Improving the Estimates for a Sequence Involving Prime Numbers

TL;DR

This paper refines estimates for a sequence involving prime numbers using new explicit bounds for the prime counting function, enhancing accuracy in prime-related calculations.

Contribution

It introduces improved explicit estimates for the sequence C_n based on recent advances in prime counting function bounds.

Findings

Sharper bounds for the sequence C_n involving primes

Enhanced accuracy in prime number sequence estimates

Application of new explicit prime counting estimates

Abstract

Based on new explicit estimates for the prime counting function, we improve the currently known estimates for the particular sequence $C_n = np_n - \sum_{k \leq n}p_k$, $n \geq 1$, involving the prime numbers.

Equations

$$C_n=n{p}_n-\sum _{k\le n}{p}_k$$

$$C_n={\int }_2^{p_n}\pi (x)dx,$$

$$C_n=\sum _{k=1}^{m-1}(k-1)!\left(1-\frac{1}{2^k}\right)\frac{p_n^2}{{\log }^k{p}_n}+O\left(\frac{p_n^2}{{\log }^m{p}_n}\right).$$

$$C_n=\frac{p_n^2}{2\log p_n}+\frac{3p_n^2}{4{\log }^2{p}_n}+\frac{7p_n^2}{4{\log }^3{p}_n}+\chi (n)+O\left(\frac{p_n^2}{{\log }^9{p}_n}\right),$$

$$\chi (n)=\frac{45p_n^2}{8{\log }^4{p}_n}+\frac{93p_n^2}{4{\log }^5{p}_n}+\frac{945p_n^2}{8{\log }^6{p}_n}+\frac{5715p_n^2}{8{\log }^7{p}_n}+\frac{80325p_n^2}{16{\log }^8{p}_n}.$$

$$C_n\ge \frac{p_n^2}{2\log p_n}+\frac{3p_n^2}{4{\log }^2{p}_n}+\frac{7p_n^2}{4{\log }^3{p}_n}+\Theta (n)$$

$$\Theta (n)=\frac{43.6p_n^2}{8{\log }^4{p}_n}+\frac{90.9p_n^2}{4{\log }^5{p}_n}+\frac{927.5p_n^2}{8{\log }^6{p}_n}+\frac{5620.5p_n^2}{8{\log }^7{p}_n}+\frac{79075.5p_n^2}{16{\log }^8{p}_n}$$

$$C_n\le \frac{p_n^2}{2\log p_n}+\frac{3p_n^2}{4{\log }^2{p}_n}+\frac{7p_n^2}{4{\log }^3{p}_n}+\Omega (n)$$

$$\Omega (n)=\frac{46.4p_n^2}{8{\log }^4{p}_n}+\frac{95.1p_n^2}{4{\log }^5{p}_n}+\frac{962.5p_n^2}{8{\log }^6{p}_n}+\frac{5809.5p_n^2}{8{\log }^7{p}_n}+\frac{118848p_n^2}{16{\log }^8{p}_n}.$$

$$C_n\ge \frac{p_n^2}{2\log p_n}+\frac{3p_n^2}{4{\log }^2{p}_n}+\frac{7p_n^2}{4{\log }^3{p}_n}+L(n),$$

$$L(n)=\frac{44.4p_n^2}{8{\log }^4{p}_n}+\frac{92.1p_n^2}{4{\log }^5{p}_n}+\frac{937.5p_n^2}{8{\log }^6{p}_n}+\frac{5674.5p_n^2}{8{\log }^7{p}_n}+\frac{79789.5p_n^2}{16{\log }^8{p}_n}.$$

$$C_n\le \frac{p_n^2}{2\log p_n}+\frac{3p_n^2}{4{\log }^2{p}_n}+\frac{7p_n^2}{4{\log }^3{p}_n}+U(n),$$

$$U(n)=\frac{45.6p_n^2}{8{\log }^4{p}_n}+\frac{93.9p_n^2}{4{\log }^5{p}_n}+\frac{952.5p_n^2}{8{\log }^6{p}_n}+\frac{5755.5p_n^2}{8{\log }^7{p}_n}+\frac{116371p_n^2}{16{\log }^8{p}_n}.$$

$$\pi (x)\sim \text{li}(x)(x\to \infty ),$$

$$\text{li}(x)={\int }_0^x\frac{dt}{\log t}=\underset{\epsilon \to 0}{\lim }\left\{{\int }_0^{1-\epsilon }\frac{dt}{\log t}+{\int }_{1+\epsilon }^x\frac{dt}{\log t}\right\}\approx {\int }_2^x\frac{dt}{\log t}+1.04516\dots .$$

