Equivalence of the Logarithmically Averaged Chowla and Sarnak Conjectures

Terence Tao ()
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Terence Tao: UCLA, Department of Mathematics

A chapter in Number Theory – Diophantine Problems, Uniform Distribution and Applications, 2017, pp 391-421 from Springer

Abstract: Abstract Let λ denote the Liouville function. The Chowla conjecture asserts that ∑ n ≤ X λ ( a 1 n + b 1 ) λ ( a 2 n + b 2 ) … λ ( a k n + b k ) = o X → ∞ ( X ) $$\displaystyle{\sum _{n\leqslant X}\lambda (a_{1}n + b_{1})\lambda (a_{2}n + b_{2})\ldots \lambda (a_{k}n + b_{k}) = o_{X\rightarrow \infty }(X)}$$ for any fixed natural numbers a 1 , a 2 , … , a k $$a_{1},a_{2},\ldots,a_{k}$$ and non-negative integer b 1 , b 2 , … , b k $$b_{1},b_{2},\ldots,b_{k}$$ with a i b j − a j b i ≠ 0 $$a_{i}b_{j} - a_{j}b_{i}\neq 0$$ for all 1 ≤ i

Keywords: Topological Entropy; Implied Constant; Pigeonhole Principle; Prime Number Theorem; Polynomial Sequence (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-55357-3_21

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