# Efficient Prime Counting and the Chebyshev Primes

Authors: Michel Planat; Patrick Solé
Journal: Journal of Discrete Mathematics
 (2013)

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## Abstract

The functionϵ(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.


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## Body

## 1. Introduction

We recall first some classical definitions and notation in prime number theory [1, 2].(i)
The Von Mangoldt functionΛ(n)=logp if n is a power of a prime p and zero otherwise.(ii)
The first and the second Chebyshev functions are, respectively,(1)θ(x)=∑p≤xlogp,(where p∈𝒫: the set of prime numbers) and
(2)ψ(x)=∑1≤n≤xΛ(n),
where n ranges over the integers.(i)
The logarithmic integral isli(x)=∫2x‍(dt/t).(ii)
The Möbius functionμ(n) is equal to 0,1,-1, respectively, if n 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 tox 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 thatθ(x) and ψ(x) are asymptotic to x (see [2, page 341]). There also exists an explicit formula, due to Von Mangoldt, relating ψ(x) to the nontrivial zeros ρ of the Riemann zeta function ζ(s) [1, 3]. One defines the normalized Chebyshev function ψ0(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
(3)ψ0(x)=x-∑ρxρρ-ζ′(0)ζ(0)-12log(1-x-2),forx>1.The functionϵ(x)=li(x)-π(x) 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 p) in the function ϵθ(x)=li[θ(x)]-π(x). Following Robin’s work on the relation between ϵθ(x) and RH (Theorem 1), this allows us to derive a new statement (Theorem 7) about the jumps of li[θ(p)] and Littlewood’s oscillation theorem.Then, we study the refined functionϵψ(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 primesChn   (see Definition 8).  The primes Chn (and the generalized primes Chn(l)) are also obtained by using an accurate calculation of the jumps of li[ψ(p)], as in Conjecture 12 (and of the jumps of the function li[ψ(pl)], as in Conjecture 14). One conjectures that the function Chn-p2n has infinitely many zeros. There exists a potential link between the nontrivial zeros ρ of ζ(s) and the position of the Chn(l)’s that is made quite explicit in Section 3.1 (Conjecture 16), and in Section 3.2 in our definition of the Riemann primes. In this context, we contribute to the Sloane’s encyclopedia with integer sequences. (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.)Finally, we introduce a new prime counting functionR(x)=∑n>1(μ(n)/n)li(x1/n), better than the standard Riemann’s one, even with three terms in the expansion.


## 2. Selected Results about the Functionsϵ,ϵθ,ϵψ

Letpn be the nth prime number and let j(pn)=li(pn)-li(pn-1) be the jump in the logarithmic integral at pn. For any n>2 one numerically observes that jpn<1. This statement is not useful for the rest of the paper. But it is enough to observe that j5=0.667  and that the sequence jpn is strictly decreasing.The next three subsections deal with the jumps in the functionsli[θ(x)] and li[ψ(x)].


