TITLE

A Heuristic for the Prime Number Theorem

AUTHOR(S)
Montgomery, Hugh L.; Wagon, Stan
PUB. DATE
June 2006
SOURCE
Mathematical Intelligencer;Summer2006, Vol. 28 Issue 3, p6
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
The article discusses a method for the prime number theorem (PNT) in understanding that the growth of primes is governed by natural logarithms. This is through an approach of convincing oneself in a computation that the convex hull satisfies the hypothesis of the theorem and using the simple proof to show that this hypothesis implies the PNT. Therefore, the given formula is asymptotic to an increasing function.
ACCESSION #
22379929

Related Articles

• The exceptional set for the distribution of primes between consecutive powers. Bazzanella, D. // Acta Mathematica Hungarica;Sep2007, Vol. 116 Issue 3, p197

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. This paper is concerned with the...

• Difference sets and shifted primes. Lucier, J. // Acta Mathematica Hungarica;Jul2008, Vol. 120 Issue 1/2, p79

We show that if A is a subset of {1, ..., n} which has no pair of elements whose difference is equal to p âˆ’ 1 with p a prime number, then the size of A is O( n(log log n)âˆ’ c(log log log log log n)) for some absolute c > 0.

• Huaï¿½s theorem with nine almost equal prime variables. L�, G.; Xu, Y. // Acta Mathematica Hungarica;Sep2007, Vol. 116 Issue 4, p309

We sharpen Huaï¿½s result by proving that each sufficiently large odd integer N can be written as , where p j are primes. This result is as good as what was previously derived from the Generalized Riemann Hypothesis.

• Principal Congruence Subgroups of Hecke Groups $$H{\left( {{\sqrt q }} \right)}$$. Özgür, Nihal Yilmaz // Acta Mathematica Sinica;Apr2006, Vol. 22 Issue 2, p383

Using the notion of quadratic reciprocity, we discuss the principal congruence subgroups of the Hecke groups $$H{\left( {{\sqrt q }} \right)}$$ , q > 5 prime number.

• On prime numbers of special kind on short intervals. Mot’kina, N. // Mathematical Notes;May/Jun2006, Vol. 79 Issue 5/6, p848

Suppose that the Riemann hypothesis holds. Suppose that where c is a real number, 1 < c â‰¤ 2. We prove that, for H> N 1/2+10Îµ, Îµ > 0, the following asymptotic formula is valid: .

• A NOTE ON THE SUM OF THE FIRST n PRIMES. MATOMÄKI, KAISA // Quarterly Journal of Mathematics;Mar2010, Vol. 61 Issue 1, p109

We show that the arithmetic mean of the first n primes is an integer for â‰ª N19/24+Îµ numbers n â‰¤ N. This follows from showing that the discrepancy of the sequence consisting of the arithmetic means is â‰ª Nâˆ’5/24+Îµ.

• Small gaps between primes. Sivak-Fischler, J. // Acta Mathematica Hungarica;Sep2007, Vol. 116 Issue 4, p327

Combining Goldston-Yildirimï¿½s method on k-correlations of the truncated von Mangoldt function with Maierï¿½s matrix method, we show that $$\Xi _r : = \lim \inf _{n \to \infty } \tfrac{{p_{n + r} - p_n }} {{\log p_n }} \leqq e^{ - \gamma } \left( {r - \tfrac{{\sqrt r }} {2}} \right)$$...

• On the prime graph of PSL(2, p) where p > 3 is a prime number. Khosravi, Bahman; Khosravi, Behnam; Khosravi, Behrooz // Acta Mathematica Hungarica;Sep2007, Vol. 116 Issue 4, p295

Let G be a finite group. We define the prime graph G( G) as follows. The vertices of G( G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying G( G) = G(...

• Finite Groups with 2pm Elements Having the Highest Order are Solvable. Youyi Jiang; Mingshu Tan // Southeast Asian Bulletin of Mathematics;2004, Vol. 28 Issue 1, p59

Finite groups with 2pm elements having the highest order are studied. It is proved that finite groups with 2pm elements having the highest order are all solvable, where p > 5 is a prime and m is a positive integer.

Share