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* Marc-Antoine Coppo, Bernard Candelpergher "The Arakawa-Kaneko zeta function" ''Ramanujan J. '' '''22''' (2010) 153-162 {{ZBL|1230.11106}}
 
* Marc-Antoine Coppo, Bernard Candelpergher "The Arakawa-Kaneko zeta function" ''Ramanujan J. '' '''22''' (2010) 153-162 {{ZBL|1230.11106}}
  
=Average order of an arithmetic function=
+
=Genus of a quadratic form=
Some simpler or better-understood function which takes the same values "on average" as an arithmetic function.
+
A classification of quadratic forms and lattices over the ring of integers.
  
Let ''f'' be a function on the [[natural number]]sWe say that ''f'' has average order ''g'' if  
+
An [[integral quadratic form]] is a quadratic form on $\mathbb{Z}^n$, or more generally a free $\mathbb{Z}$-module of finite rankTwo such forms are in the same ''genus'' if they are equivalent over the local rings $\mathbb{Z}_p$ for each prime $p$ and also equivalent over $\mathbb{R}$.
  
:<math> \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) </math>
+
Equivalent forms are in the same genus, but the converse does not hold.  For example, $X^2 + 82Y^2$ and $2X^2 + 41Y^2$ are in the same genus but not equivalent over $\mathbb{Z}$.
  
as ''x'' tends to infinity.
+
Forms in the same genus have equal [[Determinant of a quadratic form|determinant]] and hence there are only finitely many equivalence classes in a genus.
  
It is conventional to assume that the approximating function ''g'' is [[Continuous function|continuous]] and [[Monotonic function|monotone]].
+
The [[Smith–Minkowski–Siegel mass formula]] gives the ''weight'' or ''mass'' of the quadratic forms in a genus, the count of equivalence classes weighted by the reciprocals of the orders of their automorphism groups.
  
==Examples==
+
==Binary quadratic forms==
* The average order of $d(n)$, the number of divisors of $n$, is $\log n$;
+
For [[binary quadratic form]]s there is a group structure on the set $C(D)$ equivalence classes of forms with given [[Discriminant of a quadratic form|discriminant]] $D$.  The genera are defined by the ''generic characters''.  The principal genus, the genus containing the principal form, is precisely the subgroup $C(D)^2$ and the genera are the cosets of $C(D)^2$: so in this case all genera contain the same number of classes of forms.
* The average order of $\sigma(n)$, the sum of divisors of $n$, is $ \frac{\pi^2}{6} n$;
 
* The average order of $\phi(n)$, [[Euler's totient function]] of $n$, is $ \frac{6}{\pi^2} n$;
 
* The average order of $r(n)$, the number of ways of expressing $n$ as a [[sum of two squares]], is $\pi$;
 
* The [[Prime Number Theorem]] is equivalent to the statement that the [[von Mangoldt function]] $\Lambda(n)$ has average order 1.
 
  
 
==See also==
 
==See also==
* [[Divisor function]]
+
* [[Spinor genus]]
* [[Normal order of an arithmetic function]]  
 
  
 
==References==
 
==References==
* G.H. Hardy; E.M. Wright (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. ISBN 0-19-921986-5
+
* J.W.S. Cassels, ''Rational Quadratic Forms'', London Mathematical Society Monographs '''13''', Academic Press (1978) ISBN 0-12-163260-1 {{ZBL|0395.10029}}
* Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics 46. Cambridge University Press. ISBN 0-521-41261-7
 
  
[[:Category:Arithmetic functions]]
+
=Gowers norm=
  
 
+
''uniformity norm''
=Gowers norm=
 
  
 
''For the function field norm, see [[uniform norm]]; for uniformity in topology, see [[uniform space]].''
 
''For the function field norm, see [[uniform norm]]; for uniformity in topology, see [[uniform space]].''
  
In mathematics, in the field of [[additive combinatorics]], a '''Gowers norm''' or '''uniformity norm''' is a class of [[Norm (mathematics)|norm]] on functions on a finite [[Group (mathematics)|group]] or group-like object which are used in the study of arithmetic progressions in the group.  They are named after [[Timothy Gowers]] who introduced them in his work on [[Szemerédi's theorem]].
+
A class of [[Norm (mathematics)|norm]] on functions on a finite [[Group (mathematics)|group]] or group-like object which are used in the study of arithmetic progressions in the group.  They are named after Timothy Gowers who introduced them in his work on [[Szemerédi's theorem]].
 
 
Let ''f'' be a complex-valued function on a group ''G'' and let ''J'' denote complex conjugation.  The Gowers ''d''-norm is
 
 
 
:<math> \Vert f \Vert_{U^d(g)} = \mathbf{E}_{x,h_1,\ldots,h_d \in G} \prod_{\omega_1,\ldots,\omega_d \in \{0,1\}} J^{\omega_1+\cdots+\omega_d} f\left({x + h_1\omega_1 + \cdots + h_d\omega_d}\right) \ . </math>
 
  
The ''inverse conjecture'' for these norms is the statement that if ''f'' has [[L-infinity norm]] ([[uniform norm]] in the usual sense) equal to 1 then the Gowers ''s''-norm is bounded above by 1, with equality if and only if ''f'' is of the form exp(2πi ''g'') with ''g'' a polynomial of degree at most ''s''.  This can be interpreted as saying that the Gowers norm is controlled by polynomial phases.
+
Let $f$ be a complex-valued function on a group $G$ and let $J$ denote complex conjugation.  The Gowers $d$-norm is
 +
$$
 +
\Vert f \Vert_{U^d(g)} = \mathbf{E}_{x,h_1,\ldots,h_d \in G} \prod_{\omega_1,\ldots,\omega_d \in \{0,1\}} J^{\omega_1+\cdots+\omega_d} f\left({x + h_1\omega_1 + \cdots + h_d\omega_d}\right) \ .
 +
$$
 +
The ''inverse conjecture'' for these norms is the statement that if $f$ has [[L-infinity norm]] ([[uniform norm]] in the usual sense) equal to 1 then the Gowers $s$-norm is bounded above by 1, with equality if and only if $f$ is of the form $\exp(2\pi i g)$ with$g$ a polynomial of degree at most $s$.  This can be interpreted as saying that the Gowers norm is controlled by polynomial phases.
  
