Davenport H Analytic methods for Diophantine equations and

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1) |Sa,q | q 1−1/k . |A(q)| q 1−s/k , This implies that from which it follows that the singular series is absolutely convergent if s ≥ 2k + 1. 2, both hold under the same condition. 1. If a ≡ 0 (mod p) and δ = (k, p − 1) then |Sa,p | ≤ (δ − 1)p1/2 . 2) Proof. Since xk ≡ m (mod p) has the same number of solutions as xδ ≡ m (mod p), we have a δ x . Sa,p = e p x Let χ be a primitive character (mod p) of order δ. Then the number of solutions of xδ ≡ t (mod p) is 1 + χ(t) + · · · + χδ−1 (t). p = t a t , p where here (and elsewhere in this proof) summations are over a complete set of residues modulo p.

We have |T (ψ)|2 = ψ(t)ψ(u)e t u a (t − u) . p Here we can omit u = 0, since ψ(0) = 0. Changing the variable from t to v, where t ≡ vu (mod p), we obtain |T (ψ)|2 = au (v − 1) . p ψ(v)e v u=0 The inner sum is p − 1 if v = 1 and is −ψ(v) otherwise. Hence |T (ψ)|2 = pψ(1) − ψ(v) = p. v This is the result stated earlier. 3) for ψ = χ, . . 2). Note. 2) remains valid if p = 2 (so that δ = 1), but is then trivial since a = 1 and S1,2 = 1 + eiπ = 0. 2. Suppose a ≡ 0 (mod p) and p k. 4) and for ν > k, Sa,pν = pk−1 Sa,pν−k .

The factors which fluctuate most as N varies are those for which p divides k, but those for which p − 1 has a large factor in common with k may also fluctuate appreciably. In their early papers, Hardy and Littlewood worked mainly with the definition of S(N ) in terms of the exponential sums Sa,q , rather than with the expression in terms of congruences (mod pν ). N. II [38] they had to prove that S(N ) has a positive lower bound in the case k = 4, s = 21. The factors χ(p) which fluctuate most as N varies are in this case χ(2) and χ(5); the product of all the others does not differ appreciably from 1.

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