An application of the Hooley Huxley contour

Transcription

1 ACTA ARITHMETICA LXV. 993) A applicatio of the Hooley Huxley cotour by R. Balasubramaia Madras), A. Ivić Beograd) ad K. Ramachadra Bombay) To the memory of Professor Helmut Hasse ). Itroductio ad statemet of results. This paper is a cotiuatio of our paper []. We begi by statig a special case of what we prove i the preset paper. Theorem. Let k be ay complex costat ad ζs)) k = d k) s i σ 2. The ) 2) ad 3) ζ + it)) k 2 dt = T ζ + it) ζ + it) 2 d k ) Olog T ) k2 ), dt = T m log ζ + it) 2 dt = T m log p) 2 p 2m + Olog T ) 2 ), p mp m ) 2 + Olog log T ). p R e m a r k. I [] we proved ) with k = ad studied the error term i great detail. R e m a r k 2. The proof of this theorem ad Theorem 3 to follow require the use of the Hooley Huxley cotour as modified by K. Ramachadra i [2] for some explaatios see [3]). We write mhh ) for this cotour. R e m a r k 3. We have a aalogue of these results for ζ ad L-fuctios of algebraic umber fields. I fact, uder somewhat geeral coditios o

2 46 R. Balasubramaia et al. F s) = a s or eve a λ s ad so o) we ca show that 4) F + it) 2 dt = T a O log log T + ) a 2 T C where C > 0) is a large costat. The followig theorem is fairly simple to prove. Theorem 2. Let = λ < λ 2 <... be a sequece of real umbers with C0 λ + λ C 0 where C 0 ) is a costat ad let a, a 2,... be ay sequece of complex umbers satisfyig the followig coditios: i) x a = O ε x ε ) for all ε > 0 ad x. ii) a 2 λ 2 coverges for some costat λ with 0 < λ <. iii) F s) = a λ s which coverges i σ > ) is cotiuable aalytically i σ δ, t t 0 ) ad there F s) < t A, where δ 0 < δ < /0), t 0 00) ad A 2) are ay costats. 5) The t 0 +C log log T F + it) 2 dt = T a 2 λ 2 + O log log T + T C 2 a 2 ) where C ad C 2 are certai positive costats depedig o other costats which occur i the defiitio of F s). 6) 7) ad 8) We sketch a proof of this theorem. We put s = + it, t t 0, Rw) = exp si w ) 2 ), 00 u) = 2πi ) X a λ s = λ 2πi 2+i 2 i 2+i 2 i u w Rw) dw w u > 0), F s + w)x w Rw) dw w X = T C 3 ), C 3 > 0) beig a large costat. I the itegral just metioed we cut off the portio Im w C 4 log log T where C 4 > 0) is a large costat ad i the remaiig part we move the lie of itegratio to Re w = δ. Observe

3 Hooley Huxley cotour 47 that i Re w 3 we have Rw) = O exp exp Im w )) ) 00. Without much difficulty we obtai ) X 9) F s) = a λ s + OT 2 ) = As) + Es) say. λ Usig a well-kow theorem of H. L. Motgomery ad R. C. Vaugha we have 0) t 0 +C log log T = A + it) 2 dt a 2 λ 2 ) X 2T C log log T + O)). λ Now u) = Ou 2 ) always but it is also + Ou 2 ) ad usig these we are led to the theorem. However, the proof of Theorem ad also that of Theorem 3) is ot simple. It has to use the desity results Nσ, T ) = OT B σ) log T ) B ) ad Nσ, T ) = OT B σ) 3/2 log T ) B ) the former is a cosequece of the latter if we are ot particular to have a small value of B) where B > 0) ad B > 0) are costats ad δ σ. Also it has to use the zero free regio σ C 3 log t) 2/3 log log t) /3 t t 0 ) for the Riema zeta fuctio ad more geeral fuctios). Sice the costat B is uimportat i our proof, Remark 3 below Theorem holds. I fact, as will be clear from our proof, oly the portio σ δ of the mhh ) cotour will be eough for our purposes.) Also if oly the desity result Nσ, T ) = OT B σ) log T ) B ) ad the zero free regio σ C 5 log T ) are available the we ed up with O log log T + a 2 ) explog T ) 3 ) for the error term ad it is ot hard to improve this to some extet. We ow proceed to state our geeral result. Cosider the set S of all abelia L-series of all algebraic umber fields. We ca defie log Ls, χ) i the half plae Re s > by the series ) χp m )mp ms ) m p where the sum is over all positive itegers m ad p rus over all primes i the case of algebraic umber fields p rus over the orm of all prime ideals). More geerally, we ca by aalytic cotiuatio) defie log Ls, χ)

4 48 R. