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Nonlinear exponential twists of the Liouville function

Qingfeng Sun (2011)

Open Mathematics

Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum X n 2 X λ ( n ) e 2 π i α n , 0 α The main tool we use is Vaughan’s identity for λ(n).

On a sum involving the Möbius function

I. Kiuchi, M. Minamide, Y. Tanigawa (2015)

Acta Arithmetica

Let c q ( n ) be the Ramanujan sum, i.e. c q ( n ) = d | ( q , n ) d μ ( q / d ) , where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for n y ( q x c q ( n ) ) k (k = 1,2) are obtained. As an analogous problem, we evaluate n y ( n x c ̂ q ( n ) ) k (k = 1,2), where c ̂ q ( n ) : = d | ( q , n ) d | μ ( q / d ) | .

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