The inverse conjecture holds for vector spaces over a finite field.  However, for cyclic groups '''Z'''/''N'' this is not so, and the class of polynomial phases has to be extended to control the norm.
+
The inverse conjecture holds for vector spaces over a finite field.  However, for [[cyclic group]]s $C_N$ this is not so, and the class of polynomial phases has to be extended to control the norm.
  
 
==References==
 
==References==
* {{User:Richard Pinch/sandbox/Ref | zbl=pre06110460 | last=Tao | first=Terence | authorlink=Terence Tao | title=Higher order Fourier analysis | series=Graduate Studies in Mathematics | volume=142 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2012 | isbn=978-0-8218-8986-2 | url=http://terrytao.wordpress.com/books/higher-order-fourier-analysis/ }}
+
* Terence Tao, "Higher order Fourier analysis", Graduate Studies in Mathematics '''142''' American Mathematical Society (2012) ISBN 978-0-8218-8986-2 {{ZBL|1277.11010}}
  
=Ruelle zeta function=
+
=Height zeta function=
A [[Zeta-function|zeta function]] associated with a [[dynamical system]].
+
''of a set of points''
 +
A function encoding the distribution of points of given height on an [[algebraic variety]] or a subset.
  
Let ''f'' be a function defined on a [[manifold]] ''M'', such that the set of [[fixed point (mathematics)|fixed points]] Fix(''f''<sup>&nbsp;''n''</sup>) is finite for all ''n''&nbsp;>&nbsp;1.  Further let φ be a function on ''M'' with values in ''d''&nbsp;×&nbsp;''d'' complex matrices.  The zeta function of the first kind is<ref name=T28>Terras (2010) p.&nbsp;28</ref>
+
If $S$ is a set with [[Height, in Diophantine geometry|height function]] $H$, such that there are only finitely many elements of bounded height, define a ''counting function''
 +
$$
 +
N(S,H,B) = \sharp \{ x \in S : H(x) \le B \}
 +
$$
 +
and a ''zeta function''
 +
$$
 +
Z(S,H;s) = \sum_{x \in S} H(x)^{-s} \ .
 +
$$
  
:<math> \zeta(z) = \exp\left({
+
If $Z$ has [[abscissa of convergence]] $\beta$ and there is a constant $c$ such that $N$ has rate of growth
                              \sum_{m\ge1} \frac{z^m}{m} \sum_{x\in\mathrm{Fix}(f^m)}
+
$$
                              \mathrm{Tr}
+
N \sim c B^a (\log B)^{t-1}  
                                    \left({ \prod_{k=0}^{m-1} \phi(f^k(x))
+
$$
                                          }\right)
+
then a version of the [[Wiener–Ikehara theorem]] holds: $Z$ has a $t$-fold pole at $s = \beta$ with residue $c a \Gamma(t)$.
                            }\right) </math>
 
  
In the special case ''d''&nbsp;=&nbsp;1, φ&nbsp;=&nbsp;1, we have<ref name=T28/>
+
The abscissa of convergence has similar formal properties to the [[Nevanlinna invariant]] and it is conjectured that they are essentially the same. More precisely, Batyrev–Manin conjectured the following.  Let $X$ be a projective variety over a number field $K$ with [[ample divisor]] $D$ giving rise to an embedding and height function $H$, and let $U$ denote a Zariski-open subset of $X$'. Let $\alpha = \alpha(D)$ be the Nevanlinna invariant of $D$ and $\beta$ the abscissa of convergence of  $Z(U,H,s)$. Then for every $\epsilon > 0$ there is a $U$ such that $\beta < \alpha + \epsilon$: in the opposite direction, if $\alpha > 0$ then $\alpha = \beta$ for all sufficiently large fields $K$ and sufficiently small $U$.
 
 
:<math> \zeta(z) = \exp\left({ \sum_{m\ge1} \frac{z^m}{m} \left|{\mathrm{Fix}(f^m)}\right|}\right) </math>
 
 
 
which is the [[Artin–Mazur zeta function]].
 
 
 
The [[Ihara zeta function]] is an example of a Ruelle zeta function.<ref name=T29>Terras (2010) p.&nbsp;29</ref>
 
  
 
==References==
 
==References==
<references/>
+
* {{User:Richard Pinch/sandbox/Ref | zbl=0679.14008 | last1=Batyrev | first1=V.V. | last2=Manin | first2=Yu.I. | author2-link=Yuri I. Manin | title=On the number of rational points of bounded height on algebraic varieties | journal=Math. Ann. | volume=286, | pages=27–43 | year=1990 }}
* {{User:Richard Pinch/sandbox/Ref | last1=Lapidus | first1=Michel L. | last2=van Frankenhuijsen | first2=Machiel | title=Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings | series=Springer Monographs in Mathematics | location=New York, NY | publisher=[[Springer-Verlag]] | year=2006 | isbn=0-387-33285-5 | zbl=1119.28005 }}
+
* {{User:Richard Pinch/sandbox/Ref | first1=Marc | last1=Hindry | first2=Joseph H. | last2=Silverman | authorlink2=Joseph H. Silverman | title=Diophantine Geometry: An Introduction | series=Graduate Texts in Mathematics | volume=201 | year=2000 | isbn=0-387-98981-1 | zbl=0948.11023 }}
* {{User:Richard Pinch/sandbox/Ref | title=Zeta Functions of Graphs: A Stroll through the Garden | volume=128 | series=Cambridge Studies in Advanced Mathematics | first=Audrey | last=Terras | authorlink=Audrey Terras | publisher=[[Cambridge University Press]] | year=2010 | isbn=0-521-11367-9 | zbl=1206.05003 }}
+
* {{User:Richard Pinch/sandbox/Ref | first=Serge | last=Lang | authorlink=Serge Lang | title=Survey of Diophantine Geometry | publisher=Springer-Verlag | year=1997 | isbn=3-540-61223-8 | zbl=0869.11051 }}
  
