On Beurling′s Real Variable Reformulation of the Riemann Hypothesis
✍ Scribed by L Baezduarte
- Publisher
- Elsevier Science
- Year
- 1993
- Tongue
- English
- Weight
- 623 KB
- Volume
- 101
- Category
- Article
- ISSN
- 0001-8708
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✦ Synopsis
Let (\rho(x)) be the fractional part of (x, B) is the linear space of functions (\sum_{1 \leqslant k \leqslant n} a_{k} \rho\left(\theta_{k} / x\right), \theta_{k} \in(0,1], \sum a_{k} \theta_{k}=0, n) any positive integer. For (p \in(1,2]) Beurling proved that (\zeta(s) \neq 0) in (\mathscr{R} s>1 / p) if and only if -1 is in the (L_{p}(0,1))-closure of (B). For each (f \in B) the size of (|1+f|{p}) determines a possibly empty, zero-free region for (\zeta(s)). Depending on regular estimates for (M(x)) sequences (\left{f{n}\right}) in (B) are constructed such that (\left|1+f_{n}\right|{1} \rightarrow 0). Then the rate at which (\left|1+f{n}\right|{p{n}} \rightarrow 0) as (p_{n} \downarrow 1) yields explicit zero-free regions for (\zeta(s)). In particular, it is shown very simply that estimates of the form (M(x)=O\left(x \exp \left(-c(\log x)^{1 /(1+\beta)}\right)\right.) imply that (\zeta(s) \neq 0) in a region of type (\sigma>1-c_{1} \log ^{-\beta}|t|). This corresponds to an earlier result of Turán's relating the order of the error in the prime number theorem to zero-free regions for Riemann's Zeta-function. A systematic construction of Beurling functions is introduced and some related questions are discussed. 1993 Academic Press, Inc.