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By Samuel Moy

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A trivial estimate gives b = 1 (with ≤ instead of ). 47) implies that any b > 12 can be used. 60) can be split up according to c = ζc0 , with ζ ∈ O∗ , and c0 running over representatives of non-zero principal ideals in O. We write (c0 ) = AB as above. 2 to the sum over the units ζ, we have to assume that α + β > 0. 71) ζ∈O∗ S χ (r , r; ζc0 ) f |N(ζc0 )| 4π|rr |1/2 ζc0 · |N(rr )|δ/2 min b,δ CN(A)δ N(B)b−1+δ |N(rr )|α/2 N(A)β N(B)β , . N(A)α N(B)α |N(rr )|β/2 The important point is the absolute convergence of the sum of Kloosterman sums.

As before, we denote the signs by ε, ε ∈ {1, −1}d , ε j = sign(r j ). The parity of the coordinates of the weights is determined by the central character, specified by ξ ∈ {0, 1}d . It has to satisfy χ(−1) = (−1)S (ξ) = (−1)ξ1 +···+ξd . 2. Weights and auxiliary test functions. For the preliminary sum formula in this subsection, we make some more choices. We choose a weight q ∈ Zd compatible with the central character: q ≡ ξ mod 2Zd . In principle, we can derive a preliminary sum formula for any weight.

Restricted sum formula. 9. Let 1 2 < τ < 1. 61) Yξ ϑ(ν) dσr,r χ,ξ (ν) = (2π)−d |DF |1/2 α(χ, ξ; r, r ) ϑ(ν) dPlξ (ν) Yξ ε,ε + Kr,r χ Bξ ϑ , with absolute convergence of all integrals and sums. 3. 53) for the Bessel transform Bξ . 62) for j = 1, . . , d, where ν( j) ∈ Cd has coordinates ν( j) = −ν j , ν( j) = νl for l j. 42). 7), and depend holomorphically on the coordinates with | Re ν j | ≤ τ. If ε j ε j = −1, the gamma factors introduce zeros for ν j ∈ ξ j −1 2 + Z. If ε j ε j = 1, the gamma factors do not lead to zeros with | Re ν j | ≤ τ.

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