=Sesquipower=
+
=Nevanlinna invariant=
A '''sesquipower''' or '''Zimin word''' is a string over an alphabet with identical prefix and suffix.  Sesquipowers are [[unavoidable pattern]]s, in the sense that all sufficiently long strings contain one.
+
''of an ample divisor on a normal projective variety''
  
Formally, let ''A'' be an alphabet and ''A''<sup>&lowast;</sup> be the [[free monoid]] of finite strings over&nbsp;''A''.  Every non-empty word ''w'' in ''A''<sup>+</sup> is a sesquipower of order&nbsp;1. If ''u'' is a sequipower of order ''n'' then any word ''w'' = ''uvu'' is a sesquipower of order ''n''&nbsp;+&nbsp;1.<ref name=LotII135>Lothaire (2011) p.&nbsp;135</ref> The ''degree'' of a non-empty word ''w'' is the largest integer ''d'' such that ''w'' is a sesquipower of order ''d''.<ref name=LotII136>Lothaire (2011) p.&nbsp;136</ref>
+
A real number connected with the rate of growth of the number of rational points on a [[normal variety|normal]] [[projective variety]] $X$ is a with respect to the embedding defined by an [[ample divisor]] $D$.  The concept is named after Rolf Nevanlinna.
  
A '''bi-ideal sequence''' is a sequence of words ''f''<sub>''i''</sub> where ''f''<sub>1</sub> is in ''A''<sup>+</sup> and
+
Formally, $\alpha(D)$ is the [[infimum]] of the rational numbers $r$ such that $K_X + r D$ is in the closed real cone of [[effective divisor]]s in the [[Néron–Severi group]] of $X$.  If $\alpha$ is negative, then $X$ is [[pseudo-canonical variety|pseudo-canonical]].  It is expected that $\alpha(D)$ is always a [[rational number]].
  
:<math>f_{i+1} = f_i g_i f_i \ </math>
+
The Nevanlinna invariant has similar formal properties to the [[abscissa of convergence]] of the [[height zeta function]] and it is conjectured that they are essentially the same.  More precisely, Batyrev–Manin conjectured the following.  Let $X$ be a projective variety over a number field $K$ with ample divisor $D$ giving rise to an embedding and height function $H$, and let $U$ denote a Zariski-open subset of $X$.  Let $\alpha = \alpha(D)$ be the Nevanlinna invariant of $D$ and $\beta$ the abscissa of convergence of  $Z(U,H,s)$.  Then for every $\epsilon > 0$ there is a $U$ such that $\beta < \alpha + \epsilon$: in the opposite direction, if $\alpha > 0$ then $\alpha = \beta$ for all sufficiently large fields $K$ and sufficiently small $U$.
  
for some ''g''<sub>''i''</sub> in ''A''<sup>&lowast;</sup> and ''i''&nbsp;≥&nbsp;1. The degree of a word ''w'' is thus the length of the longest bi-ideal sequence ending in ''w''.<ref name=LotII136/>
+
==References==
 +
* {{User:Richard Pinch/sandbox/Ref | zbl=0679.14008 | last1=Batyrev | first1=V.V. | last2=Manin | first2=Yu.I. | author2-link=Yuri I. Manin | title=On the number of rational points of bounded height on algebraic varieties | journal=Math. Ann. | volume=286, | pages=27-43 | year=1990 }}
 +
* {{User:Richard Pinch/sandbox/Ref | first1=Marc | last1=Hindry | first2=Joseph H. | last2=Silverman | authorlink2=Joseph H. Silverman | title=Diophantine Geometry: An Introduction | series=Graduate Texts in Mathematics | volume=201 | year=2000 | isbn=0-387-98981-1 | zbl=0948.11023 }}
 +
* {{User:Richard Pinch/sandbox/Ref | first=Serge | last=Lang | authorlink=Serge Lang | title=Survey of Diophantine Geometry | publisher=Springer-Verlag | year=1997 | isbn=3-540-61223-8 | zbl=0869.11051 }}
  
For a finite alphabet ''A'' on ''k'' letters, there is an integer ''M'' depending on ''k'' and ''n'', such that any word of length ''M'' has a factor which is a sesquipower of order at least ''n''.  We express this by saying that the sesquipowers are ''unavoidable patterns''.<ref name=LotII137>Lothaire (2011) p.&nbsp;137</ref><ref name=BLRS132>Berstel et al (2009) p.132</ref>
+
=Ruelle zeta function=
 +
A [[Zeta-function|zeta function]] associated with a [[dynamical system]].
  
Given an infinite bi-ideal sequence, we note that each ''f''<sub>''i''</sub> is a prefix of ''f''<sub>''i''+1</sub> and so the ''f''<sub>''i''</sub> converge to an infinite sequence
+
Let $f$ be a function defined on a [[manifold]] $M$, such that the set of [[fixed point]]s $\mathrm{Fix}(f^n)$ is finite for all $n > 1$.  Further let $\phi$ be a function on $M$ with values in $d \times d$ complex matrices.  The zeta function of the first kind is<ref name=T28>Terras (2010) p.&nbsp;28</ref>
 +
$$
 +
\zeta(z) = \exp\left({
 +
                              \sum_{m\ge1} \frac{z^m}{m} \sum_{x\in\mathrm{Fix}(f^m)}
 +
                              \mathrm{Tr}
 +
                                    \left({ \prod_{k=0}^{m-1} \phi(f^k(x))
 +
                                          }\right)
 +
                            }\right)
 +
$$
  
:<math> f = f_1 g_1 f_2 g_2 \cdots \ </math>
+
In the special case $d=1$, $\phi = 1$, we have<ref name=T28/>
 +
$$
 +
\zeta(z) = \exp\left({ \sum_{m\ge1} \frac{z^m}{m} \left|{\mathrm{Fix}(f^m)}\right|}\right)
 +
$$
 +
which is the [[Artin–Mazur zeta function]].
  