### 2.1. The Jumps in the Function li[θ(x)]

Theorem 1 (Robin).
The statementϵθ(x)=li[θ(x)]-π(x)>0 is equivalent to RH [5, 6].Corollary 2 (related to Robin [5]).
The statementϵψ(x)=li[ψ(x)]-π(x)>0 is equivalent to RH.Proof.
If RH is true then, using the fact thatψ(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.Using the fact thatθ(pn+1-1)=θ(pn), define the jump of index n as
(4)Jn=jθ(pn)=li[θ(pn+1)]-li[θ(pn)]=∫θ(pn)θ(pn+1)dtlogt.Proposition 3.
Ifpn+1<1011, then Jn=jθ(pn)>1.Proof.
The integral definition of the jump yields(5)Jn≥θ(pn+1)-θ(pn)logθ(pn+1)=logpn+1logθ(pn+1).
The result now follows after observing that by [7, Theorem 18], we have θ(x)<x for x<108, and by using the note added in proof of [8] that establishes that θ(x)<x for x<1011.By seeing this result it would be natural to make the following conjecture.Conjecture 4.
For alln≥1 we have Jn>1.However, building on Littlewood’s oscillation theorem forθ we can prove that Jn oscillates about 1 with a small amplitude. Let us recall the Littlewood’s oscillation theorem ([9, Theorem 6.3, page 200], [10, Theorem 34])
(6)θ(x)-x=Ω±(x1/2log3x),whenx→∞,
where log3x=logloglogx. The omega notations mean that there are infinitely many numbers x and constants C+ and C-, satisfying
(7)θ(x)≤x-C-xlog3xorθ(x)≥x+C+xlog3x.
We now prepare the proof of the invalidity of Conjecture 4 by writing two lemmas.Lemma 5.
Forn≥1, we have the bounds
(8)logpn+1logθ(pn+1)≤Jn≤logpn+1logθ(pn).Proof.
This is straightforward from the integral definition of the jump.Lemma 6.
Forn large, we have
(9)θ(pn+1)=pn+1+Ω±(pn+1log3pn+1).Proof.
We know that by [9, Theorem 6.3, page 200], we have for x>0 and large
(10)θ(x)-x=Ω±(xlog3x).
The result follows by considering the primes closest to x.We can now state and prove the main result of this section.Theorem 7.
Forn large we have
(11)Jn=1+Ω±(log3pn+1pn+1logpn+1).Proof.
By Lemma6 we know there is a constant C- such that for infinitely many n’s we have
(12)θ(pn+1)<pn+1-C-pn+1log3pn+1.
By combining with the first inequality of Lemma 5 the minus part of the statement follows after some standard asymptotics. To prove the plus part write θ(pn)=θ(pn+1)-logpn+1, and proceed as before.


### 2.2. The Jumps in the Function li[ψ(x)] and the Chebyshev Primes

Definition 8.
Letp∈𝒫 be an odd prime number and the function jψ(p)=li[ψ(p)]-li[ψ(p-1)]. The primes p such that jψ(p)<1 are called here Chebyshev primesChn. (Our terminology should not be confused with that used in [11] where the Chebyshev primes are primes of the form 4n2m+1, with m>0 and n an odd prime. We used the Russian spelling Chebyshev to distinguish both meanings.) In increasing order, they are [12, Sequence A196667]{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, …}.Comment 1.
The number of Chebyshev primes less than10n, (n=1,2,…) is the sequence {0,0,42,516,4498,41423…} [12, Sequence A196671]. This sequence suggests the density (1/2)π(x) for the Chebyshev primes. The largest gaps between the Chebyshev primes are {4, 26, 42, 126, 146, 162, 176, 470, 542, 1370, 1516, 4412, 8196, 14928, 27542, 30974, 51588, 62906,…}, [12, Sequence A196672], and the Chebyshev primes that begin a record gap to the next Chebyshev prime are {109, 113, 139, 317, 887, 1327, 1913, 3089, 8297, 11177, 29761, 45707, 113383, 164893, 291377, 401417, 638371, 1045841, …} [12, Sequence A196673].The results for the jumps of the functionli[ψ(p)] are quite analogous to the results for the jumps of the function li[θ(p)] and stated below without proof.Let us define thenth jump at a prime as
(13)Kn=li[ψ(pn+1)]-li[ψ(pn+1-1)]=∫ψ(pn+1-1)ψ(pn+1)dtlogt.Theorem 9.
Forn large, we have
(14)Kn=1+Ω±(log3pn+1pn+1logpn+1).Corollary 10.
There are infinitely many Chebyshev primesChn.One observes that the sequenceChn oscillates around p2n and the largest deviations from p2n seem to be unbounded at large n. This behaviour is illustrated in Figure 1. Based on this numerical results, we are led to the following conjecture.Figure 1
A plot of the functionChn-p2n up to the 105th prime.Conjecture 11.
The functionChn-p2n possesses infinitely many zeros.Comment 2.
The first eleven zeros ofChn-p2n occur at the indices {510, 10271, 11259, 11987, 14730, 18772, 18884, 21845, 24083, 33723, 46789} [12, Sequence A1966674], where the corresponding Chebyshev primes are [12, Sequence A1966675] {164051, 231299, 255919, 274177, 343517, 447827, 450451, 528167, 587519, 847607, 1209469}.Conjecture 12.
The jump at primespn of the function li[ψ(x)] may be written as Kn-1=K~n-1+O(1/pn2), with K~n-1=logpn/log[(ψ(pn)+ψ(pn-1))/2]. In particular, the sign of K~n-1-1 is that of Kn-1.Comment 3.
The jump of indexn-1 (at the prime number pn) is
(15)Kn-1=∫ψ(pn-1)ψ(pn)dtlogt.
There exists a real cn depending on the index n, with ψ(pn-1)<cn<ψ(pn) such that
(16)Kn-1=ψ(pn)-ψ(pn-1)logcn=logpnlogcn.
Using the known locations of the Chebyshev primes of low index, it is straightforward to check that the real cn reads
(17)cn~12[ψ(pn)+ψ(pn-1)].This numerical calculations support our Conjecture12 that the Chebyshev primes may be derived from K~n-1 instead of Kn-1.


