Efficient Prime Counting and the Chebyshev Primes. (25th March 2013) - Planat, Michel; Solé, Patrick
Abstract : The function ϵ ( x ) = l i ( x ) - π ( x ), where l i 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 ϵ θ ( x ) = l i [ θ ( x ) ] - π ( x ) and ϵ ψ ( x ) = l i [ 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 j p = l i ( p ) - l i ( p - 1 ) at primes p and one investigates j p, j θ ( p ), and j ψ ( p ) . In particular, j p < 1, and j θ ( p ) > 1 for p < 1 0 11 . Besides, j ψ ( p ) < 1 for any odd p ∈ C h, 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 ( p n l ) - p n l (or of the function θ ( p n l ) - p n l ). Finally, we find a good prime counting function S N ( x ) = ∑ n = 1 N ( μ ( n ) / n )   li [ ψ ( x ) 1 / n ], that is found to be much better than the standard Riemann prime counting function.