We define an infinite word to be a sesquipower if is the limit of an infinite bi-ideal sequence.<ref name=LotII141>Lothiare (2011) p.&nbsp;141</ref>  An infinite word is a sesquipower if and only if it is a [[recurrent word]],<ref name=LotII141/><ref name=BLRS133/> that is, every factor occurs infinitely often.<ref name=LotII30>Lothaire (2011) p.&nbsp;30</ref>
+
The [[Ihara zeta function]] is an example of a Ruelle zeta function.<ref name=T29>Terras (2010) p.&nbsp;29</ref>
 
 
Fix a finite alphabet ''A'' and assume a [[total order]] on the letters.  For given integers ''p'' and ''n'', every sufficiently long word in ''A''<sup>&lowast;</sup> has either a factor which is a ''p''-power or a factor which is an ''n''-sesquipower; in the latter case the factor has an ''n''-[[Monoid factorisation|factorisation]] into [[Lyndon word]]s.<ref name=BLRS133>Berstel et al (2009) p.133</ref>
 
  
 
==References==
 
==References==
 
<references/>
 
<references/>
* {{User:Richard Pinch/sandbox/Ref | last1=Berstel | first1=Jean | last2=Lauve | first2=Aaron | last3=Reutenauer | first3=Christophe | last4=Saliola | first4=Franco V. | title=Combinatorics on words. Christoffel words and repetitions in words | series=CRM Monograph Series | volume=27 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2009 | isbn=978-0-8218-4480-9 | url=http://www.ams.org/bookpages/crmm-27 | zbl=1161.68043 }}
+
* {{User:Richard Pinch/sandbox/Ref | last1=Lapidus | first1=Michel L. | last2=van Frankenhuijsen | first2=Machiel | title=Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings | series=Springer Monographs in Mathematics | location=New York, NY | publisher=Springer-Verlag | year=2006 | isbn=0-387-33285-5 | zbl=1119.28005 }}
* {{User:Richard Pinch/sandbox/Ref | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Algebraic combinatorics on words | others=With preface by Jean Berstel and Dominique Perrin | edition=Reprint of the 2002 hardback | series=Encyclopedia of Mathematics and Its Applications | volume=90| publisher=Cambridge University Press | year=2011 | isbn=978-0-521-18071-9 | zbl=1221.68183 }}
+
* {{User:Richard Pinch/sandbox/Ref | title=Zeta Functions of Graphs: A Stroll through the Garden | volume=128 | series=Cambridge Studies in Advanced Mathematics | first=Audrey | last=Terras | authorlink=Audrey Terras | publisher=Cambridge University Press | year=2010 | isbn=0-521-11367-9 | zbl=1206.05003 }}
* {{User:Richard Pinch/sandbox/Ref | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}
+
* {{User:Richard Pinch/sandbox/Ref | title=Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion | volume=699 | series=Memoirs of the American Mathematical Society | first=Alexander  | last=Fel'shtyn | publisher=Cambridge University Press | year=2000 | isbn=0-8218-2090-7 | zbl=0963.55002 }}
 
 
  
 
=Spectrum of a sentence=
 
=Spectrum of a sentence=
Line 168: Line 175:
 
* {{User:Richard Pinch/sandbox/Ref  | first=J. W. S. | last=Cassels | authorlink=J. W. S. Cassels | title=Rational Quadratic Forms | series=London Mathematical Society Monographs | volume=13 | publisher=[[Academic Press]] | year=1978 | isbn=0-12-163260-1 | zbl=0395.10029 }}
 
* {{User:Richard Pinch/sandbox/Ref  | first=J. W. S. | last=Cassels | authorlink=J. W. S. Cassels | title=Rational Quadratic Forms | series=London Mathematical Society Monographs | volume=13 | publisher=[[Academic Press]] | year=1978 | isbn=0-12-163260-1 | zbl=0395.10029 }}
 
* {{User:Richard Pinch/sandbox/Ref | zbl=0915.52003 | last1=Conway | first1=J. H. | author1-link=John Horton Conway | last2=Sloane | first2=N. J. A. | author2-link=Neil Sloane | others=With contributions by Bannai, E.; [[Borcherds, R. E.]]; [[John Leech (mathematician)|Leech, J.]]; [[Simon P. Norton|Norton, S. P.]]; [[Odlyzko, A. M.]]; Parker, R. A.; Queen, L.; Venkov, B. B. | title=Sphere packings, lattices and groups | edition=3rd | series=Grundlehren der Mathematischen Wissenschaften | volume=290 | location=New York, NY | publisher=[[Springer-Verlag]] | isbn=0-387-98585-9 }}
 
