Estimate for exponential sums and its applications
Let \(f_{k}(n)\) be the characteristic function of \(n\) with \(\Omega(n)=k\), and \[ T_{k}(x,\alpha)=\sum_{n\leq x}f_{k}(n)e(n\alpha). \] Write \[ r(N)=\#\{N=f_{k}(n_{1})n_{1}+f_{k}(n_{2})n_{2}+f_{k}(n_{3})n_{3}\}. \] The author obtains the following result: Let \(k\leq (\log\log N)^{1-\Delta}\), then \[ r(N)=\frac{1}{2}N^{2}\mathfrak{S}(N)(\frac{(\log\log N)^{k-1}}{(k-1)!\log N})^{3}(1+O(\frac{1}{(\log\log N)^{\Delta}})), \] where \(\mathfrak{S}(N)\) is the singular series in the ternary Goldbach problem. The proof uses the circle method and a Bombieri-type estimate for the exponential sum \(T_{k}(x,\alpha)\).
scientific article; zbMATH DE number 6103008
scientific article