Explicit and Asymptotic Estimates for the Zero-Counting Function of the Riemann Zeta-Function

Let N(T) be the number of non-trivial zeros of the Riemann zeta-function with imaginary part in (0, T], counted with multiplicity, and let S(T) be the argument of ζ on the critical line, normalised so that N(T) = 1 π ϑ(T) + 1 + S(T). Two estimates are proved by the method of Turing type that Amberger introduced for Dedekind zeta-functions and that Bellotti and Wong extended to Artin L-functions. The first is explicit: for every T ≥ 1, N(T) − T 2π log T 2πe ≤ 0.08382 log T + 5.326, |S(T)| ≤ 0.08382 log T + 4.444, and a piecewise linear bound, formed by six lines, lies below the explicit bounds of Amberger and of Bellotti and Wong for every T ≥ 1. The kernels of the method are found by convex optimisation instead of by hand, the constant coming from the Euler product is computed exactly up to 108 and bounded beyond by a prime-number estimate of Rosser and Schoenfeld, and the elementary part is evaluated as one function of T. The second estimate is asymptotic: lim supT→∞ |S(T)|/ log T < 0.0748596142711. It comes from a hybrid of the kernel majorant with Jensen’s formula on discs to the right of the critical line, where the growth of ζ is controlled by the subconvexity bounds of Bourgain, of Trudgian and Yang, of Tao, Trudgian and Yang, and of Heath-Brown. The constant is smaller than the value 0.0786252 . . . that the method of Backlund gives for discs centred near s = 1 even under the Lindel.f hypothesis. Every inequality between elementary functions that enters a proof is certified in ball arithmetic, and no numerical information on the zeros is used.