### 2.3. The Generalized Chebyshev Primes

Definition 13.
Letp∈𝒫 be an odd prime number and the function jψ(pl)=li[ψ(pl)]-li[ψ(pl-1)], l≥1. The primes p such that jψ(pl)<1/l are called here generalized Chebyshev primesChn(l) (or Chebyshev primes of index l).A short list of Chebyshev primes of index 2 is as follows:{17, 29, 41, 53, 61, 71, 83, 89, 101, 103, 113, 127, 137, 149, 151, 157, 193, 211, 239, 241, …} [12, Sequence A196668].A short list of Chebyshev primes of index3 is as follows:{11, 19, 29, 61, 71, 97, 101, 107, 109, 113, 127, 131, 149, 151, 173, 181, 191, 193, 197, 199, …} [12, Sequence A196669].A short list of Chebyshev primes of index4 is as follows:{5, 7, 17, 19, 31, 37, 41, 43, 53, 59, 67, 73, 79, 83, 101, 103, 107, …} [12, Sequence A196670].Conjecture 14.
The jump at power of primespnl of the function li[ψ(x)] may be written as Kn-1(l)=K~n-1(l)+O(1/pn2l), with K~n-1(l)=logpn/log[(ψ(pnl)+ψ(pnl-1))/2]. In particular the sign of K~n-1(l)-1 is that of Kn-1(l)-1.Comment 4.
Our comment is similar to the comment given in the context of Conjecture12 but refers to the generalized Chebyshev primes Chn(l). To summarize, the jump of the function li[ψ(x)] is accurately defined by a generalized Mangoldt functionΛnnew, that is, K~n-1(l), if n=pl and 0 otherwise, with K~n-1(l) as defined in the present proposition. The sign of the function K~n-1(l)-1/l determines the position of the generalized Chebyshev primes.


