HBP | Main | 1.1 • Mittag–Leffler for Helson Zeta via Blur

Mittag–Leffler for Helson Zeta via Blur A proof without short–interval primes Aleksandar Perišić September 2025 Abstract We prove a Helson–type factorization on {Re s > 1 } by a purely blur/Fourier argument: if fis continuous on {Re s≥1}and analytic on {Re s > 1}, then f(s) = −ζ′ χ ζχ (s) + g(s) (Re s > 1), for some Helson zeta ζχ ( s ) = Pn≥1χ ( n ) n−s with completely multiplicative unimodular χ and an entire g . As in the Mittag–Leffler paradigm, we obtain prescribed zero/pole patterns for Helson zetas on domains containing {Re s > 1 } . The novelty is quantitative: we avoid short–interval prime estimates by (i) matching the target under a multiplicative Gaussian blur on the Mellin side using only PNT at relative scale, and (ii) deblurring via a time–frequency sampling principle for Gaussians. 1 Introduction and statement AHelson zeta is ζχ(s) = ∞ X n=1 χ(n) ns=Y p1−χ(p) ps−1(Re s > 1), with χ : N→C completely multiplicative and |χ ( n ) | = 1. Writing Λfor the von Mangoldt function, −ζ′ χ ζχ (s) = X n≥1 χ(n)Λ(n) ns(Re s > 1). Theorem 1.1 (Andersson–Helson factorization).Let f be continuous on {Re s≥ 1 } and analytic on {Re s > 1}. Then there exist a Helson zeta ζχand an entire function gsuch that f(s) = −ζ′ χ ζχ (s) + g(s) (Re s > 1). Corollary 1.2 (Zero-free factorization).If f is analytic and zero-free on {Re s > 1 } and f′/f extends continuously to {Re s≥ 1 } , then there exist an entire zero-free G and a Helson zeta ζχ with f(s) = G(s) ζχ(s)(Re s > 1). Corollary 1.3 (Mittag–Leffler for Helson zetas).Let U⊂C be open, connected, with {Re s > 1 } ⊂ U , and let E⊂U∩{Re s < 1 } be closed and discrete in U (accumulation only possibly on 1 + iR ). Given integers {ma}a∈E , there exists a Helson zeta ζχ meromorphic on U (unique up to a nonzero constant fixed by the Helson normalization ζχ ( s ) → 1as Re s→ + ∞ ) such that − ( ζ′ χ/ζχ )has a simple pole at each a∈E with residue ma , and extends continuously to {Re s≥ 1 } . Moreover, by adding boundary obstructions, one can force U\E to be the maximal domain of analyticity. Sign. A zero (resp. pole) of order rat agives Ress=a(−ζ′ χ/ζχ) = r(resp. −r). 1 Remark 1.4 (Idea lineage).In [ 2 ] Helson factorizations are envisioned from data on {Re s > 1 } . Our route is: blur on the multiplicative axis, match locally using phases on primes in windows, and deblur through a Gaussian Gabor sampling principle. We require no short–interval prime input. 2 Multiplicative blur and distributional Mellin extraction Fix σ > 1. For parameters ρ > 0and Y∈Rset the Gaussian Kρ(u) := 1 √2π ρe−u2/(2ρ2),Φρ,Y (x) := Kρ(log x−Y) (x > 0). For a (tempered) distribution Hon (0,∞)define the blurred Mellin sample Bσ,t;ρ,Y [H] := DH(x), x−(σ+it)Φρ,Y (x)E. We will use Hχ:= X n≥1 χ(n)Λ(n)δn,Bσ,t;ρ,Y [Hχ] = X n≥1 χ(n)Λ(n) nσ+it Φρ,Y (n). Lemma 2.1 (Mellin extraction (distributional form)).If f is continuous on {Re s≥ 1 } and analytic on {Re s > 1 } , then there exist an entire g0 and a tempered distribution q on (0 ,∞ ) supported in [1,∞)such that f(s) = Z∞ 0 q(x)x−sdx +g0(s) (Re s > 1), where the integral is the distributional pairing with the Mellin kernel. Proof. Fix η∈(0,1) and a Gaussian cutoff ψT(t) = e−t2/T2. For φ∈ S(0,∞), define ⟨qT, φ⟩:= 1 2πiZ1+η+i∞ 1+η−i∞ f(s)b φ(1 −s)ψT(Im s)ds, with b φ the Mellin transform. Then qT is supported in [1 ,∞ )and satisfies R∞ 0qT ( x ) x−sdx = f ( s ) for Re s > 1 + η . As T→ ∞ , qT→q in S′ , still supported in [1 ,∞ ), and f ( s ) = R∞ 0q ( x ) x−sdx + g0(s)on {Re s > 1}, where g0is entire (from removing the cutoff). Henceforth let Hf:= q 1 [1,∞)and f0(s) := R∞ 1q(x)x−sdx; then f=f0+g0on {Re s > 1}. 