🗺️ Map of Exact zeros of weighted sums of completely multiplicative functions

🏷️ Number theory · arithmetic functions 11A25 11K65 11L40 11P70 11R29

17 results ✅ 4 formalized in Lean ◐ 4 with a part in Lean 💻 11 checked by a program written proof only (a result can carry several tags)

Dependencies in the written proof: an arrow A → B means that this proof of B uses A. Every result rests on its written proof; the tags record what was checked besides, and each statement's “Evidence & scope” block says how far. The colour of a box is its strongest tag, in the order above; the tags in the box give the rest. Boxes with a thick border are the main results; click a box to read the statement.

What the paper contributes

For a completely multiplicative ε with values ±1, the weighted sum S_ε(m) = Σ_n≤m n ε(n) can vanish only when m ≡ 0 or 3 (mod 4).

Every such length is a zero of some ε, depending on m (Theorem A), even with the signs at finitely many primes prescribed in advance, once m is large (Proposition 3.1).
Zeros of one fixed function are rare in measure and generic in category: the ε with infinitely many zeros form a dense G_δ of probability zero (Theorem B). For the unweighted sums the picture is the reverse: they vanish infinitely often almost surely, by a theorem of Aymone.
Quadratic characters give explicit zeros. For primes q ≡ 7 (mod 8), S_χ((q^j−1)/2) = 0 for every j, and Conrey's sine series of the Legendre symbol vanishes identically on an interval (Corollary C). The case j = 1 is classical (Lebesgue; Aebi–Cairns); the iteration in j and the null interval are the corollary's content. They come from a digit recursion in base q (Theorem 4.1) and the weighted half sum (Theorem 4.2, due to Kanemitsu–Kuzumaki and Kučera).
Added after submission: for q = 179 and q = 227 the function χ_q,1 has no zero at all (Theorem N1), by a finite certificate checked in Lean.

Main tools: a second-moment bound and Sárközy's progressions in subset sums of primes (Theorem A); the anticoncentration theorem of Sárközy and Szemerédi (Theorem B).

🚀 Start here: Theorem A: every admissible length is a zero of some ε.

Main results:

Theorem A: every admissible length is a zero of some ε
Theorem B: infinitely many zeros: dense G_δ, but measure zero
Corollary C: zeros of χ at (q^j−1)/2; Conrey's series vanishes on an interval
Theorem N1 (new): no zeros at all for q = 179 and q = 227

🧭 Fields: where the paper lives and what it connects

Tools taken from other branches

Additive combinatorics → arithmetic progressions in subset sums (Sárközy, with Lev's constants in the form of Bringmann–Wellnitz); complete sequences — Theorem A, Lemma 2.2
Probability → a second-moment bound; the anticoncentration theorem of Sárközy and Szemerédi — Lemma 2.1, Theorem B
Topology (Baire category) → dense G_δ sets in the product space of sign patterns — Theorem B
Algebraic number theory → the class number formula, through the weighted half sum (Kanemitsu–Kuzumaki, Kučera) — Theorem 4.2, Corollary C
Formal verification → Lean 4 + Mathlib: the digit recursion, its propagation step, complete sequences, and the finite certificate of Theorem N1 — Theorem 4.1, Corollary 4.3, first statement, Theorem N1 (new)

What the results serve

Probabilistic number theory ← random multiplicative functions: with the weight n the picture of the unweighted sums (Aymone) is reversed — Theorem A, Theorem B
Analytic number theory (character sums) ← Conrey's sine series of the Legendre symbol, from his approach to the Riemann hypothesis: it vanishes on an interval for q ≡ 7 (mod 8); for q ≡ 3 (mod 8) the null cells are excluded under a positivity hypothesis — Corollary C, Proposition 4.6
Partial sums of Legendre symbols ← the digit recursion of Borwein–Choi–Coons and of Klurman et al., with the weight added — Theorem 4.1, Corollary 4.3