On totally multiplicative signatures of natural numbers (Q799722)

A totally multiplicative signature is a mapping \(\sigma\) from the natural numbers to \(\{\pm 1\}\) satisfying \(\sigma (ab)=\sigma (a)\sigma (b)\). Let \(\sigma\) be a totally multiplicative signature which takes the value \(-1\) on at least two primes. Then (i) \(\sigma (x)=-1=\sigma (x+1)\) has infinitely many solutions, and (ii) \(\sigma (x)=-1=\sigma (x+2)\), \(\sigma (x+1)=1\) has a solution and, if \(\sigma (2)=1\), infinitely many solutions. These are partial results toward a conjecture of Chowla, according to which \(\lambda(x+m)=\varepsilon_m\) \((1\le m\le g)\) has infinitely many solutions for any finite sequence \(\varepsilon_1,\ldots,\varepsilon_g\) \((\varepsilon_ m=\pm 1)\) where \(\lambda\) is Liouville's function.