Rational Function Multiplicative Coefficients
Michael Somos 11 Dec 2014
ms639@georgeown.edu
(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 - 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 example 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. Now consider the rational function and its power series expansion
 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 infinite family is
 f(x) = x (1 - xn-1) = x - xn
where n > 1. Also, if and only if n = pk, n > 1 and p is prime then
 f(x) = x (1 - x)-1 (1 - xn-1) (1 - xn)-1
is multiplicative. Note that 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 its 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. For example,
 g(x) = (1 - x)-1 (1 - xn)-1 (1 - xn+1)
is homogeneous multiplicative if and only if n = pk, 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.

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On 11 Dec 2014, 01:42.