* {{User:Richard Pinch/sandbox/Ref | zbl=0915.52003 | last1=Conway | first1=J. H. | author1-link=John Horton Conway | last2=Sloane | first2=N. J. A. | author2-link=Neil Sloane | others=With contributions by Bannai, E.; [[Borcherds, R. E.]]; [[John Leech (mathematician)|Leech, J.]]; [[Simon P. Norton|Norton, S. P.]]; [[Odlyzko, A. M.]]; Parker, R. A.; Queen, L.; Venkov, B. B. | title=Sphere packings, lattices and groups | edition=3rd | series=Grundlehren der Mathematischen Wissenschaften | volume=290 | location=New York, NY | publisher=[[Springer-Verlag]] | isbn=0-387-98585-9 }}
 +
 +
=Turán method=
 +
A mthod for obtaining lower bounds for [[exponential sum]]s and complex [[power sum]]s.  The method has been applied to problems in [[equidistribution]]. 
 +
 +
The method  applies to sums of the form
 +
$$
 +
s_\nu = \sum_{n=1}^N b_n z_n^\nu
 +
$$
 +
where the $b$ and $z$ are complex numbers and $v$ runs over a range of integers.  There are two main results, depending on the size of the complex numbers $z$.
 +
 +
==Turán's first theorem==
 +
The first result applies to sums $s_v$ where $|z_n| \ge 1$ for all $n$.  For any range of $v$ of length $N$, say$v = m_1,\ldots,M_N$, there is some $v$ with $|s_v|$ at least $c(M,N)|s_0|$ where
 +
$$
 +
c(M,N) = \left({ \sum_{k=0}^{N-1} \binom{M+k}{k} 2^k }\right)^{-1} \ .
 +
$$
 +
The sum here may be replaced by the weaker but simpler $\left({ \frac{N}{2e(M+N)} }\right)^{N-1}$.
 +
 +
We may deduce [[Fabry's gap theorem]] from this result.
 +
 +
==Turán's second theorem==
 +
The second result applies to sums $s_v$ where $|z_n| \le 1$ for all $n$.  Assume that the $z$ are ordered in decreasing absolute value and scaled so that $|z_1| = 1$.  Then there is some $v$ with
 +
$$
 +
|s_\nu| \ge 2 \left({ \frac{N}{8e(M+N)} }\right)^N \min_{1\le j\le N} \left\vert{\sum_{n=1}^j b_n }\right\vert \ .
 +
$$
 +
==See also==
 +
* [[Turán's theorem]] in graph theory
 +
 +
==References==
 +
* {{cite book | last=Montgomery | first=Hugh L. | authorlink=Hugh Montgomery (mathematician) | title=Ten lectures on the interface between analytic number theory and harmonic analysis | series=Regional Conference Series in Mathematics | volume=84 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0737-4 | zbl=0814.11001 }}
 +
  
 
=Zimmert set=
 
=Zimmert set=
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Fix an integer $d$ and let $D$ be the discriminant of the imaginary [[quadratic field]] $\mathbb{Q}(\sqrt{-d}$.  The ''Zimmert set'' $Z(d)$ is the set of positive integers $n$ such that $4n^2 < -D-3$ and $n \ne 2$; $D$ is a [[quadratic non-residue]] of all odd primes in $d$; $n$ is odd if $D$ is not congruent to 5 modulo 8.  The cardinality of $Z(d)$ may be denoted by $z(d)$.
 
Fix an integer $d$ and let $D$ be the discriminant of the imaginary [[quadratic field]] $\mathbb{Q}(\sqrt{-d}$.  The ''Zimmert set'' $Z(d)$ is the set of positive integers $n$ such that $4n^2 < -D-3$ and $n \ne 2$; $D$ is a [[quadratic non-residue]] of all odd primes in $d$; $n$ is odd if $D$ is not congruent to 5 modulo 8.  The cardinality of $Z(d)$ may be denoted by $z(d)$.
  
For all but a finite number of $d$ we have $z(d)>1$: indeed this is true for all $d > 10^{476}$.<ref name=MOS>{{User:Richard Pinch/sandbox/Ref | url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2095508 | doi=10.1017/S0305004100075101 | zbl=0758.20009 | last1=Mason | first1=A.W. | last2=Odoni | first2=R.W.K. | last3=Stothers | first3=W.W. | title=Almost all Bianchi groups have free, non-cyclic quotients | journal=Math. Proc. Camb. Philos. Soc. | volume=111 | number=1 | pages=1–6 | year=1992 }}</ref>
+
For all but a finite number of $d$ we have $z(d)>1$: indeed this is true for all $d > 10^{476}$.<ref name=MOS>{{User:Richard Pinch/sandbox/Ref | doi=10.1017/S0305004100075101 | zbl=0758.20009 | last1=Mason | first1=A.W. | last2=Odoni | first2=R.W.K. | last3=Stothers | first3=W.W. | title=Almost all Bianchi groups have free, non-cyclic quotients | journal=Math. Proc. Camb. Philos. Soc. | volume=111 | number=1 | pages=1–6 | year=1992 }}</ref>
  
 
Let $\Gamma_d$ Bianchi group $PSL(2,O_d)$, where $O_d$ is the [[ring of integers]] of $\mathbb{Q}(\sqrt{-d}$.  As a subgroup of $PSL(2,\mathbb{C})$, there is an action of $\Gamma_d$ on hyperbolic 3-space $H^3$, with a [[fundamental domain]].  It is a theorem that there are only finitely many values of $d$ for which$\Gamma_d$ can contain an [[arithmetic group|arithmetic subgroup]] $G$ for which the quotient $H^3/G$ is a [[link complement]].  Zimmert sets are used to obtain results in this direction: $z(d)$ is a lower bound for the rank of the largest [[Free group|free]] [[Quotient group|quotient]] of $\Gamma_d$<ref>{{User:Richard Pinch/sandbox/Ref | zbl=0254.10019 | last=Zimmert | first=R. | title=Zur SL<sub>2</sub> der ganzen Zahlen eines imaginär-quadratischen Zahlkörpers | journal=Inventiones mathematicae | volume=19 | year=1973 | pages=73–81 }}</ref> and so the result above implies that almost all Bianchi groups have non-[[Cyclic group|cyclic]] free quotients.<ref name=MOS/>
 
Let $\Gamma_d$ Bianchi group $PSL(2,O_d)$, where $O_d$ is the [[ring of integers]] of $\mathbb{Q}(\sqrt{-d}$.  As a subgroup of $PSL(2,\mathbb{C})$, there is an action of $\Gamma_d$ on hyperbolic 3-space $H^3$, with a [[fundamental domain]].  It is a theorem that there are only finitely many values of $d$ for which$\Gamma_d$ can contain an [[arithmetic group|arithmetic subgroup]] $G$ for which the quotient $H^3/G$ is a [[link complement]].  Zimmert sets are used to obtain results in this direction: $z(d)$ is a lower bound for the rank of the largest [[Free group|free]] [[Quotient group|quotient]] of $\Gamma_d$<ref>{{User:Richard Pinch/sandbox/Ref | zbl=0254.10019 | last=Zimmert | first=R. | title=Zur SL<sub>2</sub> der ganzen Zahlen eines imaginär-quadratischen Zahlkörpers | journal=Inventiones mathematicae | volume=19 | year=1973 | pages=73–81 }}</ref> and so the result above implies that almost all Bianchi groups have non-[[Cyclic group|cyclic]] free quotients.<ref name=MOS/>

Latest revision as of 12:14, 5 November 2016

Arakawa–Kaneko zeta function

A generalisation of the Riemann zeta function which generates special values of the polylogarithm function.

