On Completely multiplicative $\pm 1$ sequences that omit many consecutive $+1$ values

We investigate the construction of $\pm 1$-valued completely multiplicative functions that take the value $+1$ at at most $k$ consecutive integers, which we call length-$k$ functions. We introduce a way to extend the length based on the idea of the “rotation trick” and such an extension can be quantified by the number of modified primes. Under the assumption of Elliott’s conjecture, this method allows us to construct length-$k$ functions systematically for $k\ge 4$ which generalizes the work of I. Schur for $k=2$ and R. Hudson for $k=3$.