# On the Distribution Properties of the Smarandache Prime Part

Authors: Yahui Yu; Jiayuan Hu
Journal: Journal of Mathematics
 (2021)

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## Abstract

For each integern, denote by ppn the largest prime ≤n and by Ppn the smallest prime ≥n, called as the Smarandache inferior prime part and superior prime part of n, respectively. Define In≔1/n∑m≤nppm and Sn≔1/n∑m≤nPpm. In this short note, we proved some estimates on In−Sn and In/Sn, answering a question proposed by Kashihara and improving a result of Yan.


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## Body

## 1. Introduction

The properties of primes play a fundamental role in number theory; especially, the distribution of prime numbers in certain sequences has always been a hot topic. It is well known that ifh,k=1, the arithmetic progression(1)kn+h,n=0,1,2,…,contains infinitely many primes. This result is now known as Dirichlet’s theorem. Take another example; for any positive real numbers αandβ, the set(2)Bα,β=nα+β:n∈N,is called the associated Betty sequence (or Beatty set). For irrational α, it follows from a classical result of Ivan Vinogradov that the number πBα,β∼1/απx. In 2020, Janyarak Tongsomporn and Jorn Steuding showed that, for positive real irrational number α of finite type, arbitrary real number β, and every b>2, there exist a positive constant δb>0 and arbitrarily large xandx′ such that(3)πBα,βx+y−πBα,βx>1+δbαylogx,πBα,βx′+y′−πBα,βx′>1+δbαy′logx′,where y=yx=logxb and y′=yx′.In 1993, American-Romanian number theorist Florentin Smarandache published a book namedOnly problems, not solutions! In this book, he presented 105 unsolved arithmetical problems and conjectures about special sequences and functions, which can help us to analyze the properties of primes and the factors of integers. In 2019, Liu Miaohua studied the Smarandache dual function S∗m is defined as n, that n factorial divide m, namely, is(4)S∗m=maxn:n∈N,n!|m.They derived a formula of∏d|nS∗d; that is, for any positive integer m,(5)∏d|nS∗d=1,m=p1β1.….pjβj,pi≥3,2ββ1+1β2+1,…,βj+1,m=2βp1β1,…,pjβj,pi≥3,2ββ2+1β3+1,…,βj+13ββ1β2+1β3+1,…,βj+1,m=2β3β1p1β2,…,pj−1βj,pi≥5,β=1or2,where pi is different odd prime numbers and βi and β are the positive integers.Based on this, we now consider the following Smarandache problems. As usual, letpn be the nth prime. For each integer n, denote by ppn the largest prime ≤n and by Ppn the smallest prime ≥n (see [1]), where ppn and Ppn are called as the Smarandache inferior prime part and superior prime part of n, respectively. According to these definitions, the following are obvious:(a)
For each integern, we have ppn≤n≤Ppn.(b)
The integern is a prime if and only if ppn=Ppn.(c)
We have(6)ppm=pnPpm=pn+1,pn<m<pn+1.Define(7)In≔1n∑m≤nppm,Sn≔1n∑m≤nPpm.In [2], Problem 10, Kashihara proposed two problems about In and Sn:(A)
To determinate whetherlimn⟶∞In−Sn converges or diverges. If it converges, find the limit.(B)
To determinate whetherlimn⟶∞In/Sn converges or diverges. If it converges, find the limit.Yan [3] proved that(8)InSn=1+On−1/3,for n⟶∞, which implies an answer to problem (B):(9)limn⟶∞InSn=1.In this short note, we shall continue a such study. Our aims is double. Firstly, we shall give a complete answer for question (A). Our proof shows that this problem is closely related to the quantityDX≔∑x<pn≤2xpn+1−pn2. According to [4], it is conjectured that D(X)∼X logX as X⟶∞. Our first result shows that limn⟶∞In−Sn diverges. Precisely, we haveThis short note will focus on these two problems. The first result in this paper shows thatlimn⟶∞In−Sn diverges. Precisely, we have the following theorem.Theorem 1.
For anyε>0, there is a positive constant Cε>0 depending on ε such that the following inequalities(10)193192−εlogn−Cε≤In−Sn≤Cεn5/18+ε,hold for n≥1. In particular, we have limn⟶∞In−Sn=∞.The second result in this paper improves Yan’s equality (8) considerably.Theorem 2.
For anyε>0, we have(11)InSn=1+Oεn−13/18+ε,as n⟶∞.For comparison, we have13/18=0.722,… and 1/3=0.333,… .