Definition

The zeta function $\xi_k(s)$ is defined by $$ \xi_k(s) = \frac{1}{\Gamma(s)} \int_0^{+\infty} \frac{t^s-1}{e^t-1}\mathrm{Li}_k(1-e^{-t}) dt $$ where $\mathrm{Li}_k$ is the$k$-th polylogarithm $$ \mathrm{Li}_k(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^k} \ . $$

Properties

The integral converges for $\Re(s) > 0$ and $\xi_k(s)$ has analytic continuation to the whole complex plane as an entire function.

The special case$k=1$ gives $\xi_1(s) = s \zeta(s+1)$ where $\zeta$ is the Riemann zeta-function.

The values at integers are related to multiple zeta function values by

$$ \xi_k(m) = \zeta_m^*(k,1,\ldots,1) $$ where $$ \zeta_n^*(k_1,\dots,k_{n-1},k_n)=\sum_{0<m_1<m_2<\cdots<m_n}\frac{1}{m_1^{k_1}\cdots m_{n-1}^{k_{n-1}}m_n^{k_n}} \ . $$

References

  • Masanobou Kaneko, "Poly-Bernoulli numbers" J. Théor. Nombres Bordx 9 (1997) 221-228 Zbl 0887.11011
  • Tsuneo Arakawa, Masanobu Kaneko "Multiple zeta values, poly-Bernoulli numbers, and related zeta functions", Nagoya Math. J. 153 (1999) 189-209 Zbl 0932.11055 MR1684557 [1]
  • Marc-Antoine Coppo, Bernard Candelpergher "The Arakawa-Kaneko zeta function" Ramanujan J. 22 (2010) 153-162 Zbl 1230.11106

Genus of a quadratic form

A classification of quadratic forms and lattices over the ring of integers.

An integral quadratic form is a quadratic form on $\mathbb{Z}^n$, or more generally a free $\mathbb{Z}$-module of finite rank. Two such forms are in the same genus if they are equivalent over the local rings $\mathbb{Z}_p$ for each prime $p$ and also equivalent over $\mathbb{R}$.

Equivalent forms are in the same genus, but the converse does not hold. For example, $X^2 + 82Y^2$ and $2X^2 + 41Y^2$ are in the same genus but not equivalent over $\mathbb{Z}$.

Forms in the same genus have equal determinant and hence there are only finitely many equivalence classes in a genus.

The Smith–Minkowski–Siegel mass formula gives the weight or mass of the quadratic forms in a genus, the count of equivalence classes weighted by the reciprocals of the orders of their automorphism groups.

Binary quadratic forms

For binary quadratic forms there is a group structure on the set $C(D)$ equivalence classes of forms with given discriminant $D$. The genera are defined by the generic characters. The principal genus, the genus containing the principal form, is precisely the subgroup $C(D)^2$ and the genera are the cosets of $C(D)^2$: so in this case all genera contain the same number of classes of forms.

See also

References

  • J.W.S. Cassels, Rational Quadratic Forms, London Mathematical Society Monographs 13, Academic Press (1978) ISBN 0-12-163260-1 Zbl 0395.10029

Gowers norm

uniformity norm

For the function field norm, see uniform norm; for uniformity in topology, see uniform space.

A class of norm on functions on a finite group or group-like object which are used in the study of arithmetic progressions in the group. They are named after Timothy Gowers who introduced them in his work on Szemerédi's theorem.

Let $f$ be a complex-valued function on a group $G$ and let $J$ denote complex conjugation. The Gowers $d$-norm is $$ \Vert f \Vert_{U^d(g)} = \mathbf{E}_{x,h_1,\ldots,h_d \in G} \prod_{\omega_1,\ldots,\omega_d \in \{0,1\}} J^{\omega_1+\cdots+\omega_d} f\left({x + h_1\omega_1 + \cdots + h_d\omega_d}\right) \ . $$ The inverse conjecture for these norms is the statement that if $f$ has L-infinity norm (uniform norm in the usual sense) equal to 1 then the Gowers $s$-norm is bounded above by 1, with equality if and only if $f$ is of the form $\exp(2\pi i g)$ with$g$ a polynomial of degree at most $s$. This can be interpreted as saying that the Gowers norm is controlled by polynomial phases.

The inverse conjecture holds for vector spaces over a finite field. However, for cyclic groups $C_N$ this is not so, and the class of polynomial phases has to be extended to control the norm.

References

  • Terence Tao, "Higher order Fourier analysis", Graduate Studies in Mathematics 142 American Mathematical Society (2012) ISBN 978-0-8218-8986-2 Zbl 1277.11010

Height zeta function

of a set of points A function encoding the distribution of points of given height on an algebraic variety or a subset.

If $S$ is a set with height function $H$, such that there are only finitely many elements of bounded height, define a counting function $$ N(S,H,B) = \sharp \{ x \in S : H(x) \le B \} $$ and a zeta function $$ Z(S,H;s) = \sum_{x \in S} H(x)^{-s} \ . $$

If $Z$ has abscissa of convergence $\beta$ and there is a constant $c$ such that $N$ has rate of growth $$ N \sim c B^a (\log B)^{t-1} $$ then a version of the Wiener–Ikehara theorem holds: $Z$ has a $t$-fold pole at $s = \beta$ with residue $c a \Gamma(t)$.

