On a symmetric Diophantine equation with reciprocals (Q511439)

Let \(I_r(N)\) denote the number of solutions of the Diophantine equation \[ \frac{1}{x_1}+\cdots+\frac{1}{x_r}=\frac{1}{x_{r+1}}+\cdots+\frac{1}{x_{2r}}, \] where \(r\geq 2\) and \(x_1,\dots,x_{2r}\) are positive integers in the interval \([1,N]\). The authors derive a formula for \(I_r(N)\) for large \(N\) with a power-saving remainder term that is explicit in \(N\) and \(r\). Specifically, the authors prove in Theorem 2 that if \(N\geq 3\) then \[ I_r(N) = r! N^r+\Delta_r(N), \] where \[ |\Delta_r(N)|\leq \left(\frac{49}{50}\ln N+32 ^2\right)^{10r^3} N^{r-\delta(r)},\quad \delta(r) = \frac{1}{2(2r-1)}, \] and \(\delta(r)\) is the power-saving. The main term \(r! N^r\) is contributed mostly by those solutions \((x_1,\ldots,x_{2r})\) for which the \(r\)-tuple \((x_{r+1},\dots,x_{2r})\) is a permutation of the \(r\)-tuple \((x_1,\dots,x_r)\). Hence, the major task in the proof is to bound the number of the solutions that are not such permutations in Theorem 1. As a corollary of the above formula, the authors give a formula for the number of solutions \(K_r(N)\) of the congruence \[ x_1^*+\cdots+x_r^*\equiv x_{r+1}^*+\cdots + x_{2r}^*\pmod{p}, \] where \(1\leq x_1,\dots,x_{2r}\leq N\) are integers, \(p\) is a prime number large enough compared to \(N\) and \(r\), and \(x^*\) is defined by \(x x^*\equiv 1\pmod {p}\). The observation here is that if \(p\) is large enough, then the above congruence can be converted to an equality over the integers which is of the same type considered at the beginning. More precisely, if \(r N^{2r-1}<p\) then in fact \(K_r(N)=I_r(N)\). Finally, the authors apply these results to study the value distribution of short Kloosterman sums, so they consider the values \[ \frac{1}{\sqrt{N}}\left|\sum_{x=1}^N e^{2\pi i a x^*/p}\right| \] as \(a\) varies over \([1,p]\).