Explicit zero density estimate for the Riemann zeta-function near the critical line

The Riemann hypothesis is true up to 3·1012

“…Let count the number of zeroes with and . Explicit estimates on have been given by Kadiri [15], Kadiri, Lumley, and Ng [17], and Simonič [32]. All of these results could be improved with Theorem 1.…”

Primes between consecutive powers

“…Using H 0 = H p , Table 1 of [22] lists values of C 1 and C 2 for specific σ, after optimising over several parameters. 2 Simonič [36] recently made explicit a zero-density estimate from Selberg [35]. This estimate is applicable for σ ∈ 1 2 , 0.831 , such that for any T ≥ 2H 0 we have…”

Explicit Interval Estimates for Prime Numbers

“…Simonič [36] has given an asymptotically smaller estimate than (5) for σ near 1/2. We have not used this estimate because the zero-density estimates are only used at the Riemann height, at which point ( 4) is smaller than that of [36] for σ ∈ [1/2, 5/8] -after which ( 5) is better than both.…”