Rational Function Multiplicative Coefficients
                          Michael Somos 22 Feb 2011
                            somos@cis.csuohio.edu
                              (draft version 10)

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.