Counting Primes Rationally And Irrationally

TL;DR

This paper improves lower bounds on the prime counting function using irrationality measures of zeta function values, extending previous results and analyzing rational ratios of zeta functions.

Contribution

It enhances the lower bound estimate of c(x) to c(x) d; it also extends analysis to rational ratios of zeta functions with lower irrationality measures.

Findings

Lower bound of c(x) improved to c(x) \u2265 b1 \, ext{log} \, x

Extension of analysis to rational ratios of zeta functions

Irrationality measures b3(c) d; b3(c) d; b3(c) d; b3(c) d;

Abstract

The recent technique for estimating lower bounds of the prime counting function $\pi(x)=#\{p \leq x: p\text{ prime}\}$ by means of the irrationality measures $\mu(\zeta(s)) \geq 2$ of special values of the zeta function claims that $\pi(x) \gg \log \log x/\log \log \log x$. This note improves the lower bound to $\pi(x) \gg \log x$, and extends the analysis to the irrationality measures $\mu(\zeta(s)) \geq 1$ for rational ratios of zeta functions.

Equations

$$\pi (x)=\#\{p\le x:p\text{ is prime}\}.$$

$$\pi (x)\ge c_1\log x+c_0,$$

$$\frac{1}{\zeta (s)}=\prod _{p\ge 2}\left(1-\frac{1}{p^s}\right)\text{ and }\frac{p_x}{q_x}=\prod _{p\le x}\left(1-\frac{1}{p^s}\right),$$

$$\left|\frac{1}{\zeta (s)}-\frac{p_x}{q_x}\right|=\frac{1}{\zeta (s)}\left(1-\prod _{p>f(x)}{\left(1-\frac{1}{p^s}\right)}^{-1}\right),$$

$$\left|\frac{1}{\zeta (s)}-\frac{p_x}{q_x}\right|$$

$$\frac{1}{q_x^{\mu +\epsilon }}\le \frac{1}{2^{(\pi (x)-s-1)(\mu +\epsilon )}}\le \left|\frac{1}{\zeta (s)}-\frac{p_x}{q_x}\right|,$$

$$\frac{1}{2^{(\pi (x)-s-1)(\mu +\epsilon )}}\le \frac{1}{(s-1)\zeta (s)}\frac{1}{f(x{)}^{s-1}}.$$

$$\frac{s-1}{(\mu +\epsilon )\log 2}\log f(x)+s+1+\frac{\log (s-1)\zeta (s)}{(\mu +\epsilon )\log 2}=c_1\log f(x)+c_0\le \pi (x)$$

$$\pi (x)\ge c_3\log x+c_2,$$

$$\frac{2}{5}=\prod _{p\ge 2}\frac{p^2-1}{p^2+1}\text{ and }\frac{p_x}{q_x}=\prod _{p\le x}\frac{p^2-1}{p^2+1},$$

$$\left|\frac{2}{5}-\frac{p_x}{q_x}\right|=\frac{2}{5}\left(1-\prod _{p>f(x)}{\left(\frac{p^2-1}{p^2+1}\right)}^{-1}\right),$$

$$\prod _{p>f(x)}{\left(\frac{p^2-1}{p^2+1}\right)}^{-1}\ne 0.$$

$$\left|\frac{2}{5}-\frac{p_x}{q_x}\right|$$

$$\frac{c_5}{q_x^{\mu +\epsilon }}\le \frac{1}{2^{(\pi (x)-s-1)(\mu +\epsilon )}}\le \left|\frac{2}{5}-\frac{p_x}{q_x}\right|,$$

$$\frac{1}{2^{(\pi (x)-s-1)(\mu +\epsilon )}}\le \frac{c_4}{f(x)}.$$

$$\frac{1}{(\mu +\epsilon )\log 2}\log f(x)+s+1-\frac{\log c_4}{(1+\epsilon )\log 2}=c_3\log f(x)+c_2\le \pi (x)$$

