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Selberg sieve

A technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s.

Description

In terms of sieve theory the Selberg sieve is of combinatorial type: that is, derives from a careful use of the inclusion-exclusion principle. Selberg replaced the values of the Möbius function which arise in this by a system of weights which are then optimised to fit the given problem. The result gives an upper bound for the size of the sifted set.

Let A be a set of positive integers ≤ x and let P be a set of primes. For each p in P, let A_p denote the set of elements of A divisible by p and extend this to let A_d the intersection of the A_p for p dividing d, when d is a product of distinct primes from P. Further let A₁ denote A itself. Let z be a positive real number and P(z) denote the product of the primes in P which are ≤ z. The object of the sieve is to estimate

\[S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . \]

We assume that |A_d| may be estimated by

\[ \left\vert A_d \right\vert = \frac{1}{f(d)} X + R_d . \]

where f is a multiplicative function and X   =   |A|. Let the function g be obtained from f by Möbius inversion, that is

\[ g(n) = \sum_{d \mid n} \mu(d) f(n/d) \] \[ f(n) = \sum_{d \mid n} g(d) \]

where μ is the Möbius function. Put

\[ V(z) = \sum_{\begin{smallmatrix}d < z \\ d \mid P(z)\end{smallmatrix}} \frac{\mu^2(d)}{g(d)} . \]

Then

\[ S(A,P,z) \le \frac{X}{V(z)} + O\left({\sum_{\begin{smallmatrix} d_1,d_2 < z \\ d_1,d_2 \mid P(z)\end{smallmatrix}} \left\vert R_{[d_1,d_2]} \right\vert} \right) .\]

It is often useful to estimate V(z) by the bound

\[ V(z) \ge \sum_{d \le z} \frac{1}{f(d)} . \, \]

Applications

• The Brun-Titchmarsh theorem on the number of primes in an arithmetic progression;
• The number of n ≤ x such that n is coprime to φ(n) is asymptotic to e^-γ x / log log log (x) .

References

• Alina Carmen Cojocaru; M. Ram Murty; An introduction to sieve methods and their applications, ser. London Mathematical Society Student Texts 66 , pp. 113-134, Cambridge University Press ISBN: 0-521-61275-6
• George Greaves; Sieves in number theory, , Springer-Verlag ISBN: 3-540-41647-1
• Heini Halberstam; H.E. Richert; Sieve Methods, , Academic Press ISBN: 0-12-318250-6
• Christopher Hooley; Applications of sieve methods to the theory of numbers, , pp. 7-12, Cambridge University Press ISBN: 0-521-20915-3
• Atle Selberg; On an elementary method in the theory of primes, Norske Vid. Selsk. Forh. Trondheim, 19 (1947), pp. 64-67

Separation axioms

In topology, separation axioms describe classes of topological space according to how well the open sets of the topology distinguish between distinct points.

Terminology

A neighbourhood of a point x in a topological space X is a set N such that x is in the interior of N; that is, there is an open set U such that \(x \in U \subseteq N\). A neighbourhood of a set A in X is a set N such that A is contained in the interior of N; that is, there is an open set U such that \(A \subseteq U \subseteq N\).

Subsets U and V are separated in X if U is disjoint from the closure of V and V is disjoint from the closure of U.

A Urysohn function for subsets A and B of X is a continuous function f from X to the real unit interval such that f is 0 on A and 1 on B.

Axioms

A topological space X is

• T0 if for any two distinct points there is an open set which contains just one
• T1 if for any two points x, y there are open sets U and V such that U contains x but not y, and V contains y but not x
• T2 if any two distinct points have disjoint neighbourhoods
• T2½ if distinct points have disjoint closed neighbourhoods
• T3 if a closed set A and a point x not in A have disjoint neighbourhoods
• T3½ if for any closed set A and point x not in A there is a Urysohn function for A and {x}
• T4 if disjoint closed sets have disjoint neighbourhoods
• T5 if separated sets have disjoint neighbourhoods

• Hausdorff is a synonym for T2
• completely Hausdorff is a synonym for T2½

• regular if T0 and T3
• completely regular if T0 and T3½
• Tychonoff is completely regular and T1

• normal if T0 and T4
• completely normal if T1 and T5
• perfectly normal if normal and every closed set is a G_δ

Properties

• A space is T1 if and only if each point (singleton) forms a closed set.
• Urysohn's Lemma: if A and B are disjoint closed subsets of a T4 space X, there is a Urysohn function for A and B'.

References

• Steen, Lynn Arthur; Seebach, J. Arthur Jr.; Counterexamples in Topology, (1978), Springer-Verlag ISBN: 0-387-90312-7

Turan sieve

In mathematics, in the field of number theory, the Turán sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Pál Turán in 1934.

Description

In terms of sieve theory the Turán sieve is of combinatorial type: deriving from a rudimentary form of the inclusion-exclusion principle. The result gives an upper bound for the size of the sifted set.

Let A be a set of positive integers ≤ x and let P be a set of primes. For each p in P, let A_p denote the set of elements of A divisible by p and extend this to let A_d the intersection of the A_p for p dividing d, when d is a product of distinct primes from P. Further let A₁ denote A itself. Let z be a positive real number and P(z) denote the product of the primes in P which are ≤ z. The object of the sieve is to estimate

\[S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . \]

We assume that |A_d| may be estimated, when d is a prime p by

\[ \left\vert A_p \right\vert = \frac{1}{f(p)} X + R_p \]

and when d is a product of two distinct primes d = p q by

\[ \left\vert A_{pq} \right\vert = \frac{1}{f(p)f(q)} X + R_{p,q} \]

where X   =   |A| and f is a function with the property that 0 ≤ f(d) ≤ 1. Put

\[ U(z) = \sum_{p \mid P(z)} f(p) . \]

Then

\[ S(A,P,z) \le \frac{X}{U(z)} + \frac{2}{U(z)} \sum_{p \mid P(z)} \left\vert R_p \right\vert + \frac{1}{U(z)^2} \sum_{p,q \mid P(z)} \left\vert R_{p,q} \right\vert . \]

Applications

• The Hardy–Ramanujan theorem that the normal order of ω(n), the number of distinct prime factors of a number n, is log(log(n));
• Almost all integer polynomials (taken in order of height) are irreducible.

References

• Template:Cite book

Weierstrass preparation theorem

In algebra, the Weierstrass preparation theorem describes a canonical form for formal power series over a complete local ring.

Let O be a complete local ring and f a formal power series in O''X''. Then f can be written uniquely in the form

\[f = (X^n + b_{n-1}X^{n-1} + \cdots + b_0) u(x) , \,\]

where the b_i are in the maximal ideal m of O and u is a unit of O''X''.

The integer n defined by the theorem is the Weierstrass degree of f.

References

• Serge Lang; Algebra, (1993), pp. 208-209, Addison-Wesley ISBN: 0-201-55540-9

Zipf distribution

In probability theory and statistics, the Zipf distribution and zeta distribution refer to a class of discrete probability distributions. They have been used to model the distribution of words in words in a text , of text strings and keys in databases, and of the sizes of businesses and towns.

The Zipf distribution with parameter n assigns probability proportional to 1/r to an integer r ≤ n and zero otherwise, with normalization factor H_n, the n-th harmonic number.

A Zipf-like distribution with parameters n and s assigns probability proportional to 1/r^s to an integer r ≤ n and zero otherwise, with normalization factor \(\sum_{r=1}^n 1/r^s\).

The zeta distribution with parameter s assigns probability proportional to 1/r^s to all integers r with normalization factor given by the Riemann zeta function 1/ζ(s).

References

• Template:Cite book