Prime-counting function

Table of π(x), ⁠x/log x ⁠, and li(x)

The table shows how the three functions π(x), ⁠x/log x⁠, and li(x) compared at powers of 10. See also, and

In the On-Line Encyclopedia of Integer Sequences, the π(x) column is sequence OEIS: A006880, π(x) − ⁠x/log x⁠ is sequence OEIS: A057835, and li(x) − π(x) is sequence OEIS: A057752.

The value for π(10²⁴) was originally computed by J. Buethe, J. Franke, A. Jost, and T. Kleinjung assuming the Riemann hypothesis. It was later verified unconditionally in a computation by D. J. Platt. The value for π(10²⁵) is by the same four authors. The value for π(10²⁶) was computed by D. B. Staple. All other prior entries in this table were also verified as part of that work.

The values for 10²⁷, 10²⁸, and 10²⁹ were announced by David Baugh and Kim Walisch in 2015, 2020, and 2022, respectively.

Algorithms for evaluating π(x)

A simple way to find π(x), if x is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to x and then to count them.

A more elaborate way of finding π(x) is due to Legendre (using the inclusion–exclusion principle): given x, if p₁, p₂,…, p_n are distinct prime numbers, then the number of integers less than or equal to x which are divisible by no p_i is

${\displaystyle \lfloor x\rfloor -\sum _i\lfloor \frac{x}{p_i}\rfloor +\sum _{i<j}\lfloor \frac{x}{p_i{p}_j}\rfloor -\sum _{i<j<k}\lfloor \frac{x}{p_i{p}_j{p}_k}\rfloor +\cdots }$

(where ⌊x⌋ denotes the floor function). This number is therefore equal to

${\displaystyle \pi (x)-\pi \left(\sqrt{x}\right)+1}$

when the numbers p₁, p₂,…, p_n are the prime numbers less than or equal to the square root of x.

The Meissel–Lehmer algorithm

In a series of articles published between 1870 and 1885, Ernst Meissel described (and used) a practical combinatorial way of evaluating π(x): Let p₁, p₂,…, p_n be the first n primes and denote by Φ(m,n) the number of natural numbers not greater than m which are divisible by none of the p_i for any i ≤ n. Then

${\displaystyle \mathrm{\Phi }(m,n)=\mathrm{\Phi }(m,n-1)-\mathrm{\Phi }\left(\frac{m}{p_n},n-1\right).}$

Given a natural number m, if n = π(³√m) and if μ = π(√m) − n, then

${\displaystyle \pi (m)=\mathrm{\Phi }(m,n)+n(\mu +1)+\frac{{\mu }^2-\mu }{2}-1-\sum _{k=1}^{\mu }\pi \left(\frac{m}{p_{n+k}}\right).}$

Using this approach, Meissel computed π(x), for x equal to 5×10⁵, 10⁶, 10⁷, and 10⁸.

In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real m and for natural numbers n and k, P_k(m,n) as the number of numbers not greater than m with exactly k prime factors, all greater than p_n. Furthermore, set P₀(m,n) = 1. Then

${\displaystyle \mathrm{\Phi }(m,n)=\sum _{k=0}^{+\mathrm{\infty }}{P}_k(m,n)}$

where the sum actually has only finitely many nonzero terms. Let y denote an integer such that ³√m ≤ y ≤ √m, and set n = π(y). Then P₁(m,n) = π(m) − n and P_k(m,n) = 0 when k ≥ 3. Therefore,

${\displaystyle \pi (m)=\mathrm{\Phi }(m,n)+n-1-P_2(m,n)}$

The computation of P₂(m,n) can be obtained this way:

${\displaystyle P_2(m,n)=\sum _{y<p\le \sqrt{m}}\left(\pi \left(\frac{m}{p}\right)-\pi (p)+1\right)}$

where the sum is over prime numbers.