$$\zeta (s)=\sum _{n\ge 1}\frac{1}{n^s}=\prod _{p\ge 2}{\left(1-\frac{1}{p^s}\right)}^{-1},$$

$$L(s,\chi )=\sum _{n\ge 1}\frac{\chi (n)}{n^s}=\prod _{p\ge 2}{\left(1-\frac{\chi (p)}{p^s}\right)}^{-1},$$

$$\frac{\zeta (2n{)}^2}{\zeta (4n)}=\prod _{p\ge 2}\frac{{\left(1-p^{-2n}\right)}^{-2}}{{\left(1-p^{-2n}\right)}^{-1}}=\prod _{p\ge 2}\left(1+\frac{2}{p^{2n}-1}\right).$$

$$\frac{L(m,\chi {)}^n}{L(mn,\chi )}=\prod _{p\ge 2}\frac{{\left(1-\chi (p)p^{-m}\right)}^{-n}}{{\left(1-\chi (p)p^{-mn}\right)}^{-1}}.$$

$$\frac{\zeta (2n{)}^2}{\zeta (4n)}$$

$$\frac{5}{2}=\frac{\zeta (2{)}^2}{\zeta (4)}=\prod _{p\ge 2}\frac{{\left(1-p^{-2}\right)}^{-2}}{{\left(1-p^{-4}\right)}^{-1}}\prod _{p\ge 2}\frac{p^2+1}{p^2-1}.$$

$$\frac{1}{2}=\frac{L(1,\chi {)}^3}{L(3,\chi )}=\prod _{p\ge 2}\frac{{\left(1-\chi (p)p\right)}^{-3}}{{\left(1-\chi (p)p^{-3}\right)}^{-1}}=\prod _{p\equiv 1\mathrm{mod}4}\frac{p^3-1}{(p-1{)}^3}\prod _{p\equiv 3\mathrm{mod}4}\frac{p^3+1}{(p+1{)}^3}.$$

$$1-\prod _{p>x}{\left(1-\frac{1}{p^s}\right)}^{-1}\le \sum _{n>x}\frac{1}{n^s}=\frac{1}{s-1}\frac{1}{x^{s-1}},$$

$$\zeta (s)-\sum _{n\le x}\frac{1}{n^s}=O\left(\frac{1}{x^{s-1}}\right).$$

$$1-\prod _{p>x}{\left(\frac{p^{2s}-1}{p^{2s}+1}\right)}^{-1}\le \sum _{n>x}\frac{1}{n^{2s}}=\frac{2}{2s-1}\frac{1}{x^{2s-1}}+O\left(\frac{1}{x^{s-1}}\right).$$

$$1-\prod _{p>x}{\left(\frac{p^{2s}-1}{p^{2s}+1}\right)}^{-1}$$

$$\frac{p_x}{q_x}=\prod _{p\le x}\left(1-\frac{1}{p^s}\right),$$

$$A_x=\prod _{p\le x}\left(p^s-1\right)\text{ and }{B}_x=\prod _{p\le x}{p}^s.$$

Full text

Counting Primes Rationally And Irrationally

N. A. Carella

Abstract: The recent technique for estimating lower bounds of the prime counting function $\pi(x)=\#\{p\leq x:p\text{ prime}\}$ by means of the irrationality measures $\mu(\zeta(s))\geq 2$ of special values of the zeta function claims that $\pi(x)\gg\log\log x/\log\log\log x$. This note improves the lower bound to $\pi(x)\gg\log x$, and extends the analysis to the irrationality measures $\mu(\zeta(s))\geq 1$ for rational ratios of zeta functions. ††

AMS MSC: Primary 11N05, Secondary 11A41, 11M26.

Keywords: Distribution of prime; Prime counting function.

