Abstract
In this article, our main purpose is to introduce a new and generalized quadratic Gauss sum. By using analytic methods, the properties of classical Gauss sums, and character sums, we consider the calculating problem of its fourth power mean and give two interesting computational formulae for it.
Highlights
For any integer q > 1, let χ denote any Dirichlet character mod q
These sums have an important status in the research of analytic number theory
We introduce a generalized quadratic Gauss sum as follows: G ( χ1, χ2, · · ·, χ k ; q )
Summary
For any integer q > 1, let χ denote any Dirichlet character mod q. If q = p is an odd prime and k = h = 2, we will use the analytic method and the properties of classical Gauss sums to give an exact computational formula for (3). Let p be an odd prime, ψ be a fixed non-principal character mod p. = 3p5 + K (ψ, p), Some notes: If ψ is a non-principal even character mod p, we cannot get the exact value of the sum. It is clear that if q = p is an odd prime and k ≥ 3, maybe we can give an accurate calculating formula for mean value (3) In this case, various discussions are required based on the different characters χi mod p with 1 ≤ i ≤ k, and the situation is more complicated, so we do not discuss it further here. Does there exist a calculating formula similar to our theorems? These are open problems, which need to be further studied
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