On the other hand, the computation of Φ(m,n) can be done using the following rules:

1. ${\displaystyle \mathrm{\Phi }(m,0)=\lfloor m\rfloor }$
2. ${\displaystyle \mathrm{\Phi }(m,b)=\mathrm{\Phi }(m,b-1)-\mathrm{\Phi }\left(\frac{m}{p_b},b-1\right)}$

Using his method and an IBM 701, Lehmer was able to compute the correct value of π(10⁹) and missed the correct value of π(10¹⁰) by 1.

Further improvements to this method were made by Lagarias, Miller, Odlyzko, Deléglise, and Rivat.

Other prime-counting functions

Other prime-counting functions are also used because they are more convenient to work with.

Riemann's prime-power counting function

Riemann's prime-power counting function is usually denoted as Π₀(x) or J₀(x). It has jumps of ⁠1/n⁠ at prime powers pⁿ and it takes a value halfway between the two sides at the discontinuities of π(x). That added detail is used because the function may then be defined by an inverse Mellin transform.

Formally, we may define Π₀(x) by

${\displaystyle {\mathrm{\Pi }}_0(x)=\frac{1}{2}\left(\sum _{p^n<x}\frac{1}{n}+\sum _{p^n\le x}\frac{1}{n}\right)}$

where the variable p in each sum ranges over all primes within the specified limits.

We may also write

${\displaystyle {\mathrm{\Pi }}_0(x)=\sum _{n=2}^x\frac{\mathrm{\Lambda }(n)}{\log n}-\frac{\mathrm{\Lambda }(x)}{2\log x}=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}{\pi }_0\left(x^{1/n}\right)}$

where Λ is the von Mangoldt function and

${\displaystyle {\pi }_0(x)=\underset{\epsilon \to 0}{lim}\frac{\pi (x-\epsilon )+\pi (x+\epsilon )}{2}.}$

The Möbius inversion formula then gives

${\displaystyle {\pi }_0(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}{\mathrm{\Pi }}_0\left(x^{1/n}\right),}$

where μ(n) is the Möbius function.

Knowing the relationship between the logarithm of the Riemann zeta function and the von Mangoldt function Λ, and using the Perron formula we have

${\displaystyle \log \zeta (s)=s{\int }_0^{\mathrm{\infty }}{\mathrm{\Pi }}_0(x)x^{-s-1}\mathrm{d}x}$

Chebyshev's function

The Chebyshev function weights primes or prime powers pⁿ by log p:

${\displaystyle \begin{array}{cc}\vartheta (x)& =\sum _{p\le x}\log p\\ \psi (x)& =\sum _{p^n\le x}\log p=\sum _{n=1}^{\mathrm{\infty }}\vartheta \left(x^{1/n}\right)=\sum _{n\le x}\mathrm{\Lambda }(n).\end{array}}$

For x ≥ 2,

${\displaystyle \vartheta (x)=\pi (x)\log x-{\int }_2^x\frac{\pi (t)}{t}\mathrm{d}t}$

and

${\displaystyle \pi (x)=\frac{\vartheta (x)}{\log x}+{\int }_2^x\frac{\vartheta (t)}{t{\log }^2(t)}\mathrm{d}t.}$

Formulas for prime-counting functions

Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas. Analytic formulas for prime-counting were the first used to prove the prime number theorem. They stem from the work of Riemann and von Mangoldt, and are generally known as explicit formulae.

We have the following expression for the second Chebyshev function ψ:

${\displaystyle {\psi }_0(x)=x-\sum _{\rho }\frac{x^{\rho }}{\rho }-\log 2\pi -\frac{1}{2}\log \left(1-x^{-2}\right),}$

where

${\displaystyle {\psi }_0(x)=\underset{\epsilon \to 0}{lim}\frac{\psi (x-\epsilon )+\psi (x+\epsilon )}{2}.}$

Here ρ are the zeros of the Riemann zeta function in the critical strip, where the real part of ρ is between zero and one. The formula is valid for values of x greater than one, which is the region of interest. The sum over the roots is conditionally convergent, and should be taken in order of increasing absolute value of the imaginary part. Note that the same sum over the trivial roots gives the last subtrahend in the formula.

