Search results
Results from the WOW.Com Content Network
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. [1] [2] It is denoted by π(x) (unrelated to the number π). A symmetric variant seen sometimes is π 0 (x), which is equal to π(x) − 1 ⁄ 2 if x is exactly a prime number, and equal to π(x) otherwise.
The first such distribution found is π(N) ~ N / log(N) , where π(N) is the prime-counting function (the number of primes less than or equal to N) and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N).
It is almost certain that Euler meant that the sum of the reciprocals of the primes less than n is asymptotic to log log n as n approaches infinity. It turns out this is indeed the case, and a more precise version of this fact was rigorously proved by Franz Mertens in 1874. [3]
More precisely, they showed that there exist positive constants c and C such that for all sufficiently large numbers N, every even number less than N is the sum of two primes, with at most CN 1 − c exceptions. In particular, the set of even integers that are not the sum of two primes has density zero.
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
Let N be a positive integer, and let k be the number of primes less than or equal to N. Call those primes p 1, ... , p k. Any positive integer a which is less than or equal to N can then be written in the form = (), where each e i is either 0 or 1.
Then the triangle is in Euclidean space if the sum of the reciprocals of p, q, and r equals 1, spherical space if that sum is greater than 1, and hyperbolic space if the sum is less than 1. A harmonic divisor number is a positive integer whose divisors have a harmonic mean that is an integer. The first five of these are 1, 6, 28, 140, and 270.
There exists a natural number N such that every even integer n larger than N is a sum of a prime less than or equal to n 0.95 and a number with at most two prime factors. Tomohiro Yamada claimed a proof of the following explicit version of Chen's theorem in 2015: [ 7 ]