Efficient prime counting and the Chebyshev primes

The function $\epsilon(x)=\mbox{li}(x)-\pi(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $\epsilon_{\theta}(x)=\mbox{li}[\theta(x)]-\pi(x)$ and $\epsilon_{\psi}(x)=\mbox{li}[\psi(x)]-\pi(x)$ are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $\theta(x)=\sum_{p \le x} \log p$ and $\psi(x)=\sum_{n=1}^x \Lambda(n)$, respectively, $\mbox{li}(x)$ is the logarithmic integral, $\mu(n)$ and $\Lambda(n)$ are the M\"obius and the Von Mangoldt functions). Negative jumps in the above functions $\epsilon$, $\epsilon_{\theta}$ and $\epsilon_{\psi}$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes $j_p=\mbox{li}(p)-\mbox{li}(p-1)$ and one investigates the jumps $j_p$, $j_{\theta(p)}$ and $j_{\psi(p)}$. In particular, $j_p<1$, and $j_{\theta(p)}>1$ for $p<10^{11}$. Besides, $j_{\psi(p)}<1$ for any odd $p \in \mathcal{\mbox{Ch}}$, an infinite set of so-called {\it Chebyshev primes } with partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, \ldots\}$. We establish a few properties of the set $\mathcal{\mbox{Ch}}$, give accurate approximations of the jump $j_{\psi(p)}$ and relate the derivation of $\mbox{Ch}$ to the explicit Mangoldt formula for $\psi(x)$. In the context of RH, we introduce the so-called {\it Riemann primes} as champions of the function $\psi(p_n^l)-p_n^l$ (or of the function $\theta(p_n^l)-p_n^l$ ). Finally, we find a {\it good} prime counting function $S_N(x)=\sum_{n=1}^N \frac{\mu(n)}{n}\mbox{li}[\psi(x)^{1/n}]$, that is found to be much better than the standard Riemann prime counting function.


Introduction
Let us introduce the first and the second Chebyshev function θ(x) = p≤x log p (where p ∈ P: the set of prime numbers) and ψ(x) = x n=1 Λ(n), the logarithmic integral li(x), the Möbius function µ(n) and the Von Mangoldt function Λ(n) [1,4]. The number of primes up to x is denoted π(x). Indeed, θ(x) and ψ(x) are the logarithm of the product of all primes up to x, and the logarithm of the least common multiple of all positive integers up to x, respectively.
It has been known for a long time that θ(x) and ψ(x) are asymptotic to x (see [4], p. 341). There also exists an explicit formula, due to Von Mangoldt, relating ψ(x) to the non-trivial zeros ρ of the Riemann zeta function ζ(s) [1,2]. One defines the normalized Chebyshev function ψ 0 (x) to be ψ(x) when x is not a prime power, and ψ(x) − 1 2 Λ(x) when it is. The explicit Von Mangoldt formula reads , for x > 1.
The function ǫ(x) = li(x) − π(x) is known to be positive up to the (very large) Skewes' number [3]. In this paper we are first interested in the jumps (they occur at primes p) in the function ǫ θ(x) = li[θ(x)] − π(x). Following Robin's work on the relation between ǫ θ(x) and RH (Theorem 1.1), this allows us to derive a new statement (Theorem 1.7) about the jumps of li[θ(p)] and Littlewood's oscillation theorem.
Then, we study the refined function ǫ ψ(x) = li[ψ(x)] − π(x) and we observe that the sign of the jumps of li[ψ(p)] is controlled by an infinite sequence of primes that we call the Chebyshev primes Ch n (see proposition 1.11). The primes Ch n (and the generalized primes Ch (l) n ) are also obtained by using an accurate calculation of the jumps of li[ψ(p)], as in conjecture 1.14 (and of the jumps of the function li[ψ(p l )], as in conjecture 1.17). One conjectures that the function Ch n − p 2n has infinitely many zeros. There exists a potential link between the non-trivial zeros ρ of ζ(s) and the position of the Ch (l) n 's that is made quite explicit in Sec. 2.1 (conjecture 2.2), and in Sec. 2.2 in our definition of the Riemann primes. In this context, we contribute to the Sloane's encyclopedia with integer sequences 1 .
Finally, we introduce a new prime counting function R(x) = n>1 µ(n) n li(x 1/n ), better than the standard Riemann's one, even with three terms in the expansion. 1. Selected results about the functions ǫ, ǫ θ , ǫ ψ Let p n be the n-th prime number and j(p n ) = li(p n ) − li(p n − 1) be the jump in the logarithmic integral at p n . For any n > 2 one numerically observes that j pn < 1. This statement is not useful for the rest of the paper. But it is enough to observe that j 5 = 0.667 . . . and that the sequence j pn is strictly decreasing.
The next three subsections deal with the jumps in the function li[θ(x)] and li[ψ(x)].
Proof. If RH is true then, using the fact ψ(x) > θ(x) and that li(x) is a strictly growing function when x > 1, this follows from theorem 1 in Robin [5]. If RH is false, Lemma 2 in Robin ensures the violation of the inequality.
Proof. The integral definition of the jump yields The result now follows after observing that by [11, Theorem18], we have θ(x) < x for x < 10 8 , and by using the note added in proof of [12] that establishes that θ(x) < x for x < 10 11 . 1 The relevant sequences are A196667 to 196675 (related to the Chebyshev primes), A197185 to A197188 (related to the Riemann primes of the ψ-type and A197297 to A197300 (related to the Riemann primes of the θ-type.
where log 3 x = log log log x. The omega notations means that there are infinitely many numbers x, and constants C + and C − , satisfying We now prepare the proof of the invalidity of conjecture (1.4) by writing two lemmas.
Lemma 1.5. For n ≥ 1, we have the bounds Proof. This is straightforward from the integral definition of the jump.
Lemma 1.6. For n large, we have Proof. We know that by [9, Theorem 6.3, p.200], we have for x > 0 and large The result follows by considering the primes closest to x.
We can now state and prove the main result of this section.
Theorem 1.7. For n large we have Proof. By lemma 1.6 we know there is a constant C − such that for infinitely many n's we have By combining with the first inequality of lemma 1.5 and writing the minus part of the statement follows after some standard asymptotics. To prove the plus part write θ(p n ) = θ(p n+1 ) − log p n+1 , and proceed as before. Let us define the n-th jump at a prime as Theorem 1.10. For n large, we have Corollary 1.11. There are infinitely many Chebyshev primes Ch n .
One observes that the sequence Ch n oscillates around p 2n and the largest deviations from p 2n seem to be unbounded at large n. This behaviour is illustrated in 2 Our terminology should not be confused with that used in [7] where the Tchebychev primes are primes of the form 4n2 m + 1, with m > 0 and n an odd prime. We used the Russian spelling Chebyshev to distinguish both meanings. There exists a real c n depending of the index n, with ψ(p n − 1) < c n < ψ(p n ) such that Using the known locations of the Chebyshev primes of low index, it is straightforward to check that the real c n reads This numerical calculations support our conjecture (1.14) that the Chebyshev primes may be derived fromK n−1 instead of K n−1 .
1.3. The generalized Chebyshev primes. Definition 1.16. Let p ∈ P be a odd prime number and the function j ψ(p l ) = li[ψ(p l )] − li[ψ(p l − 1)], l ≥ 1. The primes p such that j ψ(p l ) < 1/l are called here generalized Chebyshev primes Ch (l) n (or Chebyshev primes of index l

