HBP | Main | 1.1 • Mittag–Leffler for Helson Zeta via Blur
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We prove a Helson-type factorization on $\{\Re s>1\}$ by a purely blur/Fourier argument: if $f$ is continuous on $\{\Re s\ge 1\}$ and analytic on $\{\Re s>1\}$, then
\[
f(s)= -\,\frac{\zeta'_\chi}{\zeta_\chi}(s) + g(s)\qquad(\Re s>1),
\]
for some Helson zeta $\zeta_\chi(s)=\sum_{n\ge 1}\chi(n)n^{-s}$ with completely multiplicative unimodular $\chi$ 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 \emph{multiplicative Gaussian blur} on the Mellin side using only PNT at relative scale, and (ii) deblurring via a time--frequency sampling principle for Gaussians.
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- Subtitle
- A proof without short–interval primes
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- Preprint: 10.5281/zenodo.17162608 (DOI)
References
- H. Helson, Compact groups and Dirichlet series, Ark. Mat. 8 (1969), 139–143.
- J. Andersson, Mittag–Leffler type theorems for Helson zeta-functions, arXiv:2408.15713 [math.NT], 2024.
- K. Seip, Universality and distribution of zeros and poles of some zeta functions, J. Anal. Math. 141 (2020), 331–381.
- I. Bochkov and R. Romanov, On zeroes and poles of Helson zeta functions, J. Funct. Anal. 282 (2022), 109398.
- N. U. Arakelyan, Uniform and tangential approximations by analytic functions, Izv. Akad. Nauk Arm. SSR, Mat. 3 (1968), 273–286.