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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

How to Cite This Entry:
Richard Pinch/sandbox-CZ. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-CZ&oldid=30373

@@ Line 1: / Line 1: @@
+=Selberg sieve=
+A technique for estimating the size of "sifted sets" of [[positive integer]]s which satisfy a set of conditions which are expressed by [[Congruence relation#Modular arithmetic|congruence]]s.  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 &le; ''x'' and let ''P'' be a set of primes.  For each ''p'' in ''P'', let ''A''<sub>''p''</sub> denote the set of elements of ''A'' divisible by ''p'' and extend this to let ''A''<sub>''d''</sub> the intersection of the ''A''<sub>''p''</sub> for ''p'' dividing ''d'', when ''d'' is a product of distinct primes from ''P''.  Further let A<sub>1</sub> denote ''A'' itself.  Let ''z'' be a positive real number and ''P''(''z'') denote the product of the primes in ''P'' which are &le; ''z''.  The object of the sieve is to estimate
+:<math>S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . </math>
-=ABC conjecture=
+We assume that |''A''<sub>''d''</sub>| may be estimated by
-In mathematics, the '''ABC conjecture''' relates the prime factors of two integers to those of their sum.  It was proposed by [[David Masser]] and [[Joseph Oesterlé]] in 1985.  It is connected with other problems of [[number theory]]: for example, the truth of the ABC conjecture would provide a new proof of [[Fermat's Last Theorem]].
+:<math> \left\vert A_d \right\vert = \frac{1}{f(d)} X + R_d . </math>
-==Statement==
+where ''f'' is a [[multiplicative function]] and ''X'' &nbsp; = &nbsp; |''A''|.  Let the function ''g'' be obtained from ''f'' by [[Möbius inversion formula|Möbius inversion]], that is
-Define the ''radical'' of an integer to be the product of its distinct prime factors
-:<math> r(n) = \prod_{p|n} p \ . </math>
+:<math> g(n) = \sum_{d \mid n} \mu(d) f(n/d) </math>
+:<math> f(n) = \sum_{d \mid n} g(d) </math>
-Suppose now that the equation <math>A + B + C = 0</math> holds for coprime integers <math>A,B,C</math>.  The conjecture asserts that for every <math>\epsilon > 0</math> there exists <math>\kappa(\epsilon) > 0</math> such that
+where &mu; is the [[Möbius function]].
+Put
-:<math> |A|, |B|, |C| < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . </math>
+:<math> V(z) = \sum_{\begin{smallmatrix}d < z \\ d \mid P(z)\end{smallmatrix}} \frac{\mu^2(d)}{g(d)} . </math>
-A weaker form of the conjecture states that
+Then
-:<math> (|A| \cdot |B| \cdot |C|)^{1/3} < \kappa(\epsilon) r(ABC)^{1+\epsilon} \ . </math>
+:<math> 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) .</math>
-If we define
+It is often useful to estimate ''V''(''z'') by the bound
-:<math> \kappa(\epsilon) = \inf_{A+B+C=0,\ (A,B)=1} \frac{\max\{|A|,|B|,|C|\}}{N^{1+\epsilon}} \ , </math>
+:<math> V(z) \ge \sum_{d \le z} \frac{1}{f(d)} . \, </math>
-then it is known that <math>\kappa \rightarrow \infty</math> as <math>\epsilon \rightarrow 0</math>.
+==Applications==
+* The [[Brun-Titchmarsh theorem]] on the number of primes in an arithmetic progression;
+* The number of ''n'' &le; ''x'' such that ''n'' is coprime to &phi;(''n'') is asymptotic to e<sup>-&gamma;</sup> ''x'' / log log log (''x'') .
-Baker introduced a more refined version of the conjecture in 1998.  Assume as before that <math>A + B + C = 0</math> holds for coprime integers <math>A,B,C</math>.  Let <math>N</math> be the radical of <math>ABC</math> and <math>\omega</math> the number of distinct prime factors of <math>ABC</math>.  Then there is an absolute constant <math>c</math> such that
+==References==
+* {{User:Richard Pinch/sandbox/Ref | author=Alina Carmen Cojocaru | coauthors=M. Ram Murty | title=An introduction to sieve methods and their applications | series=London Mathematical Society Student Texts | volume=66 | publisher=[[Cambridge University Press]] | isbn=0-521-61275-6 | pages=113-134 }}
+* {{User:Richard Pinch/sandbox/Ref | author=George Greaves | title=Sieves in number theory | publisher=[[Springer-Verlag]] | date=2001 | isbn=3-540-41647-1}}
