Negative values of truncations to L ( 1 , χ )

∑ n≤x λ(n) is non-positive for all x ≥ 2, which also implies the Riemann Hypothesis). In [4] Haselgrove showed that both the Turán and Pólya conjectures are false; therefore we know that truncations to L(1, χ) may get negative. Let F denote the set of all completely multiplicative functions f(·) with −1 ≤ f(n) ≤ 1 for all positive integers n, let F1 be those for which each f(n) = ±1, and F0 be those for which each f(n) = 0 or ±1. Given any x and any f ∈ F0 we may find a primitive quadratic character χ with χ(n) = f(n) for all n ≤ x (again, by using quadratic reciprocity and Dirichlet’s theorem on primes in arithmetic progressions) so that, for any x ≥ 1,