## 2.1. The Jumps in the Function li[θ(x)]

Theorem 1 (Robin).
The statementϵθ(x)=li[θ(x)]-π(x)>0 is equivalent to RH [5, 6].Corollary 2 (related to Robin [5]).
The statementϵψ(x)=li[ψ(x)]-π(x)>0 is equivalent to RH.Proof.
If RH is true then, using the fact thatψ(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.Using the fact thatθ(pn+1-1)=θ(pn), define the jump of index n as
(4)Jn=jθ(pn)=li[θ(pn+1)]-li[θ(pn)]=∫θ(pn)θ(pn+1)dtlogt.Proposition 3.
Ifpn+1<1011, then Jn=jθ(pn)>1.Proof.
The integral definition of the jump yields(5)Jn≥θ(pn+1)-θ(pn)logθ(pn+1)=logpn+1logθ(pn+1).
The result now follows after observing that by [7, Theorem 18], we have θ(x)<x for x<108, and by using the note added in proof of [8] that establishes that θ(x)<x for x<1011.By seeing this result it would be natural to make the following conjecture.Conjecture 4.
For alln≥1 we have Jn>1.However, building on Littlewood’s oscillation theorem forθ we can prove that Jn oscillates about 1 with a small amplitude. Let us recall the Littlewood’s oscillation theorem ([9, Theorem 6.3, page 200], [10, Theorem 34])
(6)θ(x)-x=Ω±(x1/2log3x),whenx→∞,
where log3x=logloglogx. The omega notations mean that there are infinitely many numbers x and constants C+ and C-, satisfying
(7)θ(x)≤x-C-xlog3xorθ(x)≥x+C+xlog3x.
We now prepare the proof of the invalidity of Conjecture 4 by writing two lemmas.Lemma 5.
Forn≥1, we have the bounds
(8)logpn+1logθ(pn+1)≤Jn≤logpn+1logθ(pn).Proof.
This is straightforward from the integral definition of the jump.Lemma 6.
Forn large, we have
(9)θ(pn+1)=pn+1+Ω±(pn+1log3pn+1).Proof.
We know that by [9, Theorem 6.3, page 200], we have for x>0 and large
(10)θ(x)-x=Ω±(xlog3x).
The result follows by considering the primes closest to x.We can now state and prove the main result of this section.Theorem 7.
Forn large we have
(11)Jn=1+Ω±(log3pn+1pn+1logpn+1).Proof.
By Lemma6 we know there is a constant C- such that for infinitely many n’s we have
(12)θ(pn+1)<pn+1-C-pn+1log3pn+1.
By combining with the first inequality of Lemma 5 the minus part of the statement follows after some standard asymptotics. To prove the plus part write θ(pn)=θ(pn+1)-logpn+1, and proceed as before.


## 2.2. The Jumps in the Function li[ψ(x)] and the Chebyshev Primes

Definition 8.
Letp∈𝒫 be an odd prime number and the function jψ(p)=li[ψ(p)]-li[ψ(p-1)]. The primes p such that jψ(p)<1 are called here Chebyshev primesChn. (Our terminology should not be confused with that used in [11] where the Chebyshev primes are primes of the form 4n2m+1, with m>0 and n an odd prime. We used the Russian spelling Chebyshev to distinguish both meanings.) In increasing order, they are [12, Sequence A196667]{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, …}.Comment 1.
The number of Chebyshev primes less than10n, (n=1,2,…) is the sequence {0,0,42,516,4498,41423…} [12, Sequence A196671]. This sequence suggests the density (1/2)π(x) for the Chebyshev primes. The largest gaps between the Chebyshev primes are {4, 26, 42, 126, 146, 162, 176, 470, 542, 1370, 1516, 4412, 8196, 14928, 27542, 30974, 51588, 62906,…}, [12, Sequence A196672], and the Chebyshev primes that begin a record gap to the next Chebyshev prime are {109, 113, 139, 317, 887, 1327, 1913, 3089, 8297, 11177, 29761, 45707, 113383, 164893, 291377, 401417, 638371, 1045841, …} [12, Sequence A196673].The results for the jumps of the functionli[ψ(p)] are quite analogous to the results for the jumps of the function li[θ(p)] and stated below without proof.Let us define thenth jump at a prime as
(13)Kn=li[ψ(pn+1)]-li[ψ(pn+1-1)]=∫ψ(pn+1-1)ψ(pn+1)dtlogt.Theorem 9.
Forn large, we have
(14)Kn=1+Ω±(log3pn+1pn+1logpn+1).Corollary 10.
There are infinitely many Chebyshev primesChn.One observes that the sequenceChn oscillates around p2n and the largest deviations from p2n seem to be unbounded at large n. This behaviour is illustrated in Figure 1. Based on this numerical results, we are led to the following conjecture.Figure 1
A plot of the functionChn-p2n up to the 105th prime.Conjecture 11.
The functionChn-p2n possesses infinitely many zeros.Comment 2.
The first eleven zeros ofChn-p2n occur at the indices {510, 10271, 11259, 11987, 14730, 18772, 18884, 21845, 24083, 33723, 46789} [12, Sequence A1966674], where the corresponding Chebyshev primes are [12, Sequence A1966675] {164051, 231299, 255919, 274177, 343517, 447827, 450451, 528167, 587519, 847607, 1209469}.Conjecture 12.
The jump at primespn of the function li[ψ(x)] may be written as Kn-1=K~n-1+O(1/pn2), with K~n-1=logpn/log[(ψ(pn)+ψ(pn-1))/2]. In particular, the sign of K~n-1-1 is that of Kn-1.Comment 3.
The jump of indexn-1 (at the prime number pn) is
(15)Kn-1=∫ψ(pn-1)ψ(pn)dtlogt.
There exists a real cn depending on the index n, with ψ(pn-1)<cn<ψ(pn) such that
(16)Kn-1=ψ(pn)-ψ(pn-1)logcn=logpnlogcn.
Using the known locations of the Chebyshev primes of low index, it is straightforward to check that the real cn reads
(17)cn~12[ψ(pn)+ψ(pn-1)].This numerical calculations support our Conjecture12 that the Chebyshev primes may be derived from K~n-1 instead of Kn-1.