$$\text{li}(x)=\frac{x}{\log x}+\frac{x}{{\log }^{2}x}+\frac{2x}{{\log }^{3}x}+\frac{6x}{{\log }^{4}x}+\frac{24x}{{\log }^{5}x}+\dots +\frac{(m-1)!x}{{\log }^{m}x}+O\left(\frac{x}{{\log }^{m+1}x}\right)$$

$$\pi (x)=\text{li}(x)+O(x\exp (-a\sqrt{\log x})),$$

$$\pi (x)=\frac{x}{\log x}+\frac{x}{{\log }^{2}x}+\frac{2x}{{\log }^{3}x}+\frac{6x}{{\log }^{4}x}+\frac{24x}{{\log }^{5}x}+\dots +\frac{(m-1)!x}{{\log }^{m}x}+O\left(\frac{x}{{\log }^{m+1}x}\right).$$

$$\pi (x)\ge \frac{x}{\log x}+\sum _{k=2}^m\frac{a_{k}x}{{\log }^{k}x}$$

$$\text{li}(x)\ge \sum _{j=1}^{m-1}\frac{(j-1)!x}{{\log }^{j}x}$$

$$C_n\ge d_0+\sum _{k=1}^{m-1}\left(\frac{(k-1)!}{2^k}(1+2t_{k-1,1})\right)\frac{p_n^2}{{\log }^k{p}_n}$$

$$t_{i,j}=(j-1)!\sum _{l=j}^i\frac{2^{l-j}{a}_{l+1}}{l!}.$$

$$d_0=d_0(m,a_2,\dots ,a_m,x_0)={\int }_2^{x_0}\pi (x)dx-(1+2t_{m-1,1})\text{li}(x_0^2)+\sum _{k=1}^{m-1}{t}_{m-1,k}\frac{x_0^2}{{\log }^k{x}_0}.$$

$$C_n\ge d_0+\frac{p_n^2}{2\log p_n}+\frac{3p_n^2}{4{\log }^2{p}_n}+\frac{7p_n^2}{4{\log }^3{p}_n}+L(n)$$

$$d_0$$

$$+\frac{89.4375x_0^2}{{\log }^4{x}_0}+\frac{165.95x_0^2}{{\log }^5{x}_0}+\frac{357.75x_0^2}{{\log }^6{x}_0}+\frac{715.5x_0^2}{{\log }^7{x}_0}.$$

$$\text{li}(x)\le \frac{x}{\log x}+\frac{x}{{\log }^{2}x}+\frac{2x}{{\log }^{3}x}+\frac{6x}{{\log }^{4}x}+\frac{24x}{{\log }^{5}x}+\frac{120x}{{\log }^{6}x}+\frac{900x}{{\log }^{7}x}$$

$$d_0$$

$$-\frac{117.1875x_0^2}{{\log }^6{x}_0}-\frac{1065.515625x_0^2}{{\log }^7{x}_0}.$$

$$d_0\ge {\int }_2^{x_0}\pi (x)dx-8.188366\cdot 10^{18}.$$

$${\int }_2^{x_0}\pi (x)dx=C_{841160647}=8188378036394419009.$$

Full text

1. Introduction

Let $p_{n}$ denotes the $n$th prime number. In this paper, we establish new explicit estimates for the sequence $(C_{n})_{n\geq 1}$ with

[TABLE]

(see [6]). In [1, Theorem 10], the present author used the identity

[TABLE]

where $\pi(x)$ denotes the number of primes not exceeding $x$, to derive that the asymptotic formula

[TABLE]

holds for each positive integer $m$. By setting $m=9$ in (2), we get

[TABLE]

where

[TABLE]

In the direction of (3), the present author [1, Theorem 3 and Theorem 4] showed that

[TABLE]

for every $n\geq 52703656$, where

[TABLE]

and that the upper bound

[TABLE]

holds for every positive integer $n$, where

[TABLE]

Using new explicit estimates for the prime counting function $\pi(x)$, which are found in [2, Proposition 3.6 and Proposition 3.12], we improve the inequalities (4) and (5) by showing the following both results.

Theorem 1.1**.**

For every positive integer $n\geq 440200309$, we have

[TABLE]

where

[TABLE]

Theorem 1.2**.**

For every positive integer $n$, we have

[TABLE]

where

[TABLE]

2. Preliminaries

In 1793, Gauß [4] stated a conjecture concerning an asymptotic magnitude of $\pi(x)$, namely

[TABLE]

where the logarithmic integral $\text{li}(x)$ defined for every real $x\geq 0$ as

[TABLE]

Using the method of integration of parts, (7) implies that

[TABLE]

for every positive integer $m$. The asymptotic formula (6) was proved independently by Hadamard [5] and by de la Vallée-Poussin [7] in 1896, and is known as the Prime Number Theorem. By proving the existence of a zero-free region for the Riemann zeta-function $\zeta(s)$ to the left of the line $\text{Re}(s)=1$, de la Vallée-Poussin [8] was able to estimate the error term in the Prime Number Theorem by

[TABLE]

where $a$ is a positive absolute constant. Together with (8), we obtain that the asymptotic formula

[TABLE]

holds for every positive integer $m$.

