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Power-full number

of type $k$

A natural number $n$ with the property that if a prime $p$ divides $n$, then $p^k$ divides $n$. A square-full number is a power-full number of type 2; a cube-full number is a power-full number of type 3.

If $N_k(x)$ counts the $k$-full numbers $\le x$, then $$ N_2(x) = \frac{\zeta\left({\frac{3}{2}}\right) }{\zeta(3) } x^{\frac{1}{2}} + \frac{\zeta\left({\frac{2}{3}}\right) }{\zeta(2) } x^{\frac{1}{3}} + o\left({x^{\frac{1}{6}}}\right) $$ where $\zeta(s)$ is the Riemann zeta function‏. Similarly, $$ N_3(x) = c_{03} x^{\frac{1}{3}} + c_{13} x^{\frac{1}{4}} + c_{23} x^{\frac{1}{5}} + o\left({ x^{\frac{1}{8}} }\right) $$ and generally $$ N_k(x) = c_{0k} x^{1/k} + O(x^{1/(k+1)}) $$ where $$ c_{0k} = \prod_p \left({ 1 + \sum_{m=k+1}^{2k-1} p^{-m/k} }\right) \ . $$ The $c_{\circ k}$ are the Bateman–Grosswald constants.

References

[BG]	Paul T. Bateman, Emil Grosswald, "On a theorem of Erdős and Szekeres" Ill. J. Math. 2 (1958) 88-98 Zbl 0079.07104
[Fi]	Steven R. Finch, "Mathematical Constants", Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001
[Gu]	Richard K. Guy, Unsolved Problems in Number Theory 3rd ed. Springer-Verlag (2004) ISBN 0-387-20860-7 Zbl 1058.11001

Formal Dirichlet series

A formal Dirichlet series over a ring $R$ is associated to a function $a$ from the positive integers to $R$

\[ D(a,s) = \sum_{n=1}^\infty a(n) n^{-s} \ \]

with addition and multiplication defined by

\[ D(a,s) + D(b,s) = \sum_{n=1}^\infty (a+b)(n) n^{-s} \ \] \[ D(a,s) \cdot D(b,s) = \sum_{n=1}^\infty (a*b)(n) n^{-s} \ \]

where

\[ (a+b)(n) = a(n)+b(n) \ \]

is the pointwise sum and

\[ (a*b)(n) = \sum_{k|n} a(k)b(n/k) \ \]

is the Dirichlet convolution of a and b.

The formal Dirichlet series form a ring $\Omega$, indeed an $R$-algebra, with the zero function as additive zero element and the function $\delta$ defined by $\delta(1)=1$, $\delta(n)=0$ for $n>1$ (so that $D(\delta,s)=1$) as multiplicative identity. An element of this ring is invertible if $a(1)$ is invertible in $R$. If $R$ is commutative, so is $\Omega$; if $R$' is an integral domain, so is $\Omega$. The non-zero multiplicative functions form a subgroup of the group of units of $\Omega$.

The ring of formal Dirichlet series over $\mathbb{C}$ is isomorphic to a ring of formal power series in countably many variables.

References

• E.D. Cashwell, C.J. Everett,. "The ring of number-theoretic functions", Pacific J. Math. 9 (1959) 975-985 Zbl d{(2k)!} B_k z^k