The abscissa of convergence has similar formal properties to the Nevanlinna invariant and it is conjectured that they are essentially the same. More precisely, Batyrev–Manin conjectured the following. Let $X$ be a projective variety over a number field $K$ with ample divisor $D$ giving rise to an embedding and height function $H$, and let $U$ denote a Zariski-open subset of $X$'. Let $\alpha = \alpha(D)$ be the Nevanlinna invariant of $D$ and $\beta$ the abscissa of convergence of $Z(U,H,s)$. Then for every $\epsilon > 0$ there is a $U$ such that $\beta < \alpha + \epsilon$: in the opposite direction, if $\alpha > 0$ then $\alpha = \beta$ for all sufficiently large fields $K$ and sufficiently small $U$.

References

  • Batyrev, V.V.; Manin, Yu.I.; On the number of rational points of bounded height on algebraic varieties, Math. Ann., 286, (1990), pp. 27–43, Zbl 0679.14008
  • Hindry, Marc; Silverman, Joseph H.; Diophantine Geometry: An Introduction, ser. Graduate Texts in Mathematics 201 (2000), Zbl 0948.11023 ISBN: 0-387-98981-1
  • Lang, Serge; Survey of Diophantine Geometry, (1997), Springer-Verlag, Zbl 0869.11051 ISBN: 3-540-61223-8

Nevanlinna invariant

of an ample divisor on a normal projective variety

A real number connected with the rate of growth of the number of rational points on a normal projective variety $X$ is a with respect to the embedding defined by an ample divisor $D$. The concept is named after Rolf Nevanlinna.

Formally, $\alpha(D)$ is the infimum of the rational numbers $r$ such that $K_X + r D$ is in the closed real cone of effective divisors in the Néron–Severi group of $X$. If $\alpha$ is negative, then $X$ is pseudo-canonical. It is expected that $\alpha(D)$ is always a rational number.

The Nevanlinna invariant has similar formal properties to the abscissa of convergence of the height zeta function and it is conjectured that they are essentially the same. More precisely, Batyrev–Manin conjectured the following. Let $X$ be a projective variety over a number field $K$ with ample divisor $D$ giving rise to an embedding and height function $H$, and let $U$ denote a Zariski-open subset of $X$. Let $\alpha = \alpha(D)$ be the Nevanlinna invariant of $D$ and $\beta$ the abscissa of convergence of $Z(U,H,s)$. Then for every $\epsilon > 0$ there is a $U$ such that $\beta < \alpha + \epsilon$: in the opposite direction, if $\alpha > 0$ then $\alpha = \beta$ for all sufficiently large fields $K$ and sufficiently small $U$.

References

  • Batyrev, V.V.; Manin, Yu.I.; On the number of rational points of bounded height on algebraic varieties, Math. Ann., 286, (1990), pp. 27-43, Zbl 0679.14008
  • Hindry, Marc; Silverman, Joseph H.; Diophantine Geometry: An Introduction, ser. Graduate Texts in Mathematics 201 (2000), Zbl 0948.11023 ISBN: 0-387-98981-1
  • Lang, Serge; Survey of Diophantine Geometry, (1997), Springer-Verlag, Zbl 0869.11051 ISBN: 3-540-61223-8

Ruelle zeta function

A zeta function associated with a dynamical system.

Let $f$ be a function defined on a manifold $M$, such that the set of fixed points $\mathrm{Fix}(f^n)$ is finite for all $n > 1$. Further let $\phi$ be a function on $M$ with values in $d \times d$ complex matrices. The zeta function of the first kind is[1] $$ \zeta(z) = \exp\left({ \sum_{m\ge1} \frac{z^m}{m} \sum_{x\in\mathrm{Fix}(f^m)} \mathrm{Tr} \left({ \prod_{k=0}^{m-1} \phi(f^k(x)) }\right) }\right) $$

In the special case $d=1$, $\phi = 1$, we have[1] $$ \zeta(z) = \exp\left({ \sum_{m\ge1} \frac{z^m}{m} \left|{\mathrm{Fix}(f^m)}\right|}\right) $$ which is the Artin–Mazur zeta function.

The Ihara zeta function is an example of a Ruelle zeta function.[2]

References

  1. 1.0 1.1 Terras (2010) p. 28
  2. Terras (2010) p. 29
  • Lapidus, Michel L.; van Frankenhuijsen, Machiel; Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings, ser. Springer Monographs in Mathematics (2006), Springer-Verlag, Zbl 1119.28005 ISBN: 0-387-33285-5
  • Terras, Audrey; Zeta Functions of Graphs: A Stroll through the Garden, ser. Cambridge Studies in Advanced Mathematics 128 (2010), Cambridge University Press, Zbl 1206.05003 ISBN: 0-521-11367-9
  • Fel'shtyn, Alexander; Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion, ser. Memoirs of the American Mathematical Society 699 (2000), Cambridge University Press, Zbl 0963.55002 ISBN: 0-8218-2090-7

Spectrum of a sentence

The set of natural numbers that occur as the size of a finite model in which the sentence is true.

Definition

Let $\psi$ be a sentence in first-order logic. The spectrum of $\psi$ is the set of natural numbers $n$ such that there is a finite model for $\psi$ with $n$ elements.

If the vocabulary for $\psi$ consists of relational symbols, then $\psi$ can be regarded as a sentence in existential second-order logic quantified over the relations, over the empty vocabulary. A generalised spectrum is the set of models of a general ESOL sentence.

Properties

Fagin's theorem is a result in descriptive complexity theory that states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP. It is remarkable since it is a characterization of the class NP that does not invoke a model of computation such as a Turing machine. The theorem was proven by Ronald Fagin in 1974 (strictly, in 1973 in his doctoral thesis).

As a corollary we have a result of Jones and Selman, that a set is a spectrum if and only if it is in the complexity class NEXPTIME.

