PRH | Inter | 5.2 • An Implementable Phase-Selection Algorithm for Helson Zeta

An Implementable Phase-Selection Algorithm for Helson Zeta Functions with Prescribed Zeros on Expanding Windows Aleksandar Perišić August 2025 Abstract A Helson zeta has unimodular, completely multiplicative coefficients: ζχ(s) = X n≥1 χ(n)n−s=Y p (1 −χ(p)p−s)−1(ℜs > 1), where χ : N→T is completely multiplicative, |χ ( n ) | = 1. Andersson (2024) proved that any prescribed discrete zero/pole multiset in the half-plane ℜs < 1(with mild local restrictions) is realized by a Helson zeta. We give a software-ready construction: a global, all-at-once optimization on the prime torus with exact analytic gradients, run under a continuation schedule in the prime cutoff and the real part, together with short rigorous guardrails (uniform short-block linearization, grid → sup upgrade, summable-error schedule, and Rouché/Hurwitz zero transfer). The method yields a completely multiplicative χ for which ζχ matches any target zero pattern on expanding windows. We also show how to enforce pole-freeness at s = 1 and (optionally) place all zeros on ℜs=1 2. Note. A blur/Fourier route to the Helson factorization on {ℜs > 1 } is given independently by the author (2025), and can serve as a conceptual backdrop for the continuation/guardrails used here. 1 Introduction and statement of contributions Write χ ( n ) = eiΘ(n) with Θcompletely additive (Θ( mn ) = Θ( m ) + Θ( n )). Thus χ ( p ) = eiθp are prime phases and the Euler product depends only on the vector θ= (θp)p∈P . Existence backdrop. Andersson shows: if f is analytic and zero-free on {ℜs > 1 } and f′/f extends continuously to {ℜs≥ 1 } , then there exist a zero-free entire g and a Helson zeta ζχ with f = g ζχ on {ℜs > 1 } . By the Mittag–Leffler paradigm, one may then prescribe arbitrary zeros/poles in {ℜs < 1 } for ζχ under mild local restrictions (no accumulation in U , at most along 1 + iR ). Earlier, Bochkov–Romanov and Seip obtained flexibility only in strips to the right of ℜs = 1 2 (unconditionally for ℜs > 21 / 40 and, assuming RH, for 1 / 2 <ℜs < 1). Our goal is effective synthesis: pick θ by an algorithm that an engineer can implement and that comes with proofs of convergence on expanding windows. Contributions. 1 • Global objective with exact gradients. For primes p≤P we optimize phases θp to minimize a loss that (i) drives ZP ( s ; θ )to 0at target points near desired zeros, (ii) penalizes spurious zeros at guard points, and (iii) enforces moment constraints at s = 1 (eliminating the pole and low-order power resonances). • Continuation schedule and guarantees. We prove small lemmas ensuring that with short prime blocks and an expanding window sequence, the log-error is summable; the partial products converge uniformly to a nonvanishing limit factor; and zeros/multiplicities are transferred by Rouché/Hurwitz. • Critical-line option. To put all zeros on ℜs = 1 2 , use a target model F whose zeros are precisely {1 2+iγ}(Hadamard product); the algorithm applies verbatim. Notation For pprime, set ap(s) := p−s. For a phase vector θ= (θp)p≤Pdefine ZP(s;θ) = Y p≤P1−eiθpap(s)−1,log ZP(s;θ) = −X p≤P log1−eiθpap(s), with a continuous branch of log fixed on each working window. 