$$ where $B_k$ is the $k$-th [[Bernoulli number]]. The multiplicative sequence with $Q$ as characteristic power series is denoted $L_j(p_1,\ldots,p_j)$. The multiplicative sequence with characteristic power series $$ Q(z) = \frac{2\sqrt z}{\sinh 2\sqrt z} $$ is denoted $A_j(p_1,\ldots,p_j)$. The multiplicative sequence with characteristic power series $$Q(z) = \frac{z}{1-\exp(-z)} = 1 + \frac{x}{2} - \sum_{k=1}^\infty (-1)^k \frac{B_k}{(2k)!} z^{2k} $$ is denoted $T_j(p_1,\ldots,p_j)$: the ''[[Todd polynomial]]s''. =='"`UNIQ--h-4--QINU`"'Genus== The '''genus''' of a multiplicative sequence is a [[ring homomorphism]], from the [[cobordism|cobordism ring]] of smooth oriented [[compact manifold]]s to another [[ring]], usually the ring of [[rational number]]s. For example, the [[Todd genus]] is associated to the Todd polynomials $T_j$ with characteristic power series $$\frac{z}{1-\exp(-z)}$$ and the [[L-genus]] is associated to the polynomials $L_j$ with charac\teristic polynomial $$\frac{\sqrt z}{\tanh \sqrt z} . $$ =='"`UNIQ--h-5--QINU`"'References== * Hirzebruch, Friedrich. ''Topological methods in algebraic geometry'', Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel. Reprint of the 2nd, corr. print. of the 3rd ed. [1978] (Berlin: Springer-Verlag, 1995). ISBN 3-540-58663-6. [https://zbmath.org/?q=an%3A0843.14009 Zbl 0843.14009]. ='"`UNIQ--h-6--QINU`"'Nagao's theorem= A result, named after Hirosi Nagao, about the structure of the [[group]] of 2-by-2 [[Invertible matrix|invertible matrices]] over the [[ring of polynomials]] over a [[field]]. It has been extended by [[Jean-Pierre Serre|Serre]] to give a description of the structure of the corresponding matrix group over the [[coordinate ring]] of a [[projective curve]]. =='"`UNIQ--h-7--QINU`"'Nagao's theorem== For a [[Ring (mathematics)|general ring]] $R$ we let $GL_2(R)$ denote the group of invertible 2-by-2 matrices with entries in $R$, and let $R^*$ denote the [[group of units]] of $R$, and let $$ B(R) = \left\lbrace{ \left({\begin{array}{*{20}c} a & b \\ 0 & d \end{array}}\right) : a,d \in R^*, ~ b \in R }\right\rbrace \ . $$ Then $B(R)$ is a subgroup of $GL_2(R)$. Nagao's theorem states that in the case that $R$ is the ring $K[t]$ of polynomials in one variable over a field $K$, the group $GL_2(R)$ is the [[amalgamated product]] of $GL_2(K)$ and $B(K[t])$ over their intersection $B(K)$. =='"`UNIQ--h-8--QINU`"'Serre's extension== In this setting, $C$ is a [[Singular point of an algebraic variety|smooth]] projective curve over a field $K$. For a [[closed point]] $P$ of $C$ let $R$ be the corresponding coordinate ring of $C$ with $P$ removed. There exists a [[graph of groups]] $(G,T)$ where $T$ is a [[tree]] with at most one non-terminal vertex, such that $GL_2(R)$ is isomorphic to the [[fundamental group]] $\pi_1(G,T)$. =='"`UNIQ--h-9--QINU`"'References== * Mason, A.. "Serre's generalization of Nagao's theorem: an elementary approach". ''Transactions of the American Mathematical Society'' '''353''' (2001) 749–767. [https://doi.org/10.1090/S0002-9947-00-02707-0 DOI 10.1090/S0002-9947-00-02707-0] [https://zbmath.org/?q=an%3A0964.20027 Zbl 0964.20027]. * Nagao, Hirosi. "On $GL(2, K[x])$". J. Inst. Polytechn., Osaka City Univ., Ser. A '''10''' (1959) 117–121. [https://mathscinet.ams.org/mathscinet/article?mr=0114866 MR0114866]. [https://zbmath.org/?q=an%3A0092.02504 Zbl 0092.02504]. * Serre, Jean-Pierre. ''Trees''. (Springer, 2003) ISBN 3-540-44237-5. ='"`UNIQ--h-10--QINU`"'Brauer–Wall group= A [[group]] classifying graded [[central simple algebra]]s over a field. It was first defined by Wall (1964) as a generalisation of the [[Brauer group]]. The Brauer group $\mathrm{B}(F)$ of a field $F$ is defined on the isomorphism classes of central simple algebras over ''F''. The analogous construction for $\mathbf{Z}/2$-[[graded algebra]]s defines the Brauer–Wall group $\mathrm{BW}(F)$.