1 Introduction

Let $x\geq 1$ be a large number and define the prime counting function by

[TABLE]

Dozens of proofs are known for estimating a lower bound of the prime counting function $\pi(x)=\#\{p\leq x:p\text{ is prime}\}$, see [11, pp. 3-11], [9]. The oldest Euclidean technique has dozens of versions and refinements, and remains a research topic today, see [1] and Section 7 for some information. The recent techniques introduced in [6] and [8] for estimating lower bounds of the prime counting function $\pi(x)=\#\{p\leq x:p\text{ prime}\}$ by means of the irrationality measures $\mu(\zeta(s))\geq 2$ of special values of the zeta function claims that $\pi(x)\gg\log\log x/\log\log\log x$. This note improves the lower bound to $\pi(x)\gg\log x$, and extends the analysis to the irrationality measures $\mu(\zeta(s))\geq 1$ for rational ratio of zeta functions. The proofs are independent of the irrationality measures $\mu(\alpha)\geq 1$ of the real numbers $\alpha\in\mathbb{R}$.

2 For Irrational Values $\zeta(s)$

Theorem 2.1**.**

Let $x\geq 2$ be a large number, and let $\pi(x)=\#\{p\leq x:p\text{ is prime}\}$. Then

[TABLE]

where $c_{0}$ and $c_{1}>0$ are constants.

Proof.

Fix an even number $s>1$. Let $x\gg 1$ be a large number, and let

[TABLE]

where $\gcd(p_{x},q_{x})=1$. Then

[TABLE]

where $f(x)=x$. By Lemma 5.1, the difference has the upper bound

[TABLE]

Let $\mu=\mu(\zeta(s))\geq 2$ be the irrationality measure of the number $\zeta(s)$, see Definition 8.1. Then, applying Lemma 6.1 yields

[TABLE]

where $\varepsilon>0$ is an arbitrary small constant. Comparing (2) and (6) yield

[TABLE]

Solving for $\pi(x)$ yields

[TABLE]

where $c_{0}$, and $c_{1}=(s-1)/(\mu+\varepsilon)\log 2>0$ are constants. ∎

Significant improvement of the lower bound $\pi(x)\gg\log x$ to $\pi(x)\gg x/\log x$ can be achieved by setting $f(x)=2^{cx/\log x+o(x/\log x)}$, for some constant $c>0$. In fact, this can be viewed as a near proof of the Prime Number Theorem by elementary methods. The earlier related analysis appear almost simultaneously in [6], and [8].

3 For Rational Values $\zeta(s)/\zeta(t)$

Theorem 3.1**.**

Let $x\geq 2$ be a large number, and let $\pi(x)=\#\{p\leq x:p\text{ is prime}\}$. Then

[TABLE]

where $c_{2}$ and $c_{3}>0$ are constants.

Proof.

Let $x\gg 1$ be a large number, and let

[TABLE]

where $\gcd(p_{x},q_{x})=1$. Then

[TABLE]

where $f(x)=x$. By the Euclidean Prime Number Theorem, there are infinitely many primes, so the product

[TABLE]

Applying Lemma 4.1 yields

[TABLE]

where $c_{4}>0$ is a constant. Let $\mu=\mu(5/2)=1$ be the irrationality measure of the number $5/2$, see Definition 8.1. Then, applying Lemma 6.2 yields

[TABLE]

where $c_{5}>0$ is a constant. Comparing (3) and (14) yield

[TABLE]

Solving for $\pi(x)$ yields

[TABLE]

where $c_{2}$, and $c_{3}=1/(\mu+\varepsilon)\log 2>0$ are constants.

∎

4 Infinite Products

The evaluations of some infinite prime products have rational values. The best known cases are generated by some ratios of zeta functions, and by some ratios of $L$-functions. The zeta function and the $L$-function are defined by

[TABLE]

and

[TABLE]

where $s\in\mathbb{C}$ is a complex variable, and $\chi$ is a character modulo $q\geq 1$, respectively.

