Unitary divisor explained

In mathematics, a natural number a is a unitary divisor (or Hall divisor) of a number b if a is a divisor of b and if a and

(b)/(a) are coprime, having no common factor other than 1. Equivalently, a divisor a of b is a unitary divisor if and only if every prime factor of a has the same multiplicity in a as it has in b.

The concept of a unitary divisor originates from R. Vaidyanathaswamy (1931),^[1] who used the term block divisor.

Example

5 is a unitary divisor of 60, because 5 and

(60)/(5) =12 have only 1 as a common factor.

On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and

(60)/(6) =10 have a common factor other than 1, namely 2.

Sum of unitary divisors

The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: σ*(n). The sum of the k-th powers of the unitary divisors is denoted by σ*_k(n):

\sigma ^{*(n)} _{k} = \sum _{d} d ^{k.}

If the proper unitary divisors of a given number add up to that number, then that number is called a unitary perfect number.

Properties

Number 1 is a unitary divisor of every natural number.

The number of unitary divisors of a number n is 2^k, where k is the number of distinct prime factors of n. This is because each integer N > 1 is the product of positive powers p^r_p of distinct prime numbers p. Thus every unitary divisor of N is the product, over a given subset S of the prime divisors of N,of the prime powers p^r_p for p ∈ S. If there are k prime factors, then there are exactly 2^k subsets S, and the statement follows.

The sum of the unitary divisors of n is odd if n is a power of 2 (including 1), and even otherwise.

Both the count and the sum of the unitary divisors of n are multiplicative functions of n that are not completely multiplicative. The Dirichlet generating function is

(\zeta(s)\zeta(s - k))/(\zeta( 2 s - k)) = \sum _{n\ge} (\sigma ^{*(n)} _{k})/(n ^{s}) .

Every divisor of n is unitary if and only if n is square-free.

Odd unitary divisors

The sum of the k-th powers of the odd unitary divisors is

\sigma ^{(o)*} _{k} (n) = \sum _{{d} \atop \gcd(d,n/d) =1 } d ^{k.}

It is also multiplicative, with Dirichlet generating function

(\zeta(s)\zeta(s - k)( 1-2 ^{k-s} ))/(\zeta( 2 s - k)( 1-2 ^{k-2s} )) = \sum _{n\ge} (\sigma ^{(o)*} _{k} (n))/(n ^{s}) .

Bi-unitary divisors

A divisor d of n is a bi-unitary divisor if the greatest common unitary divisor of d and n/d is 1. This concept originates from D. Suryanarayana (1972). [The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics 251: 273–282, New York, Springer–Verlag].

A log x where^[2] A = \prod _{p\left({1} - (p -1)/(p ^{2(p+1)}) }\right) .

A bi-unitary perfect number is one equal to the sum of its bi-unitary aliquot divisors. The only such numbers are 6, 60 and 90.^[3]

References

• Book: Richard K. Guy

. Richard K. Guy. Richard K. Guy. Unsolved Problems in Number Theory. Springer-Verlag. 2004. 0-387-20860-7 . 84. Section B3.

• Book: Paulo Ribenboim

. My Numbers, My Friends: Popular Lectures on Number Theory . Paulo Ribenboim . Paulo Ribenboim . Springer-Verlag . 2000 . 0-387-98911-0 . 352 .

• Eckford. Cohen. A class of residue systems (mod r) and related arithmetical functions. I. A generalization of Möbius inversion. Pacific J. Math.. 9. 1 . 13–23. 1959. 0109806. 10.2140/pjm.1959.9.13. free.
• Eckford. Cohen. Arithmetical functions associated with the unitary divisors of an integer. Mathematische Zeitschrift. 74. 1960. 66–80. 0112861 . 10.1007/BF01180473. 53004302.
• Eckford. Cohen. The number of unitary divisors of an integer. 67. 9. 879–880. 0122790. 1960. American Mathematical Monthly. 10.2307/2309455. 2309455.
• Graeme L.. Cohen. On an integers' infinitary divisors. 54. 189 . 395–411. 0993927. 10.1090/S0025-5718-1990-0993927-5. Math. Comp.. 1990. 1990MaCom..54..395C. free.
• Graeme L.. Cohen. Arithmetic functions associated with infinitary divisors of an integer. 16. 2. 373–383. 10.1155/S0161171293000456. Int. J. Math. Math. Sci.. 1993. free.
• Web site: Steven. Finch. Unitarism and Infinitarism. 2004.
• Book: Ivić, Aleksandar . The Riemann zeta-function. The theory of the Riemann zeta-function with applications . A Wiley-Interscience Publication . New York etc. . John Wiley & Sons . 1985 . 0-471-80634-X . 0556.10026 . 395 .
• 1106.4038 . R. J.. Mathar. 2011. Survey of Dirichlet series of multiplicative arithmetic functions. math.NT. Section 4.2
• Book: Sándor . József . Mitrinović . Dragoslav S. . Crstici . Borislav . Handbook of number theory I . Dordrecht . . 2006 . 1-4020-4215-9 . 1151.11300 .
• L. . Toth. On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function. 2009 . J. Int. Seq. . 12.

External links

• • Mathoverflow | Boolean ring of unitary divisors

Notes and References

1. R. Vaidyanathaswamy . The theory of multiplicative arithmetic functions . Transactions of the American Mathematical Society . 33 . 2 . 579-662 . 1931 . 10.1090/S0002-9947-1931-1501607-1. free .
2. Ivić (1985) p.395
3. Sandor et al (2006) p.115