Estimating 𝜋(𝑥) and related functions under partial RH assumptions

Source abstract

We give a direct interpretation of the validity of the Riemann hypothesis for all zeros with ℑ ( ρ ) ∈ ( 0 , T ] \Im (\rho )\in (0,T] in terms of the prime-counting function π ( x ) \pi (x) by proving that Schoenfeld’s explicit estimates for π ( x ) \pi (x) and the Chebyshov functions hold as long as 4.92 x / log ⁡ ( x ) ≤ T 4.92\sqrt {x/\log (x)} \leq T . We also improve some of the existing bounds of Chebyshov type for the function ψ ( x ) \psi (x) .