Research On The Mean Of Gauss Sums

The research of exponential sums is of great significance in different number theory problems and practical applications.This paper mainly discusses the properties of Gauss sums,including the calculation of classical Gauss sums,the high-power mean of Gauss sums,the calculation problem of the hybrid power mean of Gauss sums and two-term exponential sum and the high-power mean of the reciprocal of Gauss sums.In addition,this paper also studies the arithmetic properties of second-order linear recurrence sequences.Specifically,the main results of this paper are as follows:1.Solve the calculation problem of the classical Gauss sums with symmetrical forms.By using the properties of Gauss sums and analytical methods,we obtain the fourth-order linear recurrence formula of τk(ψ)+τk(ψ),where ψ is any fourth-order character mod p and k is a positive integer.This further improves the calculation results of the hybrid power mean of the quartic Gauss sum and Kloosterman sum.2.Study the calculation of the mean of Gauss sum.First,we research the high-power mean of the generalized cubic Gauss sum and obtain an exact identity;second,we study the hybrid power mean of the quartic Gauss sum and the two-term exponential sum when the modulo satisfies different conditions,the hybrid power mean of the reciprocal of the quartic Gauss sum and the two-term exponential sum,and the high-power mean of the reciprocal of the quartic Gauss sum,and the exact calculation formulas are obtained respectively;finally,we also give the lower bound estimate for the quartic Gauss sum under some certain conditions.3.Obtain the convolution formulae related to the second-order linear recurrence sequence.By using the elementary method and the property of symmetry,for any second-order linear recurrence sequence with generating function f(t)=1/1+at+bt2(a,b ∈ R),we can give the exact coefficient expression for the power series expansion of fx(t)for x ∈ R.Thus we not only can get the convolution formulas of some important polynomials such as Fibonacci polynomials,Lucas polynomials,Chebyshev polynomials,Legendre polynomials,etc.,but also further strengthen the relationship between polynomials and themselves,and more easily realize the transformation between polynomials.