Rational Function Multiplicative Coefficients
Michael Somos 11 Dec 2014
michael.somos@gmail.com
(draft version 11)
1 Rational generating functions of multiplicative sequences
Any numerical sequence has an associated generating function (GF). For example,
the Fibonacci sequence is associated with GF x / (1 - x - x^2) , a rational
function of x . Consider a multiplicative sequence. That is, a(1) = 1 and
a(n m) = a(n) a(m) for all positive integers n and m relatively prime
to each other. Can its GF f(x) = a(1) x + a(2) x^2 + a(3) x^3 + ... ever be
rational? The answer is yes if f(x) = x / (1 - x) and a(n) = 1 if n > 0 .
This is the simplest example where a(n) is non-zero for all n > 0 . Another
is f(x) = x / (1 - x^2) and a(n) = 1 if n > 0 is odd and a(n) = 0
otherwise. Now consider the rational function and its power series expansion
f(x) = x (1 - x)^e1 (1 - x^2)^e2 = x - e1 x^2 + ((e1^2 - e1)/2 - e2) x^3 + ...
where e1 and e2 are integers. A search finds that f(x) is the GF of a
multiplicative sequence for 11 pairs of integers [e1,e2] as follows:
[-4,1],[-2,0],[-1,0],[-1,1],[0,-1],[0,0],[0,1],[1,-1],[1,0],[2,-2],[4,-3].
The multiplicative integer sequences for these pairs are of a simple form. Some
algebra is enough to prove that this list is complete. Allowing more factors in
f(x) increases the difficulty of search and algebraic proof.
2 Conjecture 1
Conjecture 1: there is a finite set of rational functions of the form
f(x) = x (1 - x)^e1 (1 - x^2)^e2 ... (1 - x^n)^en
for some integers e1, ..., en which are the GF for multiplicative integer
sequences provided we exclude some infinite families which are predictable.
One example infinite family is
f(x) = x (1 - x^n-1) = x - x^n
where n > 1 . Also, if and only if n = p^k, n > 1 and p is prime then
f(x) = x (1 - x)^-1 (1 - x^n-1) (1 - x^n)^-1
is multiplicative. Note that f(x) is in the set when -f(-x) is since
(1 + x) = (1 - x^2) / (1 - x) and so on.
3 Homogeneous generalization of multiplicative sequences
Now assume a(0) and a(1) are nonzero. For example, consider the sequence
a(n) with its GF
g(x) = (1 - x)^2 / (1 - x^2) = 1 - 2 x + 2 x^2 - 2 x^3 + ... .
Then a(1) a(n m) = a(n) a(m) for all positive integers n and m relatively
prime to each other. This is a homogeneous generalization of multiplicative
sequences. As in the first section, but without a factor of x , consider
g(x) = (1 - x)^e1 (1 - x^2)^e2 = 1 - e1 x + ((e1^2 - e1)/2 - e2) x^2 + ...
where e1 and e2 are integers. A search finds that g(x) is the GF of a
homogeneous multiplicative sequence for 10 pairs of integers [e1,e2]
as follows:
[-4,2],[-2,1],[-2,2],[-1,0],[-1,1],[1,-1],[1,0],[2,-1],[2,0],[4,-2].
Again, algebra is enough to prove the list is complete.
4 Conjecture 2
Conjecture 2: there is a finite set of rational functions of the form
g(x) = (1 - x)^e1 (1 - x^2)^e2 ... (1 - x^n)^en
for some integers e1, ..., en which are the GF for homogeneous
multiplicative integer sequences provided we exclude some infinite families
which are predictable. For example,
g(x) = (1 - x)^-1 (1 - x^n)^-1 (1 - x^n+1)
is homogeneous multiplicative if and only if n = p^k, n > 1 and p is prime.
Note that g(x) is in the set when g(-x) is.
5 Further Work
The rational functions in the two conjectures have applications related to
Ramanujan's Lambert series. A study of rational functions with poles only at
roots of unity appeared in 2003 by Juan B. Gil and Sinai Robins who defined a
Hecke operator on power series. Kyoji Saito studied cyclotomic functions
related to eta-products in 2001. Rational functions of a simple form having
multiplicative coefficients is related to a paper on Multiplicative eta -
Quotients by Yves Martin in 1996.