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A hybrid mean value involving two-term exponential sums and polynomial character sums

Han Di (2014)

Czechoslovak Mathematical Journal

Let q 3 be a positive integer. For any integers m and n , the two-term exponential sum C ( m , n , k ; q ) is defined by C ( m , n , k ; q ) = a = 1 q e ( ( m a k + n a ) / q ) , where e ( y ) = e 2 π i y . In this paper, we use the properties of Gauss sums and the estimate for Dirichlet character of polynomials to study the mean value problem involving two-term exponential sums and Dirichlet character of polynomials, and give an interesting asymptotic formula for it.

A hybrid mean value of the Dedekind sums

Hai Yang (2010)

Czechoslovak Mathematical Journal

The main purpose of this paper is to use the M. Toyoizumi's important work, the properties of the Dedekind sums and the estimates for character sums to study a hybrid mean value of the Dedekind sums, and give a sharper asymptotic formula for it.

Average r-rank Artin conjecture

Lorenzo Menici, Cihan Pehlivan (2016)

Acta Arithmetica

Let Γ ⊂ ℚ * be a finitely generated subgroup and let p be a prime such that the reduction group Γₚ is a well defined subgroup of the multiplicative group ₚ*. We prove an asymptotic formula for the average of the number of primes p ≤ x for which [ₚ*:Γₚ] = m. The average is taken over all finitely generated subgroups Γ = a , . . . , a r * , with a i and a i T i , with a range of uniformity T i > e x p ( 4 ( l o g x l o g l o g x ) 1 / 2 ) for every i = 1,...,r. We also prove an asymptotic formula for the mean square of the error terms in the asymptotic formula with a similar...

Character sums in complex half-planes

Sergei V. Konyagin, Vsevolod F. Lev (2004)

Journal de Théorie des Nombres de Bordeaux

Let A be a finite subset of an abelian group G and let P be a closed half-plane of the complex plane, containing zero. We show that (unless A possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of A which belongs to P . In other words, there exists a non-trivial character χ G ^ such that a A χ ( a ) P .

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