OKADA'S THEOREM AND MULTIPLE DIRICHLET SERIES

Abstract

Let k1, . . . , kr be positive integers. Let q1, . . . , qr be pairwise coprime positive integers with qi > 2 (i = 1, . . . , r ), and set q = q1 · · · qr . For each i = 1, . . . , r , let Ti be a set of φ(qi )/2 representatives mod qi such that the union Ti ∪ (−Ti ) is a complete set of coprime residues mod qi . Let K be an algebraic number field over which the qth cyclotomic polynomial 8q is irreducible. Then, φ(q)/2r numbers r ∏ i=1 dki−1 dzi i (cot π zi )|zi=ai /qi (ai ∈ Ti , i = 1, . . . , r) are linearly independent over K . As an application, a generalization of the Baker–Birch– Wirsing theorem on the non-vanishing of the multiple Dirichlet series L(s1, . . . , sr ; f ) with periodic coefficients at (s1, . . . , sr )= (k1, . . . , kr ) is proven under a parity condition.