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1
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Nonlinear exponential twists of the Liouville function

100%
Sun Q.
Open Mathematics
|
2011
|
tom 9
|
nr 2
328-337
EN
Let λ(n) be the Liouville function. We find a nontrivial upper bound for the sum $$ \sum\limits_{X \leqslant n \leqslant 2X} {\lambda (n)e^{2\pi i\alpha \sqrt n } } ,0 \ne \alpha \in \mathbb{R} $$ The main tool we use is Vaughan’s identity for λ(n).
2
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On the Golomb’s conjecture and Lehmer’s numbers

100%
Tingting W., Xiaonan W.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1003-1009
EN
Let p be an odd prime. For each integer a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one ā with 1 ≤ ā ≤ p − 1 such that a · ā ≡ 1 mod p. Let N(p) denote the set of all primitive roots a mod p with 1 ≤ a ≤ p − 1 in which a and ā are of opposite parity. The main purpose of this paper is using the analytic method and the estimate for the hybrid exponential sums to study the solvability of the congruence a + b ≡ 1 mod p with a, b ∈ N(p), and give a sharper asymptotic formula for the number of the solutions of the congruence equation.
3
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One kind sixth power mean of the three-term exponential sums

100%
Wang X., Li X.
Open Mathematics
|
2017
|
tom 15
|
nr 1
705-710
EN
In this paper, we use the estimate for trigonometric sums and the properties of the congruence equations to study the computational problem of one kind sixth power mean of the three-term exponential sums. As a conclusion, we give an exact computational formula for it.
4
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On the fourth-order linear recurrence formula related to classical Gauss sums

76%
Zhuoyu C., Wenpeng Z.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1251-1255
EN
Let p be an odd prime with p ≡ 1 mod 4, k be any positive integer, ψ be any fourth-order character mod p. In this paper, we use the analytic method and the properties of character sums mod p to study the computational problem of G(k, p) = τk(ψ)+τk(ψ), and give an interesting fourth-order linear recurrence formula for it, where τ(ψ) denotes the classical Gauss sums.
5
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Upper bound estimate of incomplete Cochrane sum

76%
Ma Y., Peng W., Zhang T.
Open Mathematics
|
2017
|
tom 15
|
nr 1
852-858
EN
By using the properties of Kloosterman sum and Dirichlet character, an optimal upper bound estimate of incomplete Cochrane sum is given.
6
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On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions

64%
Kühleitner M., Nowak W.
Open Mathematics
|
2013
|
tom 11
|
nr 3
477-486
EN
The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.
7
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On certain arithmetic functions involving the greatest common divisor

64%
Krätzel E., Nowak W., Tóth L.
Open Mathematics
|
2012
|
tom 10
|
nr 2
761-774
EN
The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ2(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered.
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