Updating the error term in the prime number theorem
نویسندگان
چکیده
منابع مشابه
The Prime Number Theorem
The Prime Number Theorem asserts that the number of primes less than or equal to x is approximately equal to x log x for large values of x (here and for the rest of these notes, log denotes the natural logarithm). This quantitative statement about the distribution of primes which was conjectured by several mathematicians (including Gauss) early in the nineteenth century, and was finally proved ...
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In number theory, the prime number theory describes the asymptotic distribution of prime numbers. We all know that there are infinitely many primes,but how are they distributed? Dirichlet’s theorem states that for any two positive coprime integers a and d, there are infinitely many primes which are congruent to a modulo d. A stronger form of Dirichlet’s theorem states that the sum of the recipr...
متن کاملHigher Correlations of Divisor Sums Related to Primes Ii: Variations of the Error Term in the Prime Number Theorem
We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, ...
متن کاملNote on the Prime Number Theorem
Proof. First of all, we prove that if pn is the nth prime number then we have that pn ≤ 2 n−1 . Since there must be some pn+1 dividing the number p1p2 · · · pn− 1 and not exceeding it, it follows from the induction step that pn+1 ≤ 2 0 2 1 · · · 22n−1 = 220+21+···+2n−1 ≤ 22n . If x ≥ 2 is some real number, then we select the largest natural number n satisfying 22n−1 ≤ x, so that we have that 2 ...
متن کاملSimple Proof of the Prime Number Theorem
A form of this was conjectured by Gauss about 1800, [Chebyshev 1848/52] and [Chebyshev 1850/52] made notable progress with essentially elementary methods. The landmark paper Riemann 1859] made clear the intimate connection between prime numbers and the behavior of ζ(s) as a function of a complex variable. The theorem was proven independently by [Hadamard 1896] and [de la Vallée Poussin 1896] by...
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ژورنال
عنوان ژورنال: The Ramanujan Journal
سال: 2015
ISSN: 1382-4090,1572-9303
DOI: 10.1007/s11139-014-9656-6