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 - x^{2}), a rational
function of x. Consider a multiplicative sequence. That is, a(1) = 1 and
a(nm) = 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

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.

for some integers e_{1}, ..., e_{n} 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

Then a(1) a(nm) = 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

where e_{1} and e_{2} are integers. A search finds that g(x) is the GF of a
homogeneous multiplicative sequence for 10 pairs of integers [e_{1},e_{2}]
as follows:

for some integers e_{1}, ..., e_{n} which are the GF for homogeneous
multiplicative integer sequences provided we exclude some infinite families
which are predictable. For example,

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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