+* {{User:Richard Pinch/sandbox/Ref | author=Heini Halberstam | coauthors=H.E. Richert | title=Sieve Methods | publisher=[[Academic Press]] | date=1974 | isbn=0-12-318250-6}}
+* {{User:Richard Pinch/sandbox/Ref | author= Christopher Hooley | authorlink=Christopher Hooley | title=Applications of sieve methods to the theory of numbers | publisher=Cambridge University Press | date=1976 | isbn=0-521-20915-3| pages=7-12}}
+* {{User:Richard Pinch/sandbox/Ref | author=Atle Selberg | authorlink=Atle Selberg | title=On an elementary method in the theory of primes | journal=Norske Vid. Selsk. Forh. Trondheim | volume=19 | year=1947 | pages=64-67 }}
+=Separation axioms=
+In [[topology]], '''separation axioms''' describe classes of [[topological space]] according to how well the [[open set]]s 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 <math>x \in U \subseteq N</math>.
+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 <math>A \subseteq U \subseteq N</math>.
+Subsets ''U'' and ''V'' are ''separated'' in ''X'' if ''U'' is disjoint from the [[Closure (topology)|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
-:<math> |A|, |B|, |C| < c (\epsilon^{-\omega} N)^{1+\epsilon} \ . </math>
+* '''Hausdorff''' is a synonym for T2
+* '''completely Hausdorff''' is a synonym for T2½
-This form of the conjecture would give very strong bounds in the [[method of linear forms in logarithms]].
+* '''regular''' if T0 and T3
+* '''completely regular''' if T0 and T3½
+* '''Tychonoff''' is completely regular and T1
-==Results==
+* '''normal''' if T0 and T4
-It is known that there is an effectively computable <math>\kappa(\epsilon)</math> such that
+* '''completely normal''' if T1 and T5
+* '''perfectly normal''' if normal and every closed set is a [[G-delta set|G<sub>δ</sub>]]
-:<math> |A|, |B|, |C| < \exp\left({ \kappa(\epsilon) N^{1/3} (\log N)^3 }\right) \ . </math>
+==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==
-* {{User:Richard Pinch/sandbox/Ref | key=1
+* {{User:Richard Pinch/sandbox/Ref | last1=Steen | first1=Lynn Arthur | last2=Seebach | first2=J. Arthur Jr. | title=Counterexamples in Topology | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 }}
- | zbl=1046.11035 | last=Goldfeld | first=Dorian | authorlink=Dorian Goldfeld | chapter=Modular forms, elliptic curves and the abc-conjecture | editor=Wüstholz, Gisbert | title=A panorama in number theory or The view from Baker's garden. | location=Cambridge | publisher=Cambridge University Press | pages=128-147 | year=2002 | isbn=0-521-80799-9 }}
-* {{User:Richard Pinch/sandbox/Ref | key=2
+=Turan sieve=
- | last=Baker | first=Alan | authorlink=Alan Baker | chapter=Logarithmic forms and the $abc$-conjecture | pages=37-44 | editor=Győry, Kálmán (ed.) et al. | title=Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996 | location=Berlin | publisher=de Gruyter | year=1998 | isbn=3-11-015364-5 | zbl=0973.11047 }}
-* {{User:Richard Pinch/sandbox/Ref | key=3
- | last=Stewart | first=C. L. | coauthors=Yu Kunrui | title=On the ''abc'' conjecture. II | journal=Duke Math. J. | volume=108 | number=1 | pages=169-181 | year=2001 | issn=0012-7094 | zbl=1036.11032 }}
-=Albert algebra=
+In [[mathematics]], in the field of [[number theory]], the '''Turán sieve''' is a technique for estimating the size of "sifted sets" of [[positive integer]]s which satisfy a set of conditions which are expressed by [[Congruence relation#Modular arithmetic|congruence]]s.  It was developed by [[Pál Turán]] in 1934.
-The set of 3×3 [[self-adjoint]] matrices over the [[octonion]]s with binary operation
-$$
-x \star y = \frac12 (x \cdot y + y \cdot x) \,,
-$$
-where $\cdot$ denotes matrix multiplication.
-The operation is commutative but not associative.  It is an example of an exceptional [[Jordan algebra]].   Because most other exceptional Jordan algebras are constructed using this one, it is often referred to as "the" exceptional Jordan algebra.
+==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.
-==References==
