Rational Function Multiplicative Coefficients
Michael Somos 11 Dec 2014
(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(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) 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
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.
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
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.
File translated from
version 3.82. On 11 Dec 2014, 01:42.