Prime Number Theorem

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Theorem

The prime-counting function $\pi \left({n}\right)$, that is, the number of primes less than $n$, satisfies:

$\displaystyle \lim_{n \to \infty} \pi \left({n}\right) \frac {\ln \left({n}\right)} n = 1$

or equivalently:

$\pi \left({n}\right) \sim \dfrac n {\ln \left({n}\right)}$

where $\sim$ denotes asymptotic equivalence.


Proof

The proof presented here is a version of Donald J. Newman's proof. For ease of reading, the proof is broken into parts, with the goal of each part presented.

From the Von Mangoldt Equivalence, the Prime Number Theorem is logically equivalent to:

$\displaystyle \lim_{N \to \infty} \frac 1 N \sum_{n \mathop = 1}^N \Lambda \left({n}\right) = 1$

where $\Lambda$ is the von Mangoldt function.


While useful, the von Mangoldt function is a discrete function that is not very much easier to work with than $\pi \left({n}\right)$ itself.

It behooves us to find another statement equivalent to the Prime Number Theorem.


From Zeta Equivalence to Prime Number Theorem, the Prime Number Theorem is logically equivalent to the statement that:

The average of the first $N$ coefficients of $\dfrac{\zeta'}{\zeta}$ tend to $-1$ as $N$ goes to infinity.


Now we demonstrate the truth of this claim regarding $\dfrac{\zeta'}{\zeta}$.

Doing so proves the Prime Number Theorem.

We know that all of the coefficients of $\zeta$ are precisely $1$.


So the statement:

The average of the first $N$ coefficients of $\frac{\zeta'}{\zeta}$ tend to $1$ as $N$ goes to infinity

is equivalent to the statement:

The average of the first $N$ coefficients of $\frac{\zeta'}{\zeta} - \zeta$ tend to $0$ as $N$ goes to infinity.

The latter will be more convenient for our purposes.

We write:

$\dfrac{\zeta' \left({z}\right)} {\zeta \left({z}\right)} - \zeta \left({z}\right) = \dfrac 1 {\zeta \left({z}\right)} \left({\zeta' \left({z}\right) - \zeta^2 \left({z}\right)}\right)$

From Reciprocal of Riemann Zeta Function, Square of Riemann Zeta Function and Derivative of Riemann Zeta Function:

$\displaystyle \frac 1 {\zeta \left({z}\right)} \left({ \zeta' \left({z}\right) - \zeta^2 \left({z}\right)}\right) = \left({ \sum_{n \mathop = 1}^\infty \frac {\mu \left({n}\right)} {n^z} }\right) \left({ \left({ \sum_{n \mathop = 1}^\infty \frac {\ln \left({n}\right)} {n^z} }\right) - \left({ \sum_{n \mathop = 1}^\infty \frac {d \left({n}\right)} {n^z} }\right) }\right)$

where:

$\mu \left({n}\right)$ is the Möbius function
$d \left({n}\right)$ is the divisor function.

Given this form of the function, we can see that the average of the first $N$ coefficients is:

$\displaystyle \frac 1 N \sum_{a b \mathop \le N} \left({ \mu \left({a}\right) \left({ \ln \left({b}\right) - d \left({b}\right)}\right) }\right)$

Hence the Prime Number Theorem is equivalent to the statement that this expression tends to $0$ as $N \to \infty$.

At this point, we can add:

$0 = 2 \gamma / N - 2 \gamma / N$

where $\gamma$ is the Euler-Mascheroni constant:

$\displaystyle = \frac 1 N \sum_{a b \mathop \le N} \left({ \mu \left({a}\right) \left({ \ln \left({b}\right) - d \left({b}\right)}\right) }\right) + 1 \frac {2 \gamma} N - \frac {2 \gamma} N$

Using a convenient property of the Möbius function, this $1$ is just:

$\displaystyle 1 = \underbrace{ \sum_{a \mathop \backslash 1} \mu \left({a}\right)}_{= 1} + \underbrace{\sum_{a \mathop \backslash 2} \mu \left({a}\right)}_{= 0} + \dots + \underbrace{\sum_{a \mathop \backslash N} \mu \left({a}\right)}_{= 0}$
$\displaystyle = \sum_{n \mathop = 1}^N \left({ \sum_{a \mathop \backslash n} \mu \left({a}\right)}\right)$

and so:

$\displaystyle = \frac 1 N \sum_{a b \mathop \le N} \left({ \mu \left({a}\right) \left({ \ln \left({b}\right) - d( \left({b}\right)}\right) }\right) + 1 \frac{2 \gamma} N - \frac{2 \gamma} N = \frac 1 N \sum_{a b \mathop \le N} \left({ \mu \left({a}\right) \left({ \ln \left({b}\right) - d \left({b}\right)}\right) }\right) + \frac 1 N \sum_{n \mathop = 1}^N \left({ \sum_{a \mathop \backslash n} \mu \left({a}\right) 2 \gamma }\right) - \frac {2 \gamma} N$
$\displaystyle = \frac 1 N \sum_{ab \mathop \le N} \left({ \mu \left({a}\right) \left({ \ln \left({b}\right) - d \left({b}\right) + 2 \gamma }\right) }\right) - \frac {2 \gamma} N$


By Order of Divisor Function:

$\displaystyle = \frac 1 N \sum_{a \mathop \le N} \mu \left({a}\right) O \left({-\sqrt N}\right) - \frac {2 \gamma} N$

and by Order of Möbius Function:

$\displaystyle = \frac 1 N o \left({N}\right) O \left({-\sqrt N}\right) - \frac {2 \gamma} N = O \left({\frac{-1}{\sqrt N}}\right) o \left({N}\right) - \frac {2 \gamma} N$

As $N \to \infty$, we have

$\displaystyle \lim_{N \to \infty} \left({ O \left({ \frac{-1}{\sqrt N}}\right) o \left({N}\right) - \frac {2 \gamma} N}\right)$

which clearly goes to $0$ as $O \left({\dfrac{-1}{\sqrt N}}\right)$ dominates $o \left({N}\right)$.



Interpretation

The Prime Number Theorem can also be rendered as:

$\displaystyle \lim_{x \mathop \to \infty} \dfrac {\pi \left({x}\right) / x} {1 / \ln x} = 1$

where it can be interpreted as the probability that a number chosen at random will be prime.

Thus, for large $n$, that probability is approximately equal to $\dfrac 1 {\ln n}$.


Historical Note

The Prime Number Theorem (PNT) was first conjectured by Carl Friedrich Gauss when he was $14$ or $15$, but he was never able to prove it.

He also posited the suggestion that it could be approximated by the Eulerian logarithmic integral $\displaystyle \operatorname {Li} \left({x}\right) = \int_2^x \frac {\mathrm d t}{\ln \left({t}\right)}$.


It took another century before a proof was found.


Pafnuty Lvovich Chebyshev was the first one to provide any support for Gauss's conjecture when he proved in $1850$ that:

$\dfrac 7 8 < \dfrac {\pi \left({x}\right)} {x / \ln x} < \dfrac 9 8$

for all sufficiently large $x$.

He also proved that if the limit of the expression in question does exist, then its value must be $1$.


Since then, several complete proofs have been discovered.


The first proofs were given independently by Jacques Salomon Hadamard and Charles de la Vallée Poussin in $1896$.

They relied on the theory of functions of a complex variable.

The original theorem of Hadamard used in that proof is given on $\mathsf{Pr} \infty \mathsf{fWiki}$ as Ingham's Theorem on Convergent Dirichlet Series, which is used in Order of Möbius Function, an essential part of the above proof.


Atle Selberg and Paul Erdős would later give an elementary proof of the PNT, in $1948$.

Their proof did not make use of any analytic function theory, and relied entirely on basic properties of logarithms.

Dispute over whether to publish their results jointly or separately created a life-long feud between the two mathematicians.



While the PNT states that $\pi \left({n}\right)$ is asymptotic to $\dfrac n {\ln \left({n}\right)}$, it states nothing about the error of this approximation beyond:

$\pi \left({n}\right) - \dfrac n {\ln \left({n}\right)} = O \left({ n^\alpha }\right)$

where $\alpha < 1$.

This follows from the PNT because if the error was of the form $O \left({ n^\alpha }\right), \alpha \ge 1$, the PNT would be false.

Investigations into the error of the approximation to $\pi \left({n}\right)$ continue, most notably with ongoing research on the Riemann Hypothesis.

If the hypothesis is true, then the error is:

$\pi \left({n}\right) - \dfrac n {\ln \left({n}\right)} = O \left({\sqrt n}\log n\right)$

If the hypothesis is false, the error is: $O \left({ n^\alpha }\right)$ for some $\alpha > \dfrac 1 2$