Let χ be a primitive, real and even Dirichlet character with conductor q, and let s be a positive real number. An old result of H. Davenport is that the cycle sums Sν(s, χ) = ∑(ν+1)q−1 n=νq+1 χ(n) ns , ν = 0, 1, 2, . . . , are all positive at s = 1, and this has the immediate important consequence of the positivity of L(1, χ). We extend Davenport’s idea to show that in fact for ν ≥ 1, Sν(s, χ) ...

Interactions of multiple points with oscillatory kernels are widely encountered in wave analysis. For large scale problems, its direct evaluation is prohibitive since the computational cost increases quadratically number points. Various fast algorithms have been constructed by exploiting specific properties kernel function. Early algorithms, such as multipole method (FMM) and variants, H2-...