Efficient Prime Counting and the Chebyshev Primes

Abstract EN

The function ϵ(x)=li(x)-π(x), where li is the logarithm integral and π(x) the number of primes up to x, is well known to be positive up to the (very large) Skewes' number.

Likewise, according to Robin's work, the functions ϵθ(x)=li[θ(x)]-π(x) and ϵψ(x)=li[ψ(x)]-π(x), where θ and ψ are Chebyshev summatory functions, are positive if and only if Riemann hypothesis (RH) holds.

One introduces the jump function jp=li(p)-li(p-1) at primes p and one investigates jp, jθ(p), and jψ(p).

In particular, jp<1, and jθ(p)>1 for p<1011.

Besides, jψ(p)<1 for any odd p∈Ch, an infinite set of the so-called Chebyshev primes.

In the context of RH, we introduce the so-called Riemann primes as champions of the function ψ(pnl)-pnl (or of the function θ(pnl)-pnl).

Finally, we find a good prime counting function SN(x)=∑n=1N(μ(n)/n) li[ψ(x)1/n], that is found to be much better than the standard Riemann prime counting function.