Number Theory
A two-sided criterion for the Riemann hypothesis from weighted prime sums, verified up to 2·10^29
For N > 1 let C(N) = 2 Σ_{n<N} Λ(n) n^{-1/2} log(N/n) and M(N) = R(N) − C(N), where R(N) = 8√N − 5.3721834 log N − 7.4013354 + O(N^{-5/2}) is explicit. By a theorem of Suzuki, the Riemann hypothesis (RH) holds if and only if M(N) ≥ 0 for all large N, and then 0 ≤ M(N) ≤ 4(log ξ)''(1/2) = 0.18484. We prove, with computer assistance, that this two-sided inequality holds for every real 1 < N ≤ 2.15·10^29: up to 10^10 from the primes with correctly rounded arithmetic, and beyond from the first 4520 zeros, certified with ball arithmetic, together with the verification of RH to 3·10^12 and an explicit zero-density estimate. The same ingredients give, for every N and without any hypothesis, −E(N) ≤ M(N) ≤ 0.18484 + E(N) with E(N) explicit and E(10^20) < 3·10^-5. We describe the local minima of M exactly and prove an unconditional energy identity for the slope α(x) = Σ_{n≤x} Λ(n) n^{-1/2} − 2√x + c0/2. We then show that positive weights within q^{δ−1/2} of Λ(q) q^{-1/2} satisfy the energy identity, the prime number theorem and Chebyshev’s identity, yet carry an off-line zero, so any proof must use Λ = μ∗log exactly. We do not prove the Riemann hypothesis.