bibkey: glocknerlucht2011weightedinversion authors: Helge Glöckner and Lutz G. Lucht year: 2011 title: Weighted inversion of general Dirichlet series doi: null url: https://arxiv.org/abs/1112.0749v2 claim: Absolute coefficient Dirichlet convolution forms a unital Banach algebra; the classical norm perturbation inversion principle gives the dominant-head inverse norm estimate by a Neumann series. strata_touched:

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Absolute coefficient Dirichlet inversion

Glöckner and Lucht, Weighted inversion of general Dirichlet series, arXiv:1112.0749v2 (29 September 2012; first submitted 4 December 2011). Page 1 defines the absolute coefficient convolution Banach algebra and proves its submultiplicative norm inequality. Page 3 identifies ordinary Dirichlet series with the additive semigroup log N. Page 4, Theorem 2(a), states the classical principle that an element within norm distance less than one of the unit is invertible.

For a real arithmetic function b with a nonzero head c=b(1), write b=c delta+h, with h(1)=0. If every finite absolute tail is bounded by T and T<|c|, then h is absolutely summable with norm at most T. The Neumann series for the inverse of delta+h/c gives the classical quantitative consequence

$$\Vert b^{-1}{\Vert }_1\le |c{|}^{-1}\sum _{j\ge 0}(T/|c|{)}^j=\frac{1}{|c|-T}.$$

The displayed quantitative consequence is elementary Banach algebra inversion; it is not claimed to be a separately numbered theorem in the paper. The Lean theorem uses an existing algebraic right inverse and proves finite partial-sum absorption before deducing absolute summability, so it assumes no global norm bound on that inverse. Its zero coefficient is zero under the ArithmeticFunction convention.

The general Wiener criterion in the paper concerns the closure of the Dirichlet-series image excluding zero, equivalently a uniform separation from zero in its stated domain. It must not be replaced by mere pointwise nonvanishing. No originality or Riemann-hypothesis assertion is attributed to this norm estimate.

Source

The 19-page arXiv v2 PDF has SHA-256 6a5a2cd65ecd912d5e436d29a90705b67f00df3d6d3ee41e54d64e84097afe22. Remark 1 on page 2 traces the ordinary Dirichlet-series predecessor to Hewitt and Williamson, Note on absolutely convergent Dirichlet series, Proc. Amer. Math. Soc. 8 (1957), 863–868, DOI 10.1090/S0002-9939-1957-0090680-X. That predecessor is secondary attribution here; its original theorem text has not been directly verified.

Verified locator

Canonical source: https://arxiv.org/abs/1112.0749v2. The v2 PDF, Weighted inversion of general Dirichlet series, defines the absolute coefficient convolution Banach algebra and its submultiplicative norm on page 1. Page 3 identifies ordinary Dirichlet series with the additive semigroup log N. Theorem 2(a), page 4, gives invertibility for norm distance from the unit less than one. The dominant-head inverse absolute sum bound above is the elementary Neumann-series consequence of that norm perturbation principle; no separately numbered theorem in the paper is attributed to the displayed quantitative constant.