Efficient Prime Counting and the Chebyshev Primes

The function where is the logarithm integral and the number of primes up to is well known to be positive up to the (very large) Skewes' number. Likewise, according to Robin's work, the functions and , where and are Chebyshev summatory functions, are positive if and only if Riemann hypothesis (RH) holds. One introduces the jump function at primes and one investigates , , and . In particular, , and for . Besides, for any odd , 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 (or of the function ). Finally, we find a good prime counting function , that is found to be much better than the standard Riemann prime counting function. 1. Introduction We recall first some classical definitions and notation in prime number theory [1, 2].(i)The Von Mangoldt function if is a power of a prime and zero otherwise.(ii)The first and the second Chebyshev functions are, respectively, (where : the set of prime numbers) and where ranges over the integers.(i)The logarithmic integral is .(ii)The M？bius function is equal to , respectively, if is, respectively, non-square-free, square-free with an even number of divisors, square-free with an odd number of divisors.(iii)The number of primes up to is denoted .Indeed, and are the logarithm of the product of all primes up to and the logarithm of the least common multiple of all positive integers up to , respectively. It has been known for a long time that and are asymptotic to (see [2, page 341]). There also exists an explicit formula, due to Von Mangoldt, relating to the nontrivial zeros of the Riemann zeta function [1, 3]. One defines the normalized Chebyshev function to be when is not a prime power, and when it is. The explicit Von Mangoldt formula reads The function is known to be positive up to the (very large) Skewes’ number [4]. In this paper we are first interested in the jumps (they occur at primes ) in the function . Following Robin’s work on the relation between and RH (Theorem 1), this allows us to derive a new statement (Theorem 7) about the jumps of and Littlewood’s oscillation theorem. Then, we study the refined function and we observe that the sign of the jumps of is controlled by an infinite sequence of primes that we call the Chebyshev primes ？？ (see Definition 8). The primes (and the generalized primes ) are also obtained by using an accurate calculation of the jumps of , as in Conjecture 12 (and of the jumps of the function , as in Conjecture 14). One conjectures that the function has infinitely many zeros. There exists