2 Target zeros and model function F Let Z be a prescribed finite or infinite multiset of zeros in {ℜs < 1 } with no accumulation in {ℜs > 1 } ∪ (1 + iR ). Pick an entire H with zero set Z and construct a holomorphic F on {ℜs≥1 2} , for instance via a Hadamard product adjusted by a canonical zero-free factor so that F is nonzero on {ℜs≥1}and F(1) = 0. Remark 2.1 (All zeros on the critical line).To force all zeros on ℜs = 1 2 , define Z = {1 2 + iγn} with any prescribed ordinate multiset (subject to local finiteness), and build F accordingly. The algorithm below then produces χ so that ζχ shares these zeros on expanding windows (and globally if one continues the exhaustion). 3 Two constructions: (A) implementable global objective; (B) blockwise proof engine 3.1 (A) Global objective (all phases at once) Fix target points S = {sj} near the desired zeros and guard points G away from zeros, all within a working window K={σ0≤ ℜs≤1,|ℑs| ≤ T}, typically with σ0= 1 initially. Define the loss JP(θ) = X s∈S w(s)|ZP(s;θ)|2+X t∈G µ(t) |ZP(t;θ)|2 +λ1X p≤P cos θp p 2+ K0 X k=2 λkX p≤P cos(kθp) p 2. The first line drives ZP to vanish at targets and stay bounded away from 0at guards. The second line enforces moment constraints at s = 1 (removing the pole and low-order prime-power resonances). 2 Exact gradients. For each prime q≤P, ∂θqlog ZP(s;θ) = ieiθqaq(s) 1−eiθqaq(s), ∂θqZP(s;θ) = ZP(s;θ)∂θqlog ZP, hence ∂θq|ZP(s;θ)|2= 2 ℜZP(s;θ)∂θqZP(s;θ)= 2 |ZP|2ℜ ieiθqaq(s) 1−eiθqaq(s)!. The moment terms differentiate trivially: ∂θqPp≤P cos(kθp) p=−k qsin(kθq). Continuation schedule. Start at ( σ0, P ) = (1 , P0 )with small |S|,|G| ; minimize JP (e.g. by L–BFGS). Then continue: enlarge ( S, G )to discretize a larger window K , increase P , and decrease σ0toward 1 2+δ. This mirrors the proofs below. 3.2 (B) Blockwise construction (proof engine) For X≫1and 0<ϑ<1, a short prime block is B(X) := {p:X < p ≤X+Xϑ}. Theorem 3.1 (Uniform short-block linearization).Let K⊂ {ℜs≥1 2 + δ} compact. For any phases {θp}p∈B(X), X p∈B(X) log1−eiθpap(s)−1=X p∈B(X) eiθpap(s) + OKX−2δ+o(1), uniformly on s∈K. Sketch. Use log (1 −z ) −1 = z + O ( z2 )with z = eiθpp−s , |z| ≤ p−(1 2+δ) , and sum the quadratic tail Pp∈B(X)p−1−2δ≪X−2δ+o(1). Lemma 3.2 (Grid → sup upgrade).Let K = [ σ0, 1] × [ −T, T ]and {sℓ} be a grid with vertical mesh h≪ 1 /T . If H is holomorphic near K , maxℓ|H ( sℓ ) | ≤ ε , and supK|H′ ( s ) | ≤ L , then supK|H(s)| ≤ ε+CKLh. Theorem 3.3 (All-at-once linear fit on a short block).Let K be as above and W holomorphic near K . For X large let B ( X )be a short prime block. Then for any ε > 0there exist phases {θp}p∈B(X) such that sup s∈KX p∈B(X) eiθpap(s)−W(s) < ε, and simultaneously X p∈B(X) cos(kθp) p−ξk < εk,1≤k≤K0, for prescribed targets ξk(e.g. 0). Sketch. Discretize Kby a grid {sℓ}m ℓ=1 and solve the underdetermined linear system X p∈B(X) cpap(sℓ)≈W(sℓ) with linear moment constraints in least squares. Project cp to the unit circle: cp7→ cp/|cp| . By near-orthogonality of columns ( ap ( sℓ )) ℓ when p ranges over a short block and the grid is fine, this degrades the fit by o(1) for large X. Lift the grid control to Kvia Lemma 3.2. 