## 2.3. The Generalized Chebyshev Primes

Definition 13.
Letp∈𝒫 be an odd prime number and the function jψ(pl)=li[ψ(pl)]-li[ψ(pl-1)], l≥1. The primes p such that jψ(pl)<1/l are called here generalized Chebyshev primesChn(l) (or Chebyshev primes of index l).A short list of Chebyshev primes of index 2 is as follows:{17, 29, 41, 53, 61, 71, 83, 89, 101, 103, 113, 127, 137, 149, 151, 157, 193, 211, 239, 241, …} [12, Sequence A196668].A short list of Chebyshev primes of index3 is as follows:{11, 19, 29, 61, 71, 97, 101, 107, 109, 113, 127, 131, 149, 151, 173, 181, 191, 193, 197, 199, …} [12, Sequence A196669].A short list of Chebyshev primes of index4 is as follows:{5, 7, 17, 19, 31, 37, 41, 43, 53, 59, 67, 73, 79, 83, 101, 103, 107, …} [12, Sequence A196670].Conjecture 14.
The jump at power of primespnl of the function li[ψ(x)] may be written as Kn-1(l)=K~n-1(l)+O(1/pn2l), with K~n-1(l)=logpn/log[(ψ(pnl)+ψ(pnl-1))/2]. In particular the sign of K~n-1(l)-1 is that of Kn-1(l)-1.Comment 4.
Our comment is similar to the comment given in the context of Conjecture12 but refers to the generalized Chebyshev primes Chn(l). To summarize, the jump of the function li[ψ(x)] is accurately defined by a generalized Mangoldt functionΛnnew, that is, K~n-1(l), if n=pl and 0 otherwise, with K~n-1(l) as defined in the present proposition. The sign of the function K~n-1(l)-1/l determines the position of the generalized Chebyshev primes.


## 3. 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-calledRiemann primes.


### 3.1. The Chebyshev Primes and the Von Mangoldt Explicit Formula

From Corollary10, one observes that the oscillations of the function ψ(x)-x around 0 are intimely related to the existence of Chebyshev primes.Proposition 15.
Ifpn is a Chebyshev prime (of index 1), then ψ(pn)>pn. In the other direction, if ψ(pn-1)>pn, then pn is a Chebyshev prime (of index 1).Proof.
Proposition15 follows from the inequalities (analogous to that of Lemma 5)
(18)logpn+1logψ(pn+1)≤Kn≤logpn+1logψ(pn+1-1).In what regards the position of the (generalized) Chebyshev primes, our numerical experiments lead to the following conjecture.Conjecture 16.
Letψ0(x) be the normalized Chebyshev function. A prime p is a Chebyshev prime of index l iff it satisfies the inequality ψ0(pl)>pl.Comment 5.
It is straigthforward to recover the known sequence of Chebyshev primes (already obtained from Definition8 or the Conjecture 12) from the new Conjecture 16. Thus, Chebyshev primes (of index 1) Chm are those primes pm satisfying ψ0(pm)>pm. Similarly, generalized Chebyshev primes of index l>1 (obtained from Definition 13, or the Conjecture 14) also follow from the Conjecture 16.


