Generalized Gaussian sums and Bernoulli polynomials

Abstract

The conception of Generalized Gaussian Sum $$G_f(m)$$ for  a periodic arithmetical functon with a period, is equal prime number q, for integers m,n is introduce: $$ G_f(m)=\sum_{n=1}^{q-1}\left(\frac nq\right)f\left(\frac{mn}q\right). $$ Here are considered the particular cases $$f(x)=B_\nu(\{x\}), \nu\geq 1,$$ where $$B_\nu(x)$$ - Bernoulli polynomials. The paper uses the technique of finite Fourier series. If the function $$f\left(\frac{k}{q}\right)$$ is defined at $$k=0,1,\ldots,q-1,$$ it can be decomposed into a finite Fourier series $$ f\left(\frac{k}{q}\right)=\sum_{m=0}^{q-1}c_me^{2\pi i\frac{mk}{q}}, \quad c_m=\frac{1}{q}\sum_{k=0}^{q-1}f\left(\frac{k}{q}\right)e^{-2\pi i\frac{mk}{q}}. $$ By decomposition into a finite Fourier series of a generalized Gauss sum $$ G_\nu(m)=G_\nu(m;B_\nu)=\sum_{n=1}^{q-1}\left(\frac nq\right)B_\nu{\left(\left\{x+\frac{mn}q\right\}\right)} $$ for $$\nu=1$$ and $$\nu=2$$ , new formulas are found that Express the value of the Legendre symbol through the full sums of periodic functions. This circumstance makes it possible to obtain new analytical properties of the corresponding Dirichlet series and arithmetic functions, which will be the topic of the following works. An important property of the sums $$G_1$$ and $$G_2,$$ namely: $$G_1\ne 0,$$ if $$q\equiv 3\pmod 4$$ and $$G_1=0,$$ if $$q\equiv 1\pmod 4;$$ $$G_2= 0,$$ if $$q\equiv 3\pmod 4$$ and $$G_2=\frac 1{q^2}\sum\limits_{n=1}^{q-1}n^2\left(\frac nq\right),$$ if $$q\equiv 1\pmod 4.$$