+Let ''A'' be a set of positive integers &le; ''x'' and let ''P'' be a set of primes.  For each ''p'' in ''P'', let ''A''<sub>''p''</sub> denote the set of elements of ''A'' divisible by ''p'' and extend this to let ''A''<sub>''d''</sub> the intersection of the ''A''<sub>''p''</sub> for ''p'' dividing ''d'', when ''d'' is a product of distinct primes from ''P''.  Further let A<sub>1</sub> denote ''A'' itself.  Let ''z'' be a positive real number and ''P''(''z'') denote the product of the primes in ''P'' which are &le; ''z''.  The object of the sieve is to estimate
-* A. V. Mikhalev, Gunter F. Pilz, "The Concise Handbook of Algebra", (Springer, 2002) ISBN 0792370724, page 346.
-=Alternant code=
+:<math>S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . </math>
-A class of parameterised [[Error detection and correction|error-correcting codes]] which generalise the [[BCH code]]s.
-An ''alternant code'' over $GF(q)$ of length $n$ is defined by a parity check matrix $H$ of [[alternant matrix|alternant]] form $H_{i,j} = \alpha_j^i y_i$, where the $\alpha_j$ are distinct elements of the extension $GF(q^m)$, the $y_i$ are further non-zero parameters again in the extension $GF(q^m)$ and the indices range as $i$ from 0 to $\delta-1$, $j$ from 1 to $n$.
+We assume that |''A''<sub>''d''</sub>| may be estimated, when ''d'' is a prime ''p'' by
-The parameters of this alternant code are length $n$, dimension $\ge n - m\delta$ and minimum distance $\ge \delta+1$.
+:<math> \left\vert A_p \right\vert = \frac{1}{f(p)} X + R_p  </math>
-There exist long alternant codes which meet the [[Gilbert-Varshamov bound]].
-The class of alternant codes includes [[BCH code]]s, [[Goppa code]]s and [[Srivasta code]]s.
+and when ''d'' is a product of two distinct primes ''d'' = ''p'' ''q'' by
-== References ==
+:<math> \left\vert A_{pq} \right\vert = \frac{1}{f(p)f(q)} X + R_{p,q}  </math>
-* {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthors=N.J.A. Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | date=1977 | isbn=0-444-85193-3 | pages=332-338 }}
-=Szpiro's conjecture=
+where ''X'' &nbsp; = &nbsp; |''A''| and ''f'' is a function with the property that 0 &le; ''f''(''d'') &le; 1.  Put
-A conjectural relationship between the [[conductor of an elliptic curve|conductor]] and the [[discriminant of an elliptic curve|discriminant]] of an [[elliptic curve]].  In a general form, it is equivalent to the well-known [[ABC conjecture]].  It is named for [[Lucien Szpiro]] who formulated it in the 1980s.
-The conjecture states that: given &epsilon; &gt; 0, there exists a constant ''C''(&epsilon;) such that for any elliptic curve ''E'' defined over '''Q''' with minimal discriminant &Delta; and conductor ''f'', we have
+:<math> U(z) = \sum_{p \mid P(z)} f(p) . </math>
-:<math> \vert\Delta\vert \leq C(\varepsilon ) \cdot f^{6+\varepsilon }. \, </math>
+Then
-The '''modified Szpiro conjecture''' states that: given &epsilon; &gt; 0, there exists a constant ''C''(&epsilon;) such that for any elliptic curve ''E'' defined over '''Q''' with invariants ''c''<sub>4</sub>, ''c''<sub>6</sub> and conductor ''f'', we have
+:<math> 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 . </math>
-:<math> \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. \, </math>
+==Applications==
+* The [[Hardy–Ramanujan theorem]] that the [[normal order of an arithmetic function|normal order]] of &omega;(''n''), the number of distinct [[prime factor]]s of a number ''n'', is log(log(''n''));
+* Almost all integer polynomials (taken in order of height) are irreducible.
 ==References==
+* {{cite book | author=Alina Carmen Cojocaru | coauthors=M. Ram Murty | title=An introduction to sieve methods and their applications | series=London Mathematical Society Student Texts | volume=66 | publisher=[[Cambridge University Press]] | isbn=0-521-61275-6 | pages=47-62 }}
+=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
-* {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | page=51 | zbl=0869.11051 | edition=Corrected 2nd printing }}
+:<math>f = (X^n + b_{n-1}X^{n-1} + \cdots + b_0) u(x) , \,</math>