Lemma 4.1**.**

For any integer $n\geq 1$, the following primes products are rational numbers.

1. (i)

For any integer $n\geq 1$,

[TABLE] 2. (ii)

For any odd integers $m,n\geq 1$, and a character $\chi$ modulo $q$,

[TABLE]

Proof.

(i) Let $B_{2n}$ be the $2n$th Bernoulli number. Then, zeta ratio has a rational value

[TABLE]

since any $B_{k}$ is rational. On the other hand, the infinite product has the expression

[TABLE]

as claimed. (ii) The proof of this case has similar calculations but lengthier. ∎

Example 4.1**.**

1. A pair of rational primes products.
2. (1)

The simplest case occurs for $n=2$. The zeta values are $\zeta(2)=\pi^{2}/6$ and $\zeta(4)=\pi^{4}/90$. Thus,

[TABLE] 3. (2)

The simplest case occurs for $m=1$, $n=3$, and the quadratic character $\chi$ modulo $q=4$. The $L$-function values are $L(1,\chi)=\pi/4$ and $L(3)=\pi^{3}/32$. Thus,

[TABLE]

5 Partial Products

Lemma 5.1**.**

Fix a real number $s\geq 1$. Let $x\gg 1$ be a large number. Then,

1. (i)

[TABLE] 2. (ii)

[TABLE]

Proof.

These are standard results, see [12, Theorem 4.11]. ∎

Lemma 5.2**.**

Fix a real number $s\geq 1$. Let $x\gg 1$ be a large number. Then,

[TABLE]

Proof.

Routine calculations yield

[TABLE]

∎

6 Rational Approximations

The denominator of the Euler approximation (27) has a trivial bound $q_{x}\leq(x!)^{2}$. Sharper and more effective descriptions of the rational approximations generated by some Euler products are provided here.

Lemma 6.1**.**

Let $s\geq 1$ be a fixed integer, and let $x\gg 1$ be a large number. Then, the product

[TABLE]

where $\gcd(p_{x},q_{x})=1$, satisfies the followings properties.

1. (i)

The integer $p_{x}\geq 2^{\pi(x)-s-1}$ has exponential growth. 2. (ii)

The integer $q_{x}\geq 2^{\pi(x)-s-1}$ has exponential growth.

Proof.

For a large number $x\gg 1$, consider

[TABLE]

Here, the integer $A_{x}$ is divisible by an increasing high power of $2$ as $x\to\infty$, but the integer $B_{x}$ is divisible by a small fixed power of $2$:

[TABLE]

Thus, the even part of the product can be precisely factored as

[TABLE]

where $A>1$ and $B>1$ are integers such that $\gcd(2,B)=1$, and $\gcd(p_{x},q_{x})=1$.

(i) To verify this statement, observe that in (30), the integer $B$ is odd and that these integers are nearly relatively prime, $1\leq\gcd(A,B)\leq A$. Hence,

[TABLE]

(ii) To verify this statement, observe that $1/2\leq p_{x}/q_{x}\leq 1$. Equivalently,

[TABLE]

Hence,

[TABLE]

These complete the verifications of (i) and (ii). ∎

Lemma 6.2**.**

Let $s\geq 1$ be a fixed integer, and let $x\gg 1$ be a large number. Then, the product

[TABLE]

where $\gcd(p_{x},q_{x})=1$, satisfies the followings properties.

1. (i)

The integer $p_{x}\geq 2^{\pi(x)-s-1}$ has exponential growth. 2. (ii)

The integer $q_{x}\geq 2^{\pi(x)-s-1}$ has exponential growth.

Proof.

For a large number $x\gg 1$, consider

[TABLE]

Here, the integer $A_{x}$ is divisible by an increasing double high power of $2$ as $x\to\infty$, but the integer $B_{x}$ is divisible by a high power of $2$:

[TABLE]

The last expression in (36) follows from $p^{2s}+1\equiv 2\bmod 4$ for $s\in\mathbb{N}$. Thus, the even part of the product can be precisely factored as

[TABLE]

where $A>1$ and $B>1$ are integers such that $\gcd(2,B)=1$, and $\gcd(p_{x},q_{x})=1$.

