On exponential sums involving the divisor function over Z[๐‘–]

Abstract

We apply the van der Corput transform to investigate the sums of view ฮฃ๏ธ€ ๐‘Ÿ(๐‘›)๐‘”(๐‘›)๐‘’(๐‘“(๐‘›))), where ๐‘Ÿ(๐‘›) is the number of representations of ๐‘› as the sum of two squares of integer numbers. Such sums have been studied by M. Jutila, O. Gunyavy, M. Huxley and etc. Depending of differential properties of the functions ๐‘”(๐‘›) and ๐‘“(๐‘›) there have been obtained the different kinds of error terms in bounds of the considered sums. In the special case, O. Gunyavy improved the result of M. Jutila in the problem on estimate the exponential sum involving the divisor function ๐œ (๐‘›). We obtain the asymptotic formula of the sum ฮฃ๏ธ€ ๐œ (๐›ผ)๐‘’ (๏ธ ๐‘Ž ๐‘ž๐‘(๐›ผ) )๏ธ over the ring of Gaussian integers which is an analogue of the asymptotic formulas obtained by M. Jutila and O. Gunyavy. MSC: 11K45.

Authors and Affiliations

S. Varbanets

Keywords

Related Articles

About the quasiareal infinitesimal deformation of catenoid

In this paper we considered the quasiareal infinitesimal deformation of catenoid under which the deviation from the tangent plane is preserved in all directions.

On the reduction of the countable linear differential system with coefficients of the oscillating type to the some special kind at the resonance case

Jashitova V. Summary For the countable linear homogeneous differential system, whose coefficients are represented as an absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency,...

Variational method of homogeneous solutions for axisymmetric elasticity problems

A variational method of homogeneous solutions for solving of axisymmetric elasticity problems for semiinfinite and finite cylinders with free lateral surface has been developed. As an example of application of the method...

The plane mixed elastical problem for the semi-infinite strip

The loaded plane elastic semistrip, the one boundary of which is fixed and the second one is in the smooth contact condition, is considered. The solution of the problem in โˆ’sin, โˆ’cos Fourier transformantโ€™s domain is cons...

Asymptotical expansion of vanishing solutions of differential equations in one special case

Considered in this paper is a system of quasilinear differential equations with factors on the right side that disappear when ๐‘ก โ†’ โˆž ( absolutely integrable or not). Sufficient conditions are obtained under which it is po...

Download PDF file
  • EP ID EP191239
  • DOI -
  • Views 69
  • Downloads 0

How To Cite

S. Varbanets (2016). On exponential sums involving the divisor function over Z[๐‘–]. ะ”ะพัะปั–ะดะถะตะฝะฝั ะฒ ะผะฐั‚ะตะผะฐั‚ะธั†ั– ั– ะผะตั…ะฐะฝั–ั†ั–, 21(2), 88-100. https://europub.co.uk/articles/-A-191239