3 Local blur matching on a prime window Fix σ > 1, a compact t –range |t| ≤ T , and an ε –net S = {t1, . . . , tM} therein. For Y≫ 1and ρ in a fixed compact interval [ρmin, ρmax]define the prime window IY,ρ := npprime :|log p−Y| ≤ cρo, with c≥1fixed. Write the targets Ξm:= Bσ,tm;ρ,Y [Hf]. Lemma 3.1 (Smoothed PNT at relative scale).Uniformly in σ∈ [1 + δ, S ],1 ≤ρ≤ρmax and Y→ ∞, X n≥1 Λ(n)n−σKρ(log n−Y) = e−(σ−1)Yc Kρσ−1+Oe−(σ−1)Ye−c0Y, and the contribution of prime powers pkwith k≥2is O(e−(σ−1)Y/2). 2 Sketch. This is standard from the explicit formula with a Schwartz test on log n (or Mellin inversion of log ζ against Kρ ( ·−Y )). The k≥ 2terms satisfy pk≍eY⇒p≍eY/k , giving the stated decay. Lemma 3.2 (Local blur matching).Given ( σ, T, ε, M )and ρmax ≥ 1, there is Y0 such that for all Y≥Y0 and ρ∈ [ ρmin, ρmax ]one can choose phases {ϑp}p∈IY,ρ with χ ( p ) := eiϑp (and χ(pk) := χ(p)k) so that Bσ,tm;ρ,Y [Hf]−Bσ,tm;ρ,Y [Hχ]≤C ε e−(σ−1)Y(1 ≤m≤M), for a constant Cdepending only on (σ, T, c, ρmax). Proof idea. Let vp:= (log p)p−(σ+itm)Kρ(log p−Y)M m=1 ∈CM. Partition IY,ρ into J≫Mlog M thin subwindows where vp is nearly constant; set Uj := Pp∈Jjvp . By choosing subwindow centers separated at ≫ 1 /T in log p , the vectors Uj are in general position (Vandermonde structure in e−itmτj ). The map ( θ1, . . . , θJ ) 7→ PjeiθjUj fills an annular region containing a ball of radius ≍e−(σ−1)Y ; choosing θj realizes the Ξ = (Ξ m )within the stated tolerance. A randomized variant with i.i.d. Steinhaus phases and a net argument also works. 4 Diagonal construction with dense time–frequency sampling Let ρ∈ [ ρmin, ρmax ]be fixed once and for all. We build χ on disjoint prime windows while enforcing small residuals on a dense sampling set in the (Y, t)–plane. Sampling set Choose spacings ∆Y, ∆t > 0with ∆Y∆t < 2πand define the lattice Λ = (Yℓ, tk) : Yℓ=Y0+ℓ∆Y, tk=k∆t, ℓ ∈N, k ∈Z. Later we only use the tail Λ≥:= Λ ∩{Y≥Y0}. Lemma 4.1 (Gabor sampling on a tail).Let g ( u ) := Kρ ( u )for fixed ρ∈ [ ρmin, ρmax ], and define the short-time Fourier coefficients Vgh(Y, t) := ⟨h, MtTYg⟩,(Mth)(u) = eituh(u),(TYh)(u) = h(u−Y). There exists Y0such that the set Λ≥is a sampling set for the subspace HY0:= {h∈L2(R) : supp h⊂[Y0,∞)}, in the sense that Vgh|Λ≥= 0 implies h= 0 a.e. on [Y0,∞). Quantitatively, A∥h∥2 L2≤X (Y,t)∈Λ≥|Vgh(Y, t)|2≤B∥h∥2 L2(h∈ HY0) for some 0< A ≤B < ∞depending on (ρ, ∆Y, ∆t). Remark 4.2. This is a standard corollary of Gaussian Gabor frame theory (density theorems in the Bargmann–Fock model) adapted to a half-line by support reduction: with our normalization Mt = eit(·) , the lattice condition ∆ Y ∆ t < 2 π corresponds to subcritical area, yielding a frame for L2 ( R ); restricting to windows centered at Y≥Y0 still frames HY0 because TYg is rapidly concentrated near Y. See e.g. Lyubarskii (1992), Seip (1992), and Gröchenig (2001/2013). 3 Stagewise construction Pick a sequence of disjoint prime windows {IYj,ρ}j≥1 with centers Yj→ ∞ and gaps ≫ 1in log p. At stage j: • choose phases on primes in IYj,ρ via Lemma 3.2 to match the targets for all sample points (σ, t)∈Σj×Sjaround (1 + 1 j,|t| ≤ Tj)at (Y, t)∈Λwith |Y−Yj| ≤ C; • simultaneously use the many degrees of freedom in the block of disjoint windows near Yj to contract the entire residual vector at all previously enforced samples by a fixed factor (say 1 / 2). This is possible because the Gaussian tails produce small but controllable influence at earlier ( Y, t ), and one can span the finite-dimensional residual space by taking enough disjoint windows in the new block. Define χ ( p )=1for primes not used by any stage. By construction, for every compact K ⊂ {Re s > 1}, sup s∈K sup (Y,t)∈Λ≥BRe s,Im s;ρ,Y [Hf−Hχ]= 0.