(i) To verify this statement, observe that in (37), the integer $B$ is odd, (follows from $p^{2s}+1\equiv 2\bmod 4$), and the these integers are nearly relatively prime, $1\leq\gcd(A,B)\leq A$. Hence,

[TABLE]

(ii) To verify this statement, observe that $1/4\leq p_{x}/q_{x}\leq 1$. Equivalently,

[TABLE]

Hence,

[TABLE]

These complete the verifications of (i) and (ii). ∎

7 Euclidean Sequences

The Euclidean sequence $q_{n}=\max\{p\mid n!+1\}$ established the existence of infinitely many primes around 23 centuries ago. The first few terms of the sequence are these:

[TABLE]

The primes are generated in a chaotic manner. Many variations of this sequence are studied in the literature, see [1] and similar references.

The Hermite sequence $q_{n}=\min\{q\mid(p-1)!+1\}$, where $q\geq 2$ ranges over the primes, generates all the primes numbers, and the primes are generated in increasing order. These nice properties follow from Wilson theorem $(p-1)!+1\equiv 0\bmod p$, see [4, p.  303] for more details. The first few terms of the sequence are these:

[TABLE]

However, the opposite Hermite sequence $q_{n}=\max\{q\mid(p-1)!+1\}$ has more complex properties, and generates primes in a chaotic manner.

About 2 centuries ago Euler introduced a new primes counting method based on the prime harmonic sum

[TABLE]

as $x\to\infty$, see [7]. More recently, about a century ago, Hadamard and delaVallee Poussin independently proved using different methods, that the sequence of increasing prime numbers (42) up to a fixed number $x\geq 2$ has

[TABLE]

primes, confer the literature for additional details.

8 Irrationality Measures

The irrationality measure $\mu(\alpha)$ of a real number $\alpha\in\mathbb{R}$ is the infimum of the subset of real numbers $\mu(\alpha)\geq 1$ for which the Diophantine inequality

[TABLE]

has finitely many rational solutions $p$ and $q$.

Definition 8.1**.**

A measure of irrationality $\mu(\alpha)\geq 2$ of an irrational real number $\alpha\in\mathbb{R}^{\times}$ is a map $\psi:\mathbb{N}\rightarrow\mathbb{R}$ such that for any $p,q\in\mathbb{N}$ with $q\geq q_{0}$,

[TABLE]

Furthermore, any measure of irrationality of an irrational real number satisfies $\psi(q)\geq\sqrt{5}q^{\mu(\alpha)}\geq\sqrt{5}q^{2}$. **

The concept of measures of irrationality of real numbers is discussed in [13, p. 556], [2, Chapter 11], et alii.

Lemma 8.1**.**

([3, Theorem 2])* The map $\mu:\mathbb{R}\longrightarrow[2,\infty)\cup\{1\}$ is surjective. Any number in the set $[2,\infty)\cup\{1\}$ is the irrationality measure of some number.*

More precisely,

1. (1)

A rational number has an irrationality measure of $\mu(\alpha)=1$, see [5, Theorem 186]. 2. (2)

An algebraic irrational number has an irrationality measure of $\mu(\alpha)=2$, an introduction to the earlier proofs of Roth Theorem appears in [10, p. 147]. 3. (3)

Any irrational number has an irrationality measure of $\mu(\alpha)\geq 2$. 4. (4)

A Mahler number $\psi_{b}=\sum_{n\geq 1}b^{-[\tau]^{n}}$ in base $b\geq 3$ has an irrationality measure of $\mu(\psi_{b})=\tau$, for any real number $\tau\geq 2$, see [3, Theorem 2].