Zeros of weighted partial sums of completely multiplicative functions: Dirichlet characters, the Euler factor at 2 and Conrey's sine series

A completely multiplicative function ε with values ±1 can make ∑n≤m n ε(n) vanish only when m ≡ 0, 3 (mod 4). We show that some ε does for every such m. The primes of (m/2, m] enter the sum only as themselves: a second moment fixes the other signs with a small remainder, and the signed sums of those primes cover it, by interval certificates up to 108 and by Sárközy's progressions in subset sums beyond. Every finite prefix of signs extends to a zero at all large admissible m, so the functions with infinitely many zeros form a dense Gδ of probability zero. Quadratic characters, extended completely multiplicatively, give explicit points of it: a digit recursion propagates their zeros in base q, for q ≡ 1 (mod 4), for q ≡ 7 (mod 8) — where the half sum of Jacobi and Dirichlet, with its Euler factor at 2, does the work — and along loops of the recursion, as for class number one. For odd characters, the digits that propagate zeros are the intervals on which Conrey's sine series of the Legendre symbol, proposed as a route to the Riemann hypothesis, vanishes identically. The same Euler factor decides which components of the moment matrix of the ring of character values vanish on its diagonal, which is where this note comes from.