The Chebyshev and Riemann primes
The next subsection relates the definition of the Chebyshev primes to the explicit Von Mangoldt formula. The following one puts in perspective the link of the Chebyshev primes to RH through the introduction of the so-called Riemann primes.
2.1. The Chebyshev primes and the Von Mangoldt explicit formula. From corollary 1.11, one observes that the oscillations of the function ψ(x) − x around 0 are intimely related to the existence of Chebyshev primes.
Proposition 2.1. If p n is a Chebyshev prime (of index 1), then ψ(p n ) > p n . In the other direction, if ψ(p n − 1) > p n , then p n is a Chebyshev prime (of index 1).
Proof. The proposition 2.1 follows from the inequalities (analogous to that of lemma 1.5) .
In what regards the position of the (generalized) Chebyshev primes, our numerical experiments lead to   Clearly, the subset of the Riemann primes of the ψ-type such that ψ 0 (p l n ) > p l n belongs to the set of Chebyshev primes of the corresponding index l. Since the Riemann primes of the ψ-type maximize ψ(x) − x, it is useful to plot the ratio r (l) = (ψ(p l n ) − p l n )/ p l n . Fig. 2 illustrates this dependence for the Riemann primes of index 1 to 4. One finds that the absolute ratio |r (l) | decreases with the index l: this corresponds to the points of lowest amplitude in Fig. 2.  Under RH, one has the inequality [12, Theorem 10] In the following, we specialize on bounds for θ(x) − x at power of primes x = p l n . Definition 2.6. The champions (left to right maxima) of the function |θ(p l n ) − p l n | are called here Riemann primes of the θ-type and index l. The Riemann primes of the θ-type maximize θ(x)−x. In Fig. 3, we plot the ratio p l n log 2 p l n ) at the Riemann primes of index 1 to 4. Again one finds that the absolute ratio |s (l) | decreases with the index l: this corresponds to the points of lowest amplitude in Fig. 3.
In the future, it will be useful to approach the proof of RH thanks to the Riemann primes.
3. An efficient prime counting function  By definition, the negative jumps in the function η N (x) may only occur at x+1 ∈ P. For N = 1, they occur at primes p ∈ Ch (the Chebyshev primes: see definition 1.8). For N > 1, negative jumps are numerically found to occur at all x + 1 ∈ P with an amplitude decreasing to zero. We are led to the conjecture More generally, the jumps of η N (x) at power of primes are described by the following Conjecture 3.2. Let η N (x) be as in conjecture 3.1. Positive jumps of the function η N (x) occur at all power of primes x+1 = p l , p ∈ P and l > 1. Moreover, the jumps are such that η N (p l )− η N (p l − 1)− 1/l > 0 and lim p→∞ η N (p l )− η N (p l − 1)− 1/l = 0 A sketch of the function η 3 (n) (for 2 < n < 1500) is given in Fig. 2. One easily detects the large positive jumps at n = p 2 (p ∈ P), the intermediate positive jumps at n = p l (l > 2), and the (very small) negative jumps at primes p. This plot can be compared to that of the function R(n) − π(n) displayed in [13]. Comment 3.3. The arithmetical structure of η N (x) just described leads to |η N (x)| < η max when N ≥ 3. Table 1 represents the maximum value η max that is reached and the position x max of the extremum, for several small values of N and x < 10 5 . Thus, the function N n=1 µ(n) n li[ψ(x) 1/n ] is a good prime counting function with only a few terms in the summation. This is about a fivefold improvement of the accuracy obtained with the standard Riemann prime counting function R(x) (in the range x < 10 4 ) and an even better improvement when x > 10 4 , already with three terms in the expansion. Another illustration of the efficiency of the calculation based on li[ψ(x)] is given in Table 2, that displays values of η 3 (x) at multiples of 10 6 .
It is known that R(x) converges for any x and may also be written as the Gram series [13] R(x) = 1 + k=1 ∞ (log x) k k!kζ(k+1) . A similar formula is not established here.