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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

Average order of an arithmetic function

Some simpler or better-understood function which takes the same values "on average" as an arithmetic function.

Let f be a function on the natural numbers. We say that f has average order g if

\[ \sum_{n \le x} f(n) \sim \sum_{n \le x} g(n) \]

as x tends to infinity.

It is conventional to assume that the approximating function g is continuous and monotone.

Examples

• The average order of $d(n)$, the number of divisors of $n$, is $\log n$;
• The average order of $\sigma(n)$, the sum of divisors of $n$, is $ \frac{\pi^2}{6} n$;
• The average order of $\phi(n)$, the Euler 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

• Divisor function
• Normal order of an arithmetic function

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
• Gérald Tenenbaum (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics 46. Cambridge University Press. ISBN 0-521-41261-7

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

• Spinor genus

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

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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

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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 $$ \zeta(z) = \exp\left({ \sum_{m\ge1} \frac{z^m}{m} \sum_{x\in\mathrm{Fix}(f^m)} \mathrm{Tr} \left({ \prod_{k=0}^{m-1} \phi(f^k(x)) }\right) }\right) $$

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

The Ihara zeta function is an example of a Ruelle zeta function.

References

• 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

Sesquipower

A sesquipower or Zimin word is a string over an alphabet with identical prefix and suffix. Sesquipowers are unavoidable patterns, in the sense that all sufficiently long strings contain one.

Formally, let A be an alphabet and A^∗ be the free monoid of finite strings over A. Every non-empty word w in A⁺ is a sesquipower of order 1. If u is a sequipower of order n then any word w = uvu is a sesquipower of order n + 1. The degree of a non-empty word w is the largest integer d such that w is a sesquipower of order d.

A bi-ideal sequence is a sequence of words f_i where f₁ is in A⁺ and

\[f_{i+1} = f_i g_i f_i \ \]

for some g_i in A^∗ and i ≥ 1. The degree of a word w is thus the length of the longest bi-ideal sequence ending in w.

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.

Given an infinite bi-ideal sequence, we note that each f_i is a prefix of f_i+1 and so the f_i converge to an infinite sequence

\[ f = f_1 g_1 f_2 g_2 \cdots \ \]

We define an infinite word to be a sesquipower if is the limit of an infinite bi-ideal sequence. An infinite word is a sesquipower if and only if it is a recurrent word, that is, every factor occurs infinitely often.

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^∗ 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-factorisation into Lyndon words.

References

• Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V.; Combinatorics on words. Christoffel words and repetitions in words, ser. CRM Monograph Series 27 (2009), American Mathematical Society, Zbl 1161.68043URL: http://www.ams.org/bookpages/crmm-27] ISBN: 978-0-8218-4480-9
• Lothaire, M.; Algebraic combinatorics on words, ser. Encyclopedia of Mathematics and Its Applications 90 (2011), Cambridge University Press, Zbl 1221.68183 ISBN: 978-0-521-18071-9
• Pytheas Fogg, N.; Substitutions in dynamics, arithmetics and combinatorics, ser. Lecture Notes in Mathematics 1794 (2002), Springer-Verlag, Zbl 1014.11015 ISBN: 3-540-44141-7

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

• Spectrum of a theory

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

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 f_p of V_p of spinor norm 1 such that M = g f_pL_p.

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

• Turán's theorem in graph theory

References

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Unavoidable pattern

A pattern of symbols that must occur in any sufficiently long string over an alphabet. An avoidable pattern is one which for which there are infinitely many words no part of which match the pattern.

Let $A$ be an alphabet of letters and $E$ a disjoint alphabet of pattern symbols or "variables". Elements of $E^+$ are patterns. For a pattern $p$', the pattern language is that subset of $A^*$ containing all words $h(p)$ where $h$ is a non-erasing semigroup morphism from the free monoid $E^*$ to $A^*$. A word $w \in A^*$ matches or meets $p$ if it contains some word in the pattern language as a factor, otherwise $w$ avoids p.

A pattern $p$ is avoidable on $A$ if there are infinitely many words in $A^*$ that avoid p; it is unavoidable on A if all sufficiently long words in $A^*$ match $p$. We say that $p$ is $k$-unavoidable if it is unavoidable on every alphabet of size $k$ and correspondingly $k$-unavoidable if it is avoidable on an alphabet of size $k$.

There is a word $W(k)$ over an alphabet of size $4k$ which avoids every avoidable pattern with fewer than $2k$ variables.

Examples

• The Thue–Morse sequence avoids the patterns $xxx$ and $xyxyx$.
• The patterns $x$ and $xyx$ are unavoidable on any alphabet.
• The power pattern $xx$ is 3-avoidable; words avoiding this pattern are square-free.
• The power patterns $x^n$ for $n \ge 3$ are 2-avoidable: the Thue–Morse sequence is an example for $n=3$.
• Sesquipowers are unavoidable.

Avoidability index

The avoidability index of a pattern $p$ is the smallest $k$ such that $p$ is $k$-avoidable, or $\infty$ if $p$ is unavoidable. For binary patterns (two variables $x$ and $y$) we have:

• $1,x,xy,xyx$ are unavoidable;
• $xx,xxy,xyy,xxyx,xxyy,xyxx,xyxy,xyyx,xxyxx,xxyxy,xyxyy$ have avoidability index 3;
• all other patterns have avoidability index 2.

Square-free words

A square-free word is one avoiding the pattern $xx$. An example is the word over the alphabet $\{0,\pm1\}$ obtained by taking the first difference of the Thue–Morse sequence.

References

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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 $d > 10^{476}$.

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 $\Gamma_d$ and so the result above implies that almost all Bianchi groups have non-cyclic free quotients.

References

• 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