Multiplicative functions with small partial sums

I will discuss Klurman’s classification of multiplicative functions taking ±1 with bounded partial sums, and extend this result to multiplicative functions taking values outside the unit disk. I also will talk about partial sums of modified Dirichlet characters, these are completely multiplicative functions that differ from some χ at a finite set of primes.

These partial sums are not well understood and I will present conjectures and examples.