3. A proof of Theorem 1.1

Now, we use some recent obtained lower bound for the prime counting function $\pi(x)$ to give a proof of Theorem 1.1.

Proof of Theorem 1.1.

First, let $m$ be a positive integer with $m\geq 2$, and let $a_{2},\ldots,a_{m}$, $x_{0}$, and $y_{0}$ be real numbers, so that

[TABLE]

for every $x\geq x_{0}$ and

[TABLE]

for every $x\geq y_{0}$. The asymptotic formulae (9) and (8) guarantee the existence of such parameters. In [1, Theorem 13], the present author showed that

[TABLE]

for every $n\geq\max\{\pi(x_{0})+1,\pi(\sqrt{y_{0}})+1\}$, where $t_{i,j}$ is defined by

[TABLE]

and $d_{0}$ is given by

[TABLE]

Now, we choose $m=9$, $a_{2}=1$, $a_{3}=2$, $a_{4}=5.85$, $a_{5}=23.85$, $a_{6}=119.25$, $a_{7}=715.5$, $a_{8}=5008.5$, $a_{9}=0$, $x_{0}=19027490297$ and $y_{0}=4171$. By [2, Proposition 3.12], we obtain that the inequality (10) holds for every $x\geq x_{0}$ and (11) holds for every $x\geq y_{0}$ by [1, Lemma 15]. Substituting these values in (12), we get

[TABLE]

for every $n\geq 841160647=\pi(x_{0})$, where $d_{0}=d_{0}(9,1,2,5.85,23.85,119.25,715.5,5008.5,0,x_{0})$ is given by

[TABLE]

The present author [1, Lemma 16] found that

[TABLE]

for every $x\geq 10^{16}$. Applying this inequality to (14), we get

[TABLE]

Computing the right-hand side of the last inequality, we get

[TABLE]

Since $x_{0}=p_{841160647}$, we use (1) and a computer to obtain

[TABLE]

Hence, by (15), we get $d_{0}\geq 1.12\cdot 10^{13}>0$. So we obtain the desired inequality for every $n\geq 841160647$. For every $440200309\leq n\leq 841160646$ we check the inequality with a computer. ∎

4. A proof of Theorem 1.2

Next, we use a recent result concerning an upper bound for the prime counting function $\pi(x)$ to establish the required inequality stated in Theorem 1.2.

Proof of Theorem 1.2.

Let $m$ be a positive integer with $m\geq 2$, let $a_{2},\ldots,a_{m},x_{1}$ be real numbers so that

[TABLE]

for every $x\geq x_{1}$ and let $\lambda,y_{1}$ be real numbers so that

[TABLE]

for every $x\geq y_{1}$. Again, the asymptotic formulae (9) and (8) guarantee the existence of such parameters. The present author [1, Theorem 14] found that the inequality

[TABLE]

holds for every $n\geq\max\{\pi(x_{1})+1,\pi(\sqrt{y_{1}})+1\}$, where $t_{i,j}$ is defined by (13), and

[TABLE]

Next, we choose $m=9$, $a_{2}=1$, $a_{3}=2$, $a_{4}=6.15$, $a_{5}=24.15$, $a_{6}=120.75$, $a_{7}=724.5$,$a_{8}=6601$, $a_{9}=0$, $\lambda=6300$, $x_{1}=13$ and $y_{1}=10^{18}$. By [2, Proposition 3.6], we get that the inequality (16) holds for every $x\geq x_{1}$ and by [1, Lemma 19], that (17) holds for every $y\geq y_{1}$. By substituting these values (18), we get

[TABLE]

for every $n\geq 50847535$, where $d_{1}=d_{1}(9,1,2,6.15,24.15,120.75,724.5,6601,0,x_{1})$ is given by

[TABLE]

A computation shows that $d_{1}\leq 453$. We define

[TABLE]

Since $f(9187322)>0$ and $f^{\prime}(x)\geq 0$ for every $x\geq e^{4}$, we get $f(p_{n})\geq 0$ for every $n\geq\pi(9187322)+1=614124$. Now we can use (19) to obtain the desired inequality for every positive integer $n\geq 50847535$. Finally, we check the remaining cases with a computer. ∎