### 3.2. Riemann Hypothesis and the Riemann Primes

Under RH, one has the inequality [1]
(19)|ψ(x)-x|=O(x1/2+ϵ0)foreveryϵ0>0,
and alternative upper bounds exist in various ranges of values of x [8]. In the following, we specialize on bounds for ψ(x)-x at power of primes x=pnl.Definition 17.
The champions (left to right maxima) of the function|ψ(pnl)-pnl| are called here Riemann primes of theψtype and index l.Comment 6.
One numerically gets the Riemann primes of theψ type and index 1 [12, Sequence A197185]{2, 59, 73, 97, 109, 113, 199, 283, 463, 467, 661, 1103, 1109, 1123, 1129, 1321, 1327, …},the Riemann primes of the ψ type and index 2 [12, Sequence A197186]{2, 17, 31, 41, 53, 101, 109, 127, 139, 179, 397, 419, 547, 787, 997, 1031, …},the Riemann primes of the ψ type and index 3 [12, Sequence A197187]{2, 3, 5, 7, 11, 13, 17, 29, 59, 67, 97, 103, …},and the Riemann primes of the ψ type and index 4 [12, Sequence A197188]{2, 5, 7, 11, 13, 17, 31, …}.
Clearly, the subset of the Riemann primes of theψ type such that ψ0(pnl)>pnl 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(pnl)-pnl)/pnl. Figure 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 Figure 2.Figure 2
The function(ψ(x)-x)/x at the Riemann primes of the ψ type and indices 1 to 4. Points of index 1 are joined.Under RH, one has the inequality [8, Theorem 10]
(20)|θ(x)-x|<18πxlog2x.
In the following, we specialize on bounds for θ(x)-x at power of primes x=pnl.Definition 18.
The champions (left to right maxima) of the function|θ(pnl)-pnl| are called here Riemann primes of theθtype and index l.Comment 7.
One numerically gets the Riemann primes of theθ type and index 1 [12, Sequence A197297]{2, 5, 7, 11, 17, 29, 37, 41, 53, 59, 97, 127, 137, 149, 191, 223, 307, 331, 337, 347, 419, …},the Riemann primes of the θ type and index 2 [12, Sequence A197298]{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 59, 73, 97, 107, 109, 139, 179, 233, 263, …},the Riemann primes of the θ type and index 3 [12, Sequence A197299]{2, 3, 5, 7, 13, 17, 23, 31, 37, 41, 43, 47, 53, 59, 67, 73, 83, 89, 101, 103, …},and the Riemann primes of the θ type and index 4 [12, Sequence A197300]{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …}.
The Riemann primes of theθ type maximize θ(x)-x. In Figure 3, we plot the ratio s(l)=(θ(pnl)-pnl)/((1/8π)pnllog2pnl) 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 Figure 3.Figure 3
The function(θ(x)-x)/((1/8π)xlog2x) at the Riemann primes of the θ type and indices 1 to 4. Points of index 1 are joined.In the future, it will be useful to approach the proof of RH thanks to the Riemann primes.


