On the high-th mean of one special character sums modulo a prime

Abstract

<abstract><p>Using the elementary method of the classical Gauss sums and the properties of character sums, we study a linear recurrence formula about the form $ G\left(n\right) = 1+\sum_{a = 1}^{p-1}\left(\frac{a^2+n\bar{a}^2}{p}\right) $ and about the mean value of $ G(n) $. This is a further exploration of Yuan and Zhang's research in 2022, which help us to better understand the character sums wide range application.</p></abstract>