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Tham khảo tài liệu 'analytic number theory a tribute to gauss and dirichlet part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 252 H. M. STARK in the case that d a2 is large. We can easily see where the two main terms come from. We have oo oo 5.1 z s Q 3 am2 s 3 am2 bmn cn2 s . m 1 n 1 m We approximate the inner sum on the right by the integral o j at2 bnt cn2 sd t a-s o p Ín i u2 1 sdu . 2a I J -o The integral on the right evaluates to o _ I U 1 -du - 2 . J r s o This gives the approximation z s Q a sz 2s as 1 ppp rp - 1 2 z 2s - 1 R s k I 2 t r s s where R s is the error made in approximating the sum by the integral. Equivalently with Z s n s 2r 2 z s and r__s R s z I r s R s 2n we have 5.2 ----T s PP r s z s Q 2n Ị 1 as-1 . R s s Z 2s a s 1 s z 2 2s as 1 R s . The main terms interchange on the right when s is replaced by 1 s. We are entitled to suspect that we have stumbled upon the functional equation for z s Q this is indeed the truth and can be derived from this expansion if one uses the Poisson summation formula on the sum on m in 5.1 . The Poisson summation formula leads to the same main terms and an expansion of R s in K-Bessel functions in a form where Ks 1 2 appears and is invariant under s 1 s. Deuring used the Euler MacLaurin summation formula to estimate R s . On Ơ 1 2 the two main terms have the same absolute value and as t increases the arguments of the two main terms spin in opposite directions in a manner which is practically linear over short ranges in t. Deuring realized in Deu35 that this leads to the zeros of z s Q lying practically in arithmetic progressions in t. From Stirling s formula when s - it 2 THE GAUSS CLASS-NUMBER PROBLEMS 253 we have 5.3 arg yH 2s arg 2 r s z 2s t log aT t log t t arg z 1 2it O 1 . If t goes from to to t0 where is suitably small then to a first approximation the right side grows by d log t O . In particular the right side grows by n when is approximately n 5.4 For our particular Q we find that the two main terms have the same absolute values on Ơ 1 2 and the sum of the two main terms has zeros almost precisely in arithmetic progressions .
