Explicit Interval Estimates for Prime Numbers

Abstract

Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of (∆, x 0 ) such that for all x ≥ x 0 there exists at least one prime in the interval (x(1 − ∆ −1 ), x].

On the error term in the explicit formula of Riemann--von Mangoldt

“…For log x M = 10 5 we can take M = 18.83 with ω = 1, and find that for m = 177, we can take µ = 0.005664 to have (5.2) hold for all log x ≥ 10 5 . For smaller values of x, we can use the Bertrand-type intervals in [CHL21]. The intervals for x ≥ 4 • 10 18 and x ≥ e 1200 respectively verify this m th powers interval for log(4 • 10 18 ) ≤ log x ≤ 10 5 , and the calculations of [SHP14] verify the remaining range.…”

Primes between consecutive powers

“…For log x ≥ 1.5•10 5 we can take K = 2.5519, and find that (24) holds for all x ≥ e 180340 . The intervals in [9] for x ≥ 4 • 10 18 and x ≥ e 1200 are smaller than this consecutive-powers interval for 4 • 10 18 < x ≤ e 181318 , which is a result of solving…”

“…Intervals with f (x) = Cx, where C is a positive constant, are the largest in the long-run -but they can also be the smallest for sufficiently small x. A method by which we can calculate the constant C and corresponding x 0 has been refined by Schoenfeld [34], Ramaré and Saouter [32], Kadiri and Lumley [21], and most recently by the author and Lee [9]. In [9], pairs of x 0 and ∆ are given such that there exists at least one prime in…”

“…A method by which we can calculate the constant C and corresponding x 0 has been refined by Schoenfeld [34], Ramaré and Saouter [32], Kadiri and Lumley [21], and most recently by the author and Lee [9]. In [9], pairs of x 0 and ∆ are given such that there exists at least one prime in…”

“…Theorem 1 is proved in Section 4, following the proof in [12] with some additional optimisation, and using more recent explicit estimates on the Riemann zeta-function and PNT. Theorem 2 is proved in Section 5 by adapting the proof of Theorem 1, and using results from [9]. The estimates to be used are listed in the following section, and their relative impact on Theorems 1 and 2 is discussed in Section 6.…”