## 2. Proof of Theorem1

In view of facts (a), (b), and (c) mentioned above, we have(12)In=1n∑m≤nppm=1n∑pk≤npk∑ppm=pk1=1n∑pk≤npkpk−pk−1,(13)Sn=1n∑m≤nPpm=1n∑pk≤npk−1∑ppm=pk−11=1n∑pk≤npk−1pk−pk−1,where we have made the convention that p0=0. From (12) and (13), we can deduce that(14)In−Sn=1n∑pk≤npkpk−pk−1−1n∑pk≤npk−1pk−pk−1=1n∑pk≤npk2+∑pk≤npk−12−2∑pk≤npk−1pk=1n∑pk≤npk−pk−12.According to Theorem1, for any ε>0, we have(15)∑X≤pk≤2Xpk−pk−12≥193192−εXlogX,for all X≥X0ε. Combining (14) and (15), we can derive that(16)In−Sn≥1n∑1≤d≤logn/X0ε/log2∑n/2d<pk≤n/2d−1pk−pk−12≥1n∑1≤d≤logn/X0ε/log2193192−εn2dlogn2d=193192−εMlogn−Elog2,where(17)M≔∑1≤d≤logn/X0ε/log212d,E≔∑1≤d≤logn/X0ε/log2d2d⋅By elementary computation, we have(18)M=1+Oεn−1,E≪1.Then, inserting (18) into (16), we can obtain the first inequality in (10).On the contrary, according to [5], Heath-Brown proved that(19)∑pk≤npk−pk−12≪εn23/18+ε,for all n≥1. The second inequality in (10) follows from (14) and (19).The second assertion is an immediate consequence of (10). This completes the proof of Theorem 1.


## 3. Proof of Theorem2

In view of (14) and (19), we derive that(20)InSn=1+1Snn∑pk≤npk−pk−12=1+On5/18+εSn.On the contrary, formula (13) can be rewritten as(21)Sn≥1n∑pk≤npk−1=1n∑pk≤npk−1n∑pk≤npk−pk−1.By the prime number theorem, we have(22)1n∑pk≤npk≫nlognn≥2.By the Cauchy–Schwarz inequality and (19), it follows that(23)1n∑pk≤npk−pk−1≤1n∑pk≤n1⋅∑pk≤npk−pk−121/2≪ε1nn1+23/18+ε1/2≪εn5/36+ε.From (21)–(23), we deduce that(24)Sn≫nlognn≥2.Inserting this into (20), we obtain(25)InSn=1+Oεn−13/18+ε.This completes the proof.


## 4. Conclusion

The main results of this paper are two theorems involving Smarandache inferior prime partppn and superior prime part Ppn. Theorem 1 establishes an inequality for In−Sn and shows that limn⟶∞In−Sn=∞. Theorem 2 establishes an estimate on In/Sn and obtains that In/Sn=1+Oεn−13/18+ε as n⟶∞. These results represent new contributions to the research on the distribution of the Smarandache inferior prime part and improve the related results before. Of course, our methods can also be generalized to other problems of primes in integer sets. However, there are also many unsolved Smarandache problems to be explored, such as the properties of the determinant of Smarandache prime numbers. These open problems remain to be further studied.



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*Source: 9937647-2021-08-17.xml*