See also

References

  • Fagin, Ronald; Complexity of Computation, "Generalized First-Order Spectra and Polynomial-Time Recognizable Sets", ser. Proc. Syp. App. Math. SIAM-AMS Proceedings 7 (1974), pp. 27–41, Zbl 0303.68035 URL: www.almaden.ibm.com/cs/people/fagin/genspec.pdf
  • Grädel, Erich; Kolaitis, Phokion G.; Libkin, Leonid; Maarten, Marx; Spencer, Joel; Vardi, Moshe Y.; Finite model theory and its applications, ser. Texts in Theoretical Computer Science. An EATCS Series (2007), Springer-Verlag, Zbl 1133.03001 ISBN 978-3-540-00428-8
  • Immerman, Neil; Descriptive Complexity, ser. Graduate Texts in Computer Science (1999), pp. 113–119, Springer-Verlag, Zbl 0918.68031 ISBN 0-387-98600-6
  • Jones, Neil D.; Selman, Alan L.; Turing machines and the spectra of first-order formulas, J. Symb. Log., 39 (1974), pp. 139-150, Zbl 0288.02021, DOI: 10.2307/2272354

Category:Model theory

Spinor genus

A classification of quadratic forms and lattices over the ring of integers, introduced by Martin Eichler. It refines the genus but may be coarser than proper equivalence

We define two Z-lattices L and M in a quadratic space V over Q to be spinor equivalent if there exists a transformation g in the proper orthogonal group O+(V) and for every prime p there exists a local transformation fp of Vp of spinor norm 1 such that M = g fpLp.

A spinor genus is an equivalence class for this equivalence relation. Properly equivalent lattices are in the same spinor genus, and lattices in the same spinor genus are in the same genus. The number of spinor genera in a genus is a power of two, and can be determined effectively.

An important result is that for indefinite forms of dimension at least three, each spinor genus contains exactly one proper equivalence class.

References

  • Cassels, J. W. S.; Rational Quadratic Forms, ser. London Mathematical Society Monographs 13 (1978), Academic Press, Zbl 0395.10029 ISBN: 0-12-163260-1
  • Conway, J. H.; Sloane, N. J. A.; Sphere packings, lattices and groups, ser. Grundlehren der Mathematischen Wissenschaften 290 , Springer-Verlag, Zbl 0915.52003 ISBN: 0-387-98585-9

Turán method

A mthod for obtaining lower bounds for exponential sums and complex power sums. The method has been applied to problems in equidistribution.

The method applies to sums of the form $$ s_\nu = \sum_{n=1}^N b_n z_n^\nu $$ where the $b$ and $z$ are complex numbers and $v$ runs over a range of integers. There are two main results, depending on the size of the complex numbers $z$.

Turán's first theorem

The first result applies to sums $s_v$ where $|z_n| \ge 1$ for all $n$. For any range of $v$ of length $N$, say$v = m_1,\ldots,M_N$, there is some $v$ with $|s_v|$ at least $c(M,N)|s_0|$ where $$ c(M,N) = \left({ \sum_{k=0}^{N-1} \binom{M+k}{k} 2^k }\right)^{-1} \ . $$ The sum here may be replaced by the weaker but simpler $\left({ \frac{N}{2e(M+N)} }\right)^{N-1}$.

We may deduce Fabry's gap theorem from this result.

Turán's second theorem

The second result applies to sums $s_v$ where $|z_n| \le 1$ for all $n$. Assume that the $z$ are ordered in decreasing absolute value and scaled so that $|z_1| = 1$. Then there is some $v$ with $$ |s_\nu| \ge 2 \left({ \frac{N}{8e(M+N)} }\right)^N \min_{1\le j\le N} \left\vert{\sum_{n=1}^j b_n }\right\vert \ . $$

See also

References


Zimmert set

A set of positive integers associated with the structure of quotients of hyperbolic three-space by a Bianchi group.

Fix an integer $d$ and let $D$ be the discriminant of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d}$. The Zimmert set $Z(d)$ is the set of positive integers $n$ such that $4n^2 < -D-3$ and $n \ne 2$; $D$ is a quadratic non-residue of all odd primes in $d$; $n$ is odd if $D$ is not congruent to 5 modulo 8. The cardinality of $Z(d)$ may be denoted by $z(d)$.

For all but a finite number of $d$ we have $z(d)>1$: indeed this is true for all $d > 10^{476}$.[1]

Let $\Gamma_d$ Bianchi group $PSL(2,O_d)$, where $O_d$ is the ring of integers of $\mathbb{Q}(\sqrt{-d}$. As a subgroup of $PSL(2,\mathbb{C})$, there is an action of $\Gamma_d$ on hyperbolic 3-space $H^3$, with a fundamental domain. It is a theorem that there are only finitely many values of $d$ for which$\Gamma_d$ can contain an arithmetic subgroup $G$ for which the quotient $H^3/G$ is a link complement. Zimmert sets are used to obtain results in this direction: $z(d)$ is a lower bound for the rank of the largest free quotient of $\Gamma_d$[2] and so the result above implies that almost all Bianchi groups have non-cyclic free quotients.[1]

References

  1. 1.0 1.1 Mason, A.W.; Odoni, R.W.K.; Stothers, W.W.; Almost all Bianchi groups have free, non-cyclic quotients, Math. Proc. Camb. Philos. Soc., 111 no. 1 (1992), pp. 1–6, Zbl 0758.20009, DOI: 10.1017/S0305004100075101
  2. Zimmert, R.; Zur SL2 der ganzen Zahlen eines imaginär-quadratischen Zahlkörpers, Inventiones mathematicae, 19 (1973), pp. 73–81, Zbl 0254.10019
  • Maclachlan, Colin; Reid, Alan W.; The Arithmetic of Hyperbolic 3-Manifolds, ser. Graduate Texts in Mathematics 219 (2003), Springer-Verlag, Zbl 1025.57001 ISBN: 0-387-98386-4
How to Cite This Entry:
Richard Pinch/sandbox-WP2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP2&oldid=35659