In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number . It is denoted by (unrelated to the number ).
A symmetric variant seen sometimes is , which is equal to if is exactly a prime number, and equal to otherwise. That is, the number of prime numbers less than , plus half if equals a prime.
Growth rate
Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately
where is the natural logarithm, in the sense that
This statement is the prime number theorem. An equivalent statement is
where is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part independently).
More precise estimates In 1899, de la Vallée Poussin proved that See also Theorem 23 of
for some positive constant . Here, is the big notation.
More precise estimates of are now known. For example, in 2002, Kevin Ford proved that
Mossinghoff and Trudgian proved an explicit upper bound for the difference between and :
For values of that are not unreasonably large, is greater than . However, is known to change sign infinitely many times. For a discussion of this, see Skewes' number.
Exact form
For , let when is prime, and let otherwise. Riemann's explicit formula is most naturally stated in terms of the prime-power counting function . In modern notation,
where the sum is over the non-trivial zeros of the Riemann zeta function, taken in the usual symmetric order. Möbius inversion then yields
This is the form used, for example, in treatments of Riemann's prime-number formula such as Riesel and Göhl.
Writing the result instead in the frequently quoted form involving requires an additional interchange between the Möbius summation and the sum over the zeros of the Riemann zeta function. Since the relevant series are not absolutely convergent, this interchange does not follow from Möbius inversion alone and requires a separate justification. Its validity in this form has consequently been questioned in the literature. More recently, a preprint has claimed that, under the standard symmetric ordering of the non-trivial zeros, the corresponding partial sums are in fact unbounded.
Table of , , and
The table shows how the three functions , , and compared at powers of 10. See also, and
{| class="wikitable" style="text-align: right" ! ! ! ! ! ! % error |- | 10 | 4 | 0 | 2 | 2.500 | −8.57% |- | 102 | 25 | 3 | 5 | 4.000 | +13.14% |- | 103 | 168 | 23 | 10 | 5.952 | +13.83% |- | 104 | 1,229 | 143 | 17 | 8.137 | +11.66% |- | 105 | 9,592 | 906 | 38 | 10.425 | +9.45% |- | 106 | 78,498 | 6,116 | 130 | 12.739 | +7.79% |- | 107 | 664,579 | 44,158 | 339 | 15.047 | +6.64% |- | 108 | 5,761,455 | 332,774 | 754 | 17.357 | +5.78% |- | 109 | 50,847,534 | 2,592,592 | 1,701 | 19.667 | +5.10% |- | 1010 | 455,052,511 | 20,758,029 | 3,104 | 21.975 | +4.56% |- | 1011 | 4,118,054,813 | 169,923,159 | 11,588 | 24.283 | +4.13% |- | 1012 | 37,607,912,018 | 1,416,705,193 | 38,263 | 26.590 | +3.77% |- | 1013 | 346,065,536,839 | 11,992,858,452 | 108,971 | 28.896 | +3.47% |- | 1014 | 3,204,941,750,802 | 102,838,308,636 | 314,890 | 31.202 | +3.21% |- | 1015 | 29,844,570,422,669 | 891,604,962,452 | 1,052,619 | 33.507 | +2.99% |- | 1016 | 279,238,341,033,925 | 7,804,289,844,393 | 3,214,632 | 35.812 | +2.79% |- | 1017 | 2,623,557,157,654,233 | 68,883,734,693,928 | 7,956,589 | 38.116 | +2.63% |- | 1018 | 24,739,954,287,740,860 | 612,483,070,893,536 | 21,949,555 | 40.420 | +2.48% |- | 1019 | 234,057,667,276,344,607 | 5,481,624,169,369,961 | 99,877,775 | 42.725 | +2.34% |- | 1020 | 2,220,819,602,560,918,840 | 49,347,193,044,659,702 | 222,744,644 | 45.028 | +2.22% |- | 1021 | 21,127,269,486,018,731,928 | 446,579,871,578,168,707 | 597,394,254 | 47.332 | +2.11% |- | 1022 | 201,467,286,689,315,906,290 | 4,060,704,006,019,620,994 | 1,932,355,208 | 49.636 | +2.02% |- | 1023 | 1,925,320,391,606,803,968,923 | 37,083,513,766,578,631,309 | 7,250,186,216 | 51.939 | +1.93% |- | 1024 | 18,435,599,767,349,200,867,866 | 339,996,354,713,708,049,069 | 17,146,907,278 | 54.243 | +1.84% |- | 1025 | 176,846,309,399,143,769,411,680 | 3,128,516,637,843,038,351,228 | 55,160,980,939 | 56.546 | +1.77% |- | 1026 | 1,699,246,750,872,437,141,327,603 | 28,883,358,936,853,188,823,261 | 155,891,678,121 | 58.850 | +1.70% |- | 1027 | 16,352,460,426,841,680,446,427,399 | 267,479,615,610,131,274,163,365 | 508,666,658,006 | 61.153 | +1.64% |- | 1028 | 157,589,269,275,973,410,412,739,598 | 2,484,097,167,669,186,251,622,127 | 1,427,745,660,374 | 63.456 | +1.58% |- | 1029 | 1,520,698,109,714,272,166,094,258,063 | 23,130,930,737,541,725,917,951,446 | 4,551,193,622,464 | 65.759 | +1.52%
In the On-Line Encyclopedia of Integer Sequences, the column is sequence , is sequence , and is sequence .
