Rational Function Multiplicative Coefficients
Michael Somos 23 Sep 2009
somos@cis.csuohio.edu
(draft version 8)

1  Rational generating functions of multiplicative sequences

A numerical sequence has an associated generating function (GF). For example, the Fibonacci sequence is associated with GF x / (1 - x - x2), 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) x2 + a(3) x3 + ... ever be rational? The answer is yes if f(x) = x / (1 - x) and a(n) = 1 if n > 0. This is the simplest case where a(n) is non-zero for all n > 0. Another is f(x) = x / (1 - x2) and a(n) = 1 if n > 0 is odd and a(n) = 0 otherwise. Consider the rational function and its power series
f(x) = x (1 - x)e1 (1 - x2)e2 = x - e1 x2 + ((e12 - e1)/2 - e2) x3 + ...
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 - x2)e2 ... (1 - xn)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 is
f(x) = x (1 - xn-1) = x - xn
where n > 1 is any prime power. Another is
f(x) = x (1 - x)-1 (1 - xp-1) (1 - xp)-1
where p is any prime. Note f(x) is in the set when -f(-x) is since (1 + x) = (1 - x2) / (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 the GF
g(x) = (1 - x)2 / (1 - x2) = 1 - 2 x + 2 x2 - 2 x3 + ... .
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 - x2)e2 = 1 - e1 x + ((e12 - e1)/2 - e2) x2 + ...
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 - x2)e2 ... (1 - xn)en
for some integers e1, ..., en which are the GF for homogeneous multiplicative integer sequences provided we exclude some infinite families which are predictable. One example of two families are
g(x) = (1 - x)-1 (1 - xn)-1 (1 - xn+1)
and
g(x) = (1 - x) (1 - x2)-1 (1 - xn) (1 - xn+1) (1 - x2n)-1
where n is any prime power. Another is
g(x) = (1 - x) (1 - x2)-1 (1 - xn)-1 (1 - xn+1)-1 (1 - x2n+1)
where n = 2k for integer k > 0. Note 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 h- Quotients by Yves Martin in 1996.

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On 23 Sep 2009, 10:59.