The Tsukano conjectures on exponential sums (Q782375)

Let \(p>2\) be a prime, \(\zeta=\exp(2πi/p)\), \(\psi(a)=(ap)\) be the Legendre symbol modular \(p\), and \(\tau(\psi)=\sum_{n=0}^{p-1}\psi(n)\zeta^n\) be the Gauss sum. For polynomials \(P,Q,R\) with integer coefficients, define the Lee-Weintraub sums \[ S[P,Q,R]=-\sum_{k,j\pmod p}\frac{\psi(k)}{(\zeta^{kP(j)}-1)(\zeta^{kQ(j)}-1)(\zeta^{kR(j)}-1)}, \] where the summation goes through a complete residue system modular \(p\) and the meaningless terms are excluded. In this paper the author proves three conjectures of \textit{S. Tsukano} in [Exponential sums and Bernoulli numbers. Osaka: Osaka University (Master Thesis) (2000)]. The first one is: \[ S[-2x,2(x+1),x(x+1)]=\tau(\psi)\left(\frac{-p+2+\psi(2)(5p+2)}{24}B_{1,\psi}+\frac{1+6\psi(2)}{4}B_{2,\psi}+\frac{1+16\psi(2)}{72}B_{3,\psi}\right). \] for any \(p>2\) prime. Here \[ B_{k,\psi}=p^{k-1}\sum_{a(p)}P_k(a/p)\psi(a) \] and the \(k\)-th periodic Bernoulli function in which \(P_k\) is defined as follows: \(P_k(x)=0\) if \(k=1\) and \(x\in \mathbb{Z}\) and \(P_k(x)=B_k(\{x\})\) otherwise, where \(\{x\}\) denotes the fractional part of \(x\), and \(B_k(x)\) is the \(k\)-th Bernoulli polynomial.