For Π₀(x) we have a more complicated formula

${\displaystyle {\mathrm{\Pi }}_0(x)=\mathrm{li}(x)-\sum _{\rho }\mathrm{li}\left(x^{\rho }\right)-\log 2+{\int }_x^{\mathrm{\infty }}\frac{\mathrm{d}t}{t\left(t^2-1\right)\log t}.}$

Again, the formula is valid for x > 1, while ρ are the nontrivial zeros of the zeta function ordered according to their absolute value. The first term li(x) is the usual logarithmic integral function; the expression li(x^ρ) in the second term should be considered as Ei(ρ log x), where Ei is the analytic continuation of the exponential integral function from negative reals to the complex plane with branch cut along the positive reals. The final integral is equal to the series over the trivial zeros:

${\displaystyle {\int }_x^{\mathrm{\infty }}\frac{\mathrm{d}t}{t\left(t^2-1\right)\log t}={\int }_x^{\mathrm{\infty }}\frac{1}{t\log t}\left(\sum _m{t}^{-2m}\right)\mathrm{d}t=\sum _m{\int }_x^{\mathrm{\infty }}\frac{t^{-2m}}{t\log t}\mathrm{d}t\stackrel{\left(u=t^{-2m}\right)}{=}-\sum _m\mathrm{li}\left(x^{-2m}\right)}$

Applying Möbius inversion to the explicit formula for ${\displaystyle {\mathrm{\Pi }}_0}$ gives

$${\displaystyle {\pi }_0(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}{\mathrm{\Pi }}_0\left(x^{1/n}\right).}$$

Introducing Riemann's ${\displaystyle R}$-function,

$${\displaystyle \mathrm{R}(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}\mathrm{li}\left(x^{1/n}\right),}$$

and formally interchanging the Möbius summation with the sums over the non-trivial and trivial zeros leads to the commonly quoted expression

$${\displaystyle \mathrm{R}(x)-\sum _{\rho }\mathrm{R}(x^{\rho })-\sum _{m=1}^{\mathrm{\infty }}\mathrm{R}(x^{-2m}).}$$

As noted above, this rearrangement would require an additional justification and cannot be a consequence of Möbius inversion alone. In fact, more recently it has shown that, with the standard symmetric ordering of the non-trivial zeros, the partial sums ${\displaystyle \sum _{|\mathrm{Im}\rho |\le T}\mathrm{R}(x^{\rho })}$ are unbounded for every fixed ${\displaystyle x>1}$, and that the corresponding series over the trivial zeros is likewise divergent.

Folkmar Bornemann proved, when assuming the conjecture that all zeros of the Riemann zeta function are simple, that

${\displaystyle \mathrm{R}\left(e^{-2\pi t}\right)=\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k-1}{t}^{-2k-1}}{(2k+1)\zeta (2k+1)}+\frac{1}{2}\sum _{\rho }\frac{t^{-\rho }}{\rho \cos \frac{\pi \rho }{2}{\zeta }^{\prime }(\rho )}}$

where ρ runs over the non-trivial zeros of the Riemann zeta function and t > 0.

The Riemann ${\displaystyle R}$-function itself provides a very good smooth approximation to the prime-counting function. Riesel and Göhl considered the refined approximation

$${\displaystyle g(x)=\mathrm{R}(x)-\frac{1}{\log x}+\frac{1}{\pi }\arctan \frac{\pi }{\log x},}$$

which differs from ${\displaystyle \mathrm{R}(x)}$ by a term tending to zero as ${\displaystyle x\to \mathrm{\infty }}$. The difference between ${\displaystyle {\pi }_0(x)}$ and this smooth approximation is heuristically expected to have size about ${\displaystyle \sqrt{x}/\log x}$. This motivates representing the fluctuations in the distribution of primes by

$${\displaystyle \mathrm{\Delta }(x)=\left({\pi }_0(x)-\mathrm{R}(x)+\frac{1}{\log x}-\frac{1}{\pi }\arctan \frac{\pi }{\log x}\right)\frac{\log x}{\sqrt{x}}.}$$

As explained above, the formally rearranged expression ${\displaystyle \sum _{m=1}^{\mathrm{\infty }}\mathrm{R}(x^{-2m})}$ should not be identified with the smooth correction.