(1) (We use analyticity in ( σ, t )and a standard mesh-to-compact argument to pass from finite nets to the whole compact; see Lemma 4.3 below.) Lemma 4.3 (Net-to-compact propagation).Let F be entire in s = σ + it on a rectangle R⊂ {Re s > 1}. If |F|is ≤εon a δ-net of R, then sup s∈R|F(s)| ≤ C(R)ε+oδ→0(1). Proof. Cover R by disks whose boundaries lie inside the net to within O ( δ ). Apply Cauchy’s integral formula and the maximum modulus principle. 5 Deblurring via Gabor sampling Set, for σ > 1and u= log x, hσ(u) := e−(σ−1)uHf−Hχ(eu), a tempered distribution supported in u≥0. Observe that BRe s,Im s;ρ,Y [Hf−Hχ] = hσ∗Kρ(Y)and its t–modulations. Thus the samples in (1) are exactly the Gaussian short-time Fourier coefficients V Kρhσ ( Y, t )on Λ≥. Lemma 5.1 (Deblurring).There exists an entire g1such that Z∞ 1 q(x)x−sdx =−ζ′ χ ζχ (s) + g1(s) (Re s > 1). Proof. By (1) and Lemma 4.1, we have V Kρhσ≡ 0on Λ ≥ , hence hσ = 0 on [ Y0,∞ )for each fixed σ > 1. Therefore Hf−Hχ is supported in [1 , eY0 ]. The Mellin transform of a compactly supported distribution is entire, so f0(s) := Z∞ 1 q(x)x−sdx =−ζ′ χ ζχ (s) + g1(s) with g1entire. Proof of Theorem 1.1. By Lemma 2.1, f = f0 + g0 with g0 entire. Apply Lemma 5.1 to f0 and set g:= g0+g1. 4 6 Corollaries Proof of Corollary 1.2. Apply Theorem 1.1 to f′/f : we get f′ f = −ζ′ χ ζχ + H with H entire. Integrate on {Re s > 1}, exponentiate, and set G=eH. Proof sketch of Corollary 1.3. Construct g on U\E with principal part ma/ ( s−a )at each a∈E and continuous extension to {Re s≥ 1 } (by standard Mittag–Leffler on U plus a collar around 1 + iR ). Apply Theorem 1.1 to f := g to obtain g = −ζ′ χ/ζχ + G on {Re s > 1 } with G entire. Then −ζ′ χ/ζχ = g−G on U\E with the required residues. Integer residues imply trivial monodromy of ( g−G ) ds on U\E , so one integrates globally to a single-valued meromorphic ζχ (unique up to a constant). Maximality follows from the engineered boundary obstruction. 7 What blur buys us (and what it avoids) • Scale matching. In a multiplicative window, both the q –integral and the prime sum have size ≍e−(σ−1)Y (by Lemma 3.1), making local matching well-conditioned and independent of Hoheisel-type inputs. • Local ⇒ global. Equality of blurred Mellin samples on a dense ( Y, t )–set and a mesh in (σ, t)forces equality up to an entire term by deblurring and analyticity. • Prime freedom. Disjoint windows give independent phase choice; later stages never revisit earlier primes. References [1] H. Helson, Compact groups and Dirichlet series, Ark. Mat. 8(1969), 139–143. [2] J. Andersson, Mittag–Leffler type theorems for Helson zeta-functions, arXiv:2408.15713 [math.NT], 2024. [3] K. Seip, Density theorems for sampling and interpolation in the Bargmann–Fock space I, J. Reine Angew. Math. 429 (1992), 91–106. [4] Y. Lyubarskii, Frames in the Bargmann space of entire functions, Ark. Mat. 32 (1994), 157–193. [5] K. Gröchenig, Foundations of Time–Frequency Analysis, Birkhäuser, 2001 (reprint 2013). [6] K. Seip, Universality and distribution of zeros and poles of some zeta functions, J. Anal. Math. 141 (2020), 331–381. [7] I. Bochkov and R. Romanov, On zeroes and poles of Helson zeta functions, J. Funct. Anal. 282 (2022), 109398. 5