Character-like functions

Borwein and Choi define a chararacter-like function to be a function [math]\displaystyle{ \lambda_p }[/math] determined by the following properties: it is completely multiplicative, p maps to 1, all other primes map to 1 if they are quadratic residues mod p and -1 otherwise. This function agrees with the Legendre character except at multiples of p (where the Legendre character equals zero).

One can get better bounds in the Erdős discrepancy problem by sending p to -1 instead -- indeed, the best construction we know that works for all n is this modification in the case p=3.

From a theoretical point of view, it may be better to define a generalized character-like function to be any completely multiplicative function with the following properties: there exists p such that the value at any non-multiple n of p is [math]\displaystyle{ \lambda_p(n) }[/math]; the sequence [math]\displaystyle{ (x_{pn})_{n=1}^\infty }[/math] is a generalized character-like function. In other words, instead of going up by a factor of p every time, one goes up by different primes at each stage.

In general, the larger the primes are, the worse the discrepancy of the sequence will be. (It is not hard to show that the discrepancy of the Legendre character itself is around [math]\displaystyle{ \sqrt{p} }[/math].) Therefore, it may be possible to devise a measure of how character-like a function is that is better when the primes used are smaller. So far we do not have such a measure.