The value for 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 is by the same four authors. Includes 600,000 value of for The value for 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
A simple way to find , if is not too large, is to use the sieve of Eratosthenes to produce the primes less than or equal to and then to count them.
A more elaborate way of finding is due to Legendre (using the inclusion–exclusion principle): given , if are distinct prime numbers, then the number of integers less than or equal to which are divisible by no is
(where denotes the floor function). This number is therefore equal to
when the numbers are the prime numbers less than or equal to the square root of .
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 : Let be the first primes and denote by the number of natural numbers not greater than which are divisible by none of the for any . Then
Given a natural number , if and if , then
Using this approach, Meissel computed , for equal to , 106, 107, and 108.
In 1959, Derrick Henry Lehmer extended and simplified Meissel's method. Define, for real and for natural numbers and , as the number of numbers not greater than with exactly prime factors, all greater than . Furthermore, set . Then
where the sum actually has only finitely many nonzero terms. Let denote an integer such that , and set . Then and when . Therefore,
The computation of can be obtained this way:
where the sum is over prime numbers.
On the other hand, the computation of can be done using the following rules:
Using his method and an IBM 701, Lehmer was able to compute the correct value of and missed the correct value of 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 or . It has jumps of at prime powers and it takes a value halfway between the two sides at the discontinuities of . That added detail is used because the function may then be defined by an inverse Mellin transform.
Formally, we may define by
where the variable in each sum ranges over all primes within the specified limits.
We may also write
where is the von Mangoldt function and
The Möbius inversion formula then gives
where is the Möbius function.
Knowing the relationship between the logarithm of the Riemann zeta function and the von Mangoldt function , and using the Perron formula we have
Chebyshev's function The Chebyshev function weights primes or prime powers by :
For ,
and
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 :
where
Here are the zeros of the Riemann zeta function in the critical strip, where the real part of is between zero and one. The formula is valid for values of 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 we have a more complicated formula
Again, the formula is valid for , while are the nontrivial zeros of the zeta function ordered according to their absolute value. The first term is the usual logarithmic integral function; the expression in the second term should be considered as , where 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:
Applying Möbius inversion to the explicit formula for gives
Introducing Riemann's -function,
and formally interchanging the Möbius summation with the sums over the non-trivial and trivial zeros leads to the commonly quoted expression
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 are unbounded for every fixed , 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,Montgomery showed that (assuming the Riemann hypothesis) at least two thirds of all zeros are simple. that
where runs over the non-trivial zeros of the Riemann zeta function and .
The Riemann -function itself provides a very good smooth approximation to the prime-counting function. Riesel and Göhl considered the refined approximation
which differs from by a term tending to zero as . The difference between and this smooth approximation is heuristically expected to have size about . This motivates representing the fluctuations in the distribution of primes by
As explained above, the formally rearranged expression should not be identified with the smooth correction.
Inequalities Ramanujan proved that the inequality
holds for all sufficiently large values of .
Here are some useful inequalities for .
The left inequality holds for and the right inequality holds for . The constant is to 5 decimal places, as has its maximum value at .
Pierre Dusart proved in 2010:
More recently, Dusart has proved (Theorem 5.1) that
for and , respectively.
Going in the other direction, an approximation for the th prime, , is
Here are some inequalities for the th prime. The lower bound is due to Dusart (1999) and the upper bound to Rosser (1941).
The left inequality holds for and the right inequality holds for . A variant form sometimes seen substitutes An even simpler lower bound is
which holds for all , but the lower bound above is tighter for .
In 2010 Dusart proved (Propositions 6.7 and 6.6) that
for and , respectively.