[[#Lam (2005) pp.98–99|[Lam (2005) pp.98–99]]] =='"`UNIQ--h-11--QINU`"'Properties== * The Brauer group $\mathrm{B}(F)$ injects into $\mathrm{BW}(F)$ by mapping a CSA $A$ to the graded algebra which is $A$ in grade zero. There is an exact sequence $$ 0 \rightarrow \mathrm{B}(F) \rightarrow \mathrm{BW}(F) \rightarrow Q(F) \rightarrow 0 $$ where $Q(F)$ is the group of graded quadratic extensions of $F$, defined as $\mathbf{Z}/2 \times F^*/(F^*)^2$ with multiplication $(e,x)(f,y) = (e+f,(-1)^{ef}xy$. The map from W to BW is the '''[[Clifford invariant]]''' defined by mapping an algebra to the pair consisting of its grade and [[Determinant of a quadratic form|determinant]]. There is a map from the additive group of the [[Witt–Grothendieck ring]] to the Brauer–Wall group obtained by sending a quadratic space to its [[Clifford algebra]]. The map factors through the [[Witt group]][[#Lam (2005) p.113|[Lam (2005) p.113]]] which has kernel $I^3$, where $I$ is the fundamental ideal of $W(F)$.[[#Lam (2005) p.115|[Lam (2005) p.115]]] =='"`UNIQ--h-12--QINU`"'Examples== * $\mathrm{BW}(\mathbf{R})$ is isomorphic to $\mathbf{Z}/8$. This is an algebraic aspect of [[Bott periodicity]]. =='"`UNIQ--h-13--QINU`"'References== * Lam, Tsit-Yuen, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''', (American Mathematical Society, 2005) ISBN 0-8218-1095-2 [https://mathscinet.ams.org/mathscinet/article?mr=2104929 MR2104929], [https://zbmath.org/?q=an%3A1068.11023 Zbl 1068.11023] * Wall, C. T. C., "Graded Brauer groups", ''Journal für die reine und angewandte Mathematik'' '''213''' (1964) 187–199, ISSN 0075-4102, [https://zbmath.org/?q=an%3A0125.01904 Zbl 0125.01904], [https://mathscinet.ams.org/mathscinet/article?mr=0167498 MR0167498] ='"`UNIQ--h-14--QINU`"'Factor system= A function on a [[group]] giving the data required to construct an [[algebra]]. A factor system constitutes a realisation of the cocycles in the second [[cohomology group]] in [[group cohomology]]. Let $G$ be a group and $L$ a field on which $G$ acts as automorphisms. A ''cocycle'' or ''factor system'' is a map $c : G \times G \rightarrow L^*$ satisfying $$ c(h,k)^g c(hk,g) = c(h,kg) c(k,g) \ . $$ Cocycles are ''equivalent'' if there exists some system of elements $a : G \rightarrow L^*$ with $$ c'(g,h) = c(g,h) (a_g^h a_h a_{gh}^{-1}) \ . $$ Cocycles of the form $$ c(g,h) = a_g^h a_h a_{gh}^{-1} $$ are called ''split''. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group $H^2(G,L^*)$. =='"`UNIQ--h-15--QINU`"'Crossed product algebras== Let us take the case that $G$ is the [[Galois group]] of a [[field extension]] $L/K$. A factor system $c$ in $H^2(G,L^*)$ gives rise to a ''crossed product algebra'' $A$, which is a $K$-algebra containing $L$ as a subfield, generated by the elements $\lambda \in L$ and $u_g$ with multiplication $$ \lambda u_g = u_g \lambda^g \ , $$ $$ u_g u_h = u_{gh} c(g,h) \ . $$ Equivalent factor systems correspond to a change of basis in $A$ over $K$. We may write $$ A = (L,G,c) \ .$$ Every [[central simple algebra]] over$K$ that splits over $L$ arises in this way. The tensor product of algebras corresponds to multiplication of the corresponding elements in$H^2$. We thus obtain an identification of the [[Brauer group]], where the elements are classes of CSAs over $K$, with $H^2$.[[#Saltman (1999) p.44|[Saltman (1999) p.44]]] =='"`UNIQ--h-16--QINU`"'Cyclic algebra== Let us further restrict to the case that $L/K$ is [[Cyclic extension|cyclic]] with Galois group $G$ of order $n$ generated by $t$. Let $A$ be a crossed product $(L,G,c)$ with factor set $c$. Let $u=u_t$ be the generator in $A$ corresponding to $t$. We can define the other generators $$ u_{t^i} = u^i $$ and then we have $u^n = a$ in $K$. This element $a$ specifies a cocycle $c$ by $$ c(t^i,t^j) = \begin{cases} 1 & \text{if } i+j < n, \\ a & \text{if } i+j \ge n. \end{cases} $$ It thus makes sense to denote $A$ simply by $(L,t,a)$. However $a$ is not uniquely specified by $A$ since we can multiply $u$ by any element $\lambda$ of $L^*$ and then $a$ is multiplied by the product of the conjugates of λ. Hence $A$ corresponds to an element of the norm residue group $(K^*/N_{L/K}L^*$. We obtain the isomorphisms $$ \mathop{Br}(L/K) \equiv K^*/\mathrm{N}_{L/K} L^* \equiv \mathrm{H}^2(G,L^*) \ . $$

References

• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Universitext. Translated from the German by Silvio Levy. With the collaboration of the translator. Springer-Verlag. ISBN 978-0-387-72487-4. Zbl 1130.12001.
• Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.