GCD sums from Poisson integrals and systems of dilated functions
By Christoph Aistleitner, István Berkes, Kristian Seip. 2012
53 citations recorded by the source corpus.
Upper bounds for GCD sums of the form [\sum_{k,{\ell}=1}^N\frac{(\gcd(n_k,n_{\ell}))^{2\alpha}}{(n_k n_{\ell})^\alpha}] are proved, where $(n_k)_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0<\alpha \le 1$; the estimate for $\alpha=1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $\alpha=1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish a Carleson--Hunt-type inequality for systems of dilated functions of bounded variation or belonging to $\lip12$, a result that in turn settles two longstanding problems on the a.e.\ behavior of systems of dilated functions: the a.e. growth of sums of the form $\sum_{k=1}^N f(n_k x)$ and the a.e.\ convergence of $\sum_{k=1}^\infty c_k f(n_kx)$ when $f$ is 1-periodic and of bounded variation or in $\lip12$.
Subfields: Algebra and Number Theory, Applied Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/65ae15534b0563df2654cb286e4ef5d4407a128c
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In Algebra and Number Theory
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Citations in the corpus
- Lower bounds for the maximum of the Riemann zeta function along vertical lines (2014, background): “several other problems, fromDiophantine approximation to linear algebra and convergence problems for series of dilated functions; see for example [1,9,11].”
- Large greatest common divisor sums and extreme values of the Riemann zeta function (2015, methodology): “The better bound exp( C p log N loglog N ) was found by Aistleitner, Berkes, and Seip in [2] by a method that also led to a full solution when 1/2 σ 1.”