## 3.1. The Chebyshev Primes and the Von Mangoldt Explicit Formula

From Corollary10, one observes that the oscillations of the function ψ(x)-x around 0 are intimely related to the existence of Chebyshev primes.Proposition 15.
Ifpn is a Chebyshev prime (of index 1), then ψ(pn)>pn. In the other direction, if ψ(pn-1)>pn, then pn is a Chebyshev prime (of index 1).Proof.
Proposition15 follows from the inequalities (analogous to that of Lemma 5)
(18)logpn+1logψ(pn+1)≤Kn≤logpn+1logψ(pn+1-1).In what regards the position of the (generalized) Chebyshev primes, our numerical experiments lead to the following conjecture.Conjecture 16.
Letψ0(x) be the normalized Chebyshev function. A prime p is a Chebyshev prime of index l iff it satisfies the inequality ψ0(pl)>pl.Comment 5.
It is straigthforward to recover the known sequence of Chebyshev primes (already obtained from Definition8 or the Conjecture 12) from the new Conjecture 16. Thus, Chebyshev primes (of index 1) Chm are those primes pm satisfying ψ0(pm)>pm. Similarly, generalized Chebyshev primes of index l>1 (obtained from Definition 13, or the Conjecture 14) also follow from the Conjecture 16.


## 3.2. Riemann Hypothesis and the Riemann Primes

Under RH, one has the inequality [1]
(19)|ψ(x)-x|=O(x1/2+ϵ0)foreveryϵ0>0,
and alternative upper bounds exist in various ranges of values of x [8]. In the following, we specialize on bounds for ψ(x)-x at power of primes x=pnl.Definition 17.
The champions (left to right maxima) of the function|ψ(pnl)-pnl| are called here Riemann primes of theψtype and index l.Comment 6.
One numerically gets the Riemann primes of theψ type and index 1 [12, Sequence A197185]{2, 59, 73, 97, 109, 113, 199, 283, 463, 467, 661, 1103, 1109, 1123, 1129, 1321, 1327, …},the Riemann primes of the ψ type and index 2 [12, Sequence A197186]{2, 17, 31, 41, 53, 101, 109, 127, 139, 179, 397, 419, 547, 787, 997, 1031, …},the Riemann primes of the ψ type and index 3 [12, Sequence A197187]{2, 3, 5, 7, 11, 13, 17, 29, 59, 67, 97, 103, …},and the Riemann primes of the ψ type and index 4 [12, Sequence A197188]{2, 5, 7, 11, 13, 17, 31, …}.
Clearly, the subset of the Riemann primes of theψ type such that ψ0(pnl)>pnl 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(pnl)-pnl)/pnl. Figure 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 Figure 2.Figure 2
The function(ψ(x)-x)/x at the Riemann primes of the ψ type and indices 1 to 4. Points of index 1 are joined.Under RH, one has the inequality [8, Theorem 10]
(20)|θ(x)-x|<18πxlog2x.
In the following, we specialize on bounds for θ(x)-x at power of primes x=pnl.Definition 18.
The champions (left to right maxima) of the function|θ(pnl)-pnl| are called here Riemann primes of theθtype and index l.Comment 7.
One numerically gets the Riemann primes of theθ type and index 1 [12, Sequence A197297]{2, 5, 7, 11, 17, 29, 37, 41, 53, 59, 97, 127, 137, 149, 191, 223, 307, 331, 337, 347, 419, …},the Riemann primes of the θ type and index 2 [12, Sequence A197298]{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 43, 47, 59, 73, 97, 107, 109, 139, 179, 233, 263, …},the Riemann primes of the θ type and index 3 [12, Sequence A197299]{2, 3, 5, 7, 13, 17, 23, 31, 37, 41, 43, 47, 53, 59, 67, 73, 83, 89, 101, 103, …},and the Riemann primes of the θ type and index 4 [12, Sequence A197300]{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …}.
The Riemann primes of theθ type maximize θ(x)-x. In Figure 3, we plot the ratio s(l)=(θ(pnl)-pnl)/((1/8π)pnllog2pnl) 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 Figure 3.Figure 3
The function(θ(x)-x)/((1/8π)xlog2x) at the Riemann primes of the θ type and indices 1 to 4. Points of index 1 are joined.In the future, it will be useful to approach the proof of RH thanks to the Riemann primes.