3 Theorem 3.4 (Summable-error schedule and uniform convergence).Choose an exhaustion Kj = [ 1 2 + δj, 1] × [ −Tj, Tj ]with δj↓ 0, Tj↑ ∞ , and tolerances εj↓ 0with Pjεj<∞ . Let Xj+1 ≫Xj and Bj:= B(Xj). Define recursively Wj(s) := log F(s)−X p≤Xj log1−eiθpap(s). Pick phases on Bj by Theorem 3.3 so that sups∈KjPp∈Bjeiθpap(s)−Wj(s)< εj and enforce the moment constraints at s= 1. Then by Theorem 3.1 sup s∈KjX p∈Bj log1−eiθpap(s)−1−Wj(s) ≪εj. Hence the partial products ZPj ( s ) := Qp≤Xj (1 −eiθpap ( s )) −1 satisfy ZPj ( s ) /F ( s ) →G ( s )uniformly on each Km, where Gis holomorphic and nonvanishing. Corollary 3.5 (Zero transfer).Let D be a finite union of disjoint discs contained in some Km , each centered at a zero of Fwith multiplicity. Then for jlarge, ZPjand Fhave the same number of zeros (with multiplicity) in each disc. The uniform limit ζχ = limjZPj satisfies ζχ = F·G on D with Gholomorphic and nonvanishing; thus ζχinherits the prescribed zeros in D. Proof. Rouché’s theorem on each disc. 4 Eliminating the pole at s= 1 and power resonances Define prime moments Mk ( P ; θ ) := Pp≤P cos(kθp) p . These control the singularity order at s = 1 and low-order prime-power resonances. Enforce Mk ( P ; θ ) ≈ 0along the continuation for 1 ≤k≤K0 (either as hard linear constraints within Theorem 3.3 or as quadratic penalties in JP ). In the limit this removes the s = 1 pole (and first K0 resonances) while preserving the Euler product and complete multiplicativity. 5 How to specialize to “all zeros on ℜs=1 2” To realize a critical-line zero set, set Z = {1 2 + iγn} and build F with precisely these zeros. Run Construction (A) or (B). No further changes are required; the only place where “ 1 2 ” appears is in the choice of Z used to define F . (Practically, keep ℜs≥1 2 + δ during fitting and let δ↓ 0along the continuation.) 6 Practical notes (for implementation) • Windows and grids. Start with K = [1 , 1] × [ −T0, T0 ], then increase T , enlarge S, G , and lower σ0. • Optimization. Use L–BFGS/Adam on θ∈ ( −π, π ] π(P) with modular wrap; vectorize evaluations of ZPand its gradients. • Derivative terms (optional). To encourage simple zeros at sj add −ηjlog |Z′ P ( sj ) | or ηj(|Z′ P(sj)| − cj)2to the loss; compute ∂slog ZP(s) = Pp≤P eiθp(−log p)ap(s) 1−eiθpap(s). •Tail beyond P.Keep ℜs > 1 2while Pgrows; tails are small and controlled. 4 7 Discussion and relation to prior work Our method turns Andersson’s existence theorem into an explicit recipe that an engineer can implement without reproducing long proofs. The guarantees here are local-to-global: they ensure uniform approximation on expanding windows and zero transfer, and can be pushed to global realization by continuing the exhaustion toward ℜs = 1 2 and |ℑs|→∞ , or by appealing directly to Andersson for the global statement. Data and code. This paper provides the mathematical spec for a future software package implementing Construction (A). Code is deferred to a separate repository. References [1] J. Andersson, Mittag–Leffler type theorems for Helson zeta-functions, arXiv:2408.15713, 2024. [2] I. Bochkov and R. Romanov, On zeroes and poles of Helson zeta functions, J. Funct. Anal. 282 (2022), 109398. [3] K. Seip, Universality and distribution of zeros and poles of some zeta functions, J. Anal. Math. 141 (2020), 331–381. [4] H. Helson, Compact groups and Dirichlet series, Ark. Mat. 8(1969), 139–143. [5] E. C. Titchmarsh (revised by D. R. Heath-Brown), The Theory of the Riemann Zeta-Function, 2nd ed., Oxford Univ. Press, 1986. [6] A. Perišić, Mittag–Leffler for Helson Zeta via Blur: a short proof without short-interval primes, preprint, Sept. 2025. 5