-* {{cite journal | author=L. Szpiro | title=Seminaire sur les pinceaux des courbes de genre au moins deux | journal=Astérisque | volume=86 | issue=3 | year=1981 | pages=44-78 | zbl=0463.00009 }}
-* {{cite journal | author=L. Szpiro | title=Présentation de la théorie d'Arakelov | journal=Contemp. Math. | volume=67 | year=1987 | pages=279-293 | zbl=0634.14012  }}
+where the ''b''<sub>''i''</sub> 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==
+* {{User:Richard Pinch/sandbox/Ref | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=208-209 }}
-=Wiener–Ikehara theorem=
-A [[Tauberian theorem]] relating the behaviour of a real sequence to the analytic properties of the associated [[Dirichlet series]].  It is used in the study of [[arithmetic function]]s and yields a proof of the [[Prime number theorem]].  It was proved by [[Norbert Wiener]] and his student [[Shikao Ikehara]] in 1932.
+=Zipf distribution=
+In [[probability theory]] and [[statistics]], the '''Zipf distribution''' and '''zeta distribution''' refer to a class of [[discrete probability distribution]]s.  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.
-Let $F(x)$ be a non-negative, [[monotonic function|monotonic]] decreasing function of the positive [[real number|real]] variable $x$.  Suppose that the [[Laplace transform]]
+The Zipf distribution with parameter ''n'' assigns probability proportional to 1/''r'' to an integer ''r'' &le; ''n'' and zero otherwise, with [[normalization]] factor ''H''<sub>''n''</sub>, the ''n''-th [[harmonic number]].
-$$
-\int_0^\infty F(x)\exp(-xs) dx
-$$
-converges for $\Re s >1$ to the function $f(s)$ and that $f(s)$ is [[analytic function|analytic]] for $\Re s \ge 1$, except for a simple [[pole]] at $s=1$ with residue 1. Then the [[Limit of a function|limit]] as $x$ goes to infinity of $e^{-x} F(x)$ is equal to 1.
-An important number-theoretic application of the theorem is to [[Dirichlet series]] of the form $\sum_{n=1}^\infty a(n) n^{-s}$ where $a(n)$ is non-negative.  If the series converges to an analytic function in $\Re s \ge b$ with a simple pole of residue $c$ at $s = b$, then $\sum_{n\le X}a(n) \sim c \cdot X^b$.
+A Zipf-like distribution with parameters ''n'' and ''s'' assigns probability proportional to 1/''r''<sup>''s''</sup> to an integer ''r'' &le; ''n'' and zero otherwise, with normalization factor <math>\sum_{r=1}^n 1/r^s</math>.
-Applying this to the logarithmic derivative of the [[Riemann zeta function]], where the coefficients in the Dirichlet series are values of the [[von Mangoldt function]], it is possible to deduce the prime number theorem from the fact that the zeta function has no zeroes on the line $\Re (s)=1$.
+The zeta distribution with parameter ''s'' assigns probability proportional to 1/''r''<sup>''s''</sup> to all integers ''r'' with normalization factor given by the [[Riemann zeta function]] 1/ζ(''s'').
 ==References==
-*{{cite journal | author=S. Ikehara | authorlink=Shikao Ikehara | title=An extension of Landau's theorem in the analytic theory of numbers | journal=J. Math. Phys. | year=1931 | volume=10 | pages=1–12 }}
+* {{cite book | author=Michael Woodroofe | coauthors=Bruce Hill | title=On Zipf's law | journal=J. Appl. Probab. | volume=12 | pages=425-434 | year=1975 | id=Zbl 0343.60012 }}
-*{{cite journal | author=N. Wiener | authorlink=Norbert Wiener | title=Tauberian theorems | journal=[[Annals of Mathematics]] | year=1932 | volume=33 | pages=1–100 }}
-* {{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=[[Robert Charles Vaughan (mathematician)|Robert C. Vaughan]] | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=0-521-84903-9 | pages=259–266 }}

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Difference between revisions of "User:Richard Pinch/sandbox-CZ"

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Contents

Selberg sieve

Description

Applications

References

Separation axioms

Terminology

Axioms

Properties

References

Turan sieve

Description

Applications

References

Weierstrass preparation theorem

References

Zipf distribution

References