## 4. An Efficient Prime Counting Function

In this section, one finds that the Riemann prime counting function [13] R(x)=∑n=1∞(μ(n)/n)li(x1/n)~π(x) may be much improved by replacing it by R[ψ(x)]. One denotes ηN(x)=∑n=1N(μ(n)/n)li[ψ(x)1/n]-π(x), N≥1, the offset in the new prime counting function. Indeed, η1(x)=ϵψ(x).By definition, the negative jumps in the functionηN(x) may only occur at x+1∈𝒫. For N=1, they occur at primes p∈Ch (the Chebyshev primes; see Definition 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 following conjecture.Conjecture 19.
LetηN(x)=∑n=1N(μ(n)/n)li[ψ(x)1/n]-π(x), N>1. Negative jumps of the function ηN(x) occur at all primes x+1∈𝒫 and limp→∞[ηN(p)-ηN(p-1)]=0.More generally, the jumps ofηN(x) at power of primes are described by the following conjecture.Conjecture 20.
LetηN(x) be as in Conjecture 19. Positive jumps of the function ηN(x) occur at all powers of primes x+1=pl, p∈𝒫, and l>1. Moreover, the jumps are such that ηN(pl)-ηN(pl-1)-1/l>0 and limp→∞ηN(pl)-ηN(pl-1)-1/l=0.A sketch of the functionη3(n) (for 2<n<5000) is given in Figure 4. One easily detects the large positive jumps at n=p2 (p∈𝒫), the intermediate positive jumps at n=pl (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].Figure 4
A plot of the functionη3(x).Comment 8.
The arithmetical structure ofηN(x) we have just described leads to |ηN(x)|<ηmax when N≥3. Table 1 represents the maximum value ηmax that is reached and the position xmax of the extremum, for several small values of N and x<105. Thus, the function ∑n=1N(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<104) and an even better improvement when x>104, 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 η3(x) at multiples of 106. See also Figure 4.
It is known thatR(x) converges for any x and may also be written as the Gram series [13] R(x)=1+∑k=1∞((logx)k/k!kζ(k+1)). A similar formula is not established here.Table 1
Upper part of the table: maximum errorηmax in the new prime counting function for x<104 (left-hand part) in comparison to the maximum error using the Riemann prime counting function (right-hand part). Lower part of the table: as above in the range x<105.
N
x
max
η
max
x
max
(
R
-
π
)
max
3
6889
1.118
7450
6.174
4
6889
1.118
7450
6.174
5
1330
-
1.061
9859
-
5.506
6
7
-
0.862
7450
5.879
7
1330
-
0.936
9949
-
5.609
10
7
-
0.884
7450
5.661
50
1330
-
0.885
9949
-
5.557
3
80090
1.840
87888
15.304
10
49727
-
1.158
59797
-
15.729Table 2
Gauss’s and Riemann’s approximation and the approximationη3(x). Compare to table III, page 35 in [1].
x
Planat and Sole´ error
Riemann’s error
Gauss’s error
1,000,000
0.79
30
130
2,000,000
−0.13
−8.0
121
3,000,000
1.83
1.8
121
4,000,000
1.28
35
130
5,000,000
0.36
−62
121
6,000,000
2.91
25
121
7,000,000
0.03
−36
130
8,000,000
2.99
−4.7
121
9,000,000
1.73
−51
121
10,000,000
−0.37
90
339


## 5. Conclusion

This work sheds light on the structure and the distribution of the generalized Chebyshev primesChnl arising from the jumps of the function li[ψ(x)]. It is inspired by Robin’s work [5] relating the sign of the functions ϵθ(x) and ϵψ(x) to RH [5]. Our most puzzling observation is that the nontrivial zeros ρ of the Riemann zeta function are mirrored in the (generalized) Chebyshev primes, whose existence at infinity crucially depends on the Littlewood’s oscillation theorem. In addition, a new accurate prime counting function, based on li[ψ(x)] has been proposed. Future work should concentrate on an effective analytic map between the the zeros ρ and the sequence Chn, in the spirit of our Conjecture 16, and of our approach of RH through the Riemann primes.



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*Source: 491627-2013-03-25.xml*