  Consider a multiplicative function f(n) taking values on the unit circle. Is
it possible that the partial sums of this function are bounded? We show that if
we weaken the notion of multiplicativity so that f(pn)=f(p)f(n) for all primes
p in some finite set P, then the answer is yes. We also discuss a result of
Bronstein that shows that functions modified from characters at a finite number
of places.
