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Analytic Number Theory A Tribute to Gauss and Dirichlet Episode 1 Part 5

Tham khảo tài liệu 'analytic number theory a tribute to gauss and dirichlet episode 1 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 72 JORG BRUDERN AND TREVOR D. WOOLEY equation 6.1 is assured and it is this observation that permits us to conclude that t m 1. Our discussion thus far permits us to conclude that when A is a positive number sufficiently small in terms of t c and n then for each m G v3P3 P3 one has Yt m M 2APt-3. But Yt m 0 1 Yt m M Yt m m and so it follows from 6.2 and 6.3 that for each n G Et P one has 6.6 Yt dn3 m APt-3. When n G Et P we now define ơn via the relation Yt dn3 m ơnYt dn3 m and then put Kt a ơne dn3a . nESt P Here in the event that Yt dn3 m 0 we put ơn 0. Consequently on abbreviating card Et P to Et we find that by summing the relation 6.6 over n G Et P one obtains 6.7 EtAPt 3 g c1a g c2a h c3a h c4a . .h cta Kt a da. dm An application of Lemma 6 within 6.7 reveals that EtAPt-3 c max max Mciayhca. t-2Kt a da. 1 1 2 3 j t Jm j On making a trivial estimate for h cja in case t 6 we find by applying Schwarz s inequality that there are indices i G 1 2 and j G 3 4 . t for which EtAPt-3 sup cia Pt-6T11 2T21 2 Qjm where we write y 1 r 1 T1 lg cịà 2h cja 4 da and T h Cja 4Kt a 2 da. The first of the latter integrals can plainly be estimated via 6.4 and a consideration of the underlying Diophantine equation reveals that the second may be estimated in similar fashion. Thus on making use of the enhanced version of Weyl s inequality Lemma 1 of V86 by now familiar to the reader we arrive at the estimate EtAPt-3 P3 4 e Pt-6 P3 ị e Pt-2-2r 2e The upper bound Et P1 T now follows whenever P is sufficiently large in terms of t c n A and T. This completes the proof of the theorem. We may now complete the proof of Theorem 2 for systems of type II. From the discussion in 3 we may suppose that s 13 that 7 q0 s 6 and that amongst the forms Aj 1 i s there are precisely 3 equivalence classes one of which has multiplicity 1. By taking suitable linear combinations of the equations 1.1 and by relabelling the variables if necessary it thus suffices to consider the pair of equations 6.8 0 1X3- .