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Unitary divisor

A natural number $d$ is a unitary divisor of a number $n$ if $d$ is a divisor of $n$ and $d$ and $n/d$ are coprime, having no common factor other than 1. Equivalently, $d$ is a unitary divisor of $n$ if and only if every prime factor of $d$ appears to the same power in $d$ as in $n$.

The sum of unitary divisors function is denoted by $\sigma^*(n)$. The sum of the $k$-th powers of the unitary divisors is denoted by $\sigma_k^*(n)$. These functions are multiplicative arithmetic functions of $n$ that are not totally multiplicative. The Dirichlet series generating function is

$$ \sum_{n\ge 1}\sigma_k^*(n) n^{-s} = \frac{\zeta(s)\zeta(s-k)}{\zeta(2s-k)} . $$

The number of unitary divisors of $n$ is $\sigma_0(n) = 2^{\omega(n)}$, where $\omega(n)$ is the number of distinct prime factors of $n$.

A unitary or unitarily perfect number is equal to the sum of its aliquot unitary divisors:equivalen tly, it is n such that $\sigma^*(n) = 2n$. A unitary perfect number must be even. It is not known whether or not there are infinitely many unitary perfect numbers, or indeed whether there are any further examples beyond the five already known. A sixth such number would have at least nine odd prime factors. The five known are

$$ 6 = 2\cdot3,\ 60 = 2^2\cdot3\cdot5,\ 90 = 2\cdot3^3\cdot5,\ 87360 = 2^6\cdot3\cdot5\cdot7\cdot13, $$ and $$ 146361946186458562560000 = 2^{18}\cdot3\cdot5^4\cdot7\cdot11\cdot13\cdot19\cdot37\cdot79\cdot109\cdot157\cdot313\ . $$

References

• Guy, Richard K. Unsolved Problems in Number Theory, Problem Books in Mathematics, 3rd ed. (Springer-Verlag, 2004) p.84, section B3. ISBN 0-387-20860-7 Zbl 1058.11001
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. (Dordrecht: Kluwer Academic, 2004) pp. 179–327. ISBN 1-4020-2546-7. Zbl 1079.11001
• Wall, Charles R. "The fifth unitary perfect number", Can. Math. Bull. 18 (1975) 115-122. ISSN 0008-4395. Zbl 0312.10004
• Wall, Charles R. "New unitary perfect numbers have at least nine odd components". Fibonacci Quarterly 26 no.4 (1988) ISSN 0015-0517. MR967649. Zbl 0657.10003

Descartes number

A number which is close to being a perfect number. They are named for René Descartes who observed that the number

$$D= 198585576189 = 3^2⋅7^2⋅11^2⋅13^2⋅22021 $$

would be an odd perfect number if only 22021 were a prime number, since the sum-of-divisors function for $D$ satisfies

$$\sigma(D) = (3^2+3+1)\cdot(7^2+7+1)\cdot(11^2+11+1)\cdot(13^3+13+1)\cdot(22021+1) \ . $$

A Descartes number is defined as an odd number $n = m p$ where $m$ and $p$ are coprime and $2n = \sigma(m)\cdot(p+1)$. The example given is the only one currently known.

If $m$ is an odd almost perfect number, that is, $\sigma(m) = 2m-1$, then $m(2m−1)$ is a Descartes number.

References

• Banks, William D.; Güloğlu, Ahmet M.; Nevans, C. Wesley; Saidak, Filip (2008). "Descartes numbers". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (edd). Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006. CRM Proceedings and Lecture Notes 46 (Providence, RI: American Mathematical Society) pp. 167–173. ISBN 978-0-8218-4406-9. Zbl 1186.11004.

Multiplicative sequence

Also m-sequence, a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.

Definition

Let $K_n$ be polynomials over a ring $A$ in indeterminates $p_1,\ldots$ weighted so that $p_i$ has weight $i$ (with $p_0=1$) and all the terms in $K_n$ have weight $n$ (so that $K_n$ is a polynomial in $p_1,\ldots,p_n$). The sequence $K_n$ is multiplicative if an identity

$$\sum_i p_i z^i = \sum p'_i z^i \cdot \sum_i p''_i z^i $$

implies

$$\sum_i K_i(p_1,\ldots,p_i) z^i = \sum_j K_j(p'_1,\ldots,p'_j) z^j \cdot \sum_k K_k(p''_1,\ldots,p''_k) z^k . $$ The power series

$$\sum K_n(1,0,\ldots,0) z^n $$

is the characteristic power series of the $K_n$. A multiplicative sequence is determined by its characteristic power series $Q(z)$, and every power series with constant term 1 gives rise to a multiplicative sequence.

To recover a multiplicative sequence from a characteristic power series $Q(z)$ we consider the coefficient of z^j in the product

$$ \prod_{i=1}^m Q(\beta_i z) $$

for any $m>j$. This is symmetric in the $\beta_i$ and homogeneous of weight j: so can be expressed as a polynomial $K_j(p_1,\ldots,p_j)$ in the elementary symmetric functions $p$ of the $\beta$. Then $K_j$ defines a multiplicative sequence.

Examples

As an example, the sequence $K_n = p_n$ is multiplicative and has characteristic power series $1+z$.

Consider the power series

$$ Q(z) = \frac{\sqrt z}{\tanh \sqrt z} = 1 - \sum_{k=1}^\infty (-1)^k \frac{2^{2k}}{(2k)!} B_k z^k $$ 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 polynomials.

Genus

The genus of a multiplicative sequence is a ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers.

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 $$\frac{\sqrt z}{\tanh \sqrt z} . $$

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. Zbl 0843.14009.

Nagao's theorem

A result, named after Hirosi Nagao, about the structure of the group of 2-by-2 invertible matrices over the ring of polynomials over a field. It has been extended by Serre to give a description of the structure of the corresponding matrix group over the coordinate ring of a projective curve.

Nagao's theorem

For a 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 $B(K)$.

Serre's extension

In this setting, $C$ is a 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 $\pi_1(G,T)$.

References

• Mason, A.. "Serre's generalization of Nagao's theorem: an elementary approach". Transactions of the American Mathematical Society 353 (2001) 749–767. DOI 10.1090/S0002-9947-00-02707-0 Zbl 0964.20027.
• Nagao, Hirosi. "On $GL(2, K[x])$". J. Inst. Polytechn., Osaka City Univ., Ser. A 10 (1959) 117–121. MR0114866. Zbl 0092.02504.
• Serre, Jean-Pierre. Trees. (Springer, 2003) ISBN 3-540-44237-5.