Inequalities

Ramanujan proved that the inequality

${\displaystyle \pi (x{)}^2<\frac{ex}{\log x}\pi \left(\frac{x}{e}\right)}$

holds for all sufficiently large values of x.

Here are some useful inequalities for π(x).

${\displaystyle \frac{x}{\log x}<\pi (x)<1.25506\frac{x}{\log x}\text{for }x\ge 17.}$

The left inequality holds for x ≥ 17 and the right inequality holds for x > 1. The constant 1.25506 is 30⁠log 113/113⁠ to 5 decimal places, as π(x) ⁠log x/x⁠ has its maximum value at x = p₃₀ = 113.

Pierre Dusart proved in 2010:

${\displaystyle \frac{x}{\log x-1}<\pi (x)<\frac{x}{\log x-1.1}\text{for }x\ge 5393\text{ and }x\ge 60184,\text{ respectively.}}$

More recently, Dusart has proved (Theorem 5.1) that

${\displaystyle \frac{x}{\log x}\left(1+\frac{1}{\log x}+\frac{2}{{\log }^{2}x}\right)\le \pi (x)\le \frac{x}{\log x}\left(1+\frac{1}{\log x}+\frac{2}{{\log }^{2}x}+\frac{7.59}{{\log }^{3}x}\right),}$

for x ≥ 88789 and x > 1, respectively.

Going in the other direction, an approximation for the nth prime, p_n, is

${\displaystyle p_n=n\left(\log n+\log \log n-1+\frac{\log \log n-2}{\log n}+O\left(\frac{(\log \log n{)}^2}{(\log n{)}^2}\right)\right).}$

Here are some inequalities for the nth prime. The lower bound is due to Dusart (1999) and the upper bound to Rosser (1941).

${\displaystyle n(\log n+\log \log n-1)<p_n<n(\log n+\log \log n)\text{for }n\ge 6.}$

The left inequality holds for n ≥ 2 and the right inequality holds for n ≥ 6. A variant form sometimes seen substitutes ${\displaystyle \log n+\log \log n=\log (n\log n).}$ An even simpler lower bound is

${\displaystyle n\log n<p_n,}$

which holds for all n ≥ 1, but the lower bound above is tighter for n > e^e ≈15.154.

In 2010 Dusart proved (Propositions 6.7 and 6.6) that

${\displaystyle n\left(\log n+\log \log n-1+\frac{\log \log n-2.1}{\log n}\right)\le p_n\le n\left(\log n+\log \log n-1+\frac{\log \log n-2}{\log n}\right),}$

for n ≥ 3 and n ≥ 688383, respectively.

In 2024, Axler further tightened this (equations 1.12 and 1.13) using bounds of the form

${\displaystyle f(n,g(w))=n\left(\log n+\log \log n-1+\frac{\log \log n-2}{\log n}-\frac{g(\log \log n)}{2{\log }^{2}n}\right)}$

proving that

${\displaystyle f(n,w^2-6w+11.321)\le p_n\le f(n,w^2-6w)}$

for n ≥ 2 and n ≥ 3468, respectively. The lower bound may also be simplified to f(n, w²) without altering its validity. The upper bound may be tightened to f(n, w² − 6w + 10.667) if n ≥ 46254381.

There are additional bounds of varying complexity.

The Riemann hypothesis

The Riemann hypothesis implies a much tighter bound on the error in the estimate for π(x), and hence to a more regular distribution of prime numbers,

${\displaystyle \pi (x)=\mathrm{li}(x)+O(\sqrt{x}\log x).}$

Specifically,

${\displaystyle |\pi (x)-\mathrm{li}(x)|<\frac{\sqrt{x}}{8\pi }\log x,\text{for all }x\ge 2657.}$

Dudek (2015) proved that the Riemann hypothesis implies that for all x ≥ 2 there is a prime p satisfying

${\displaystyle x-\frac{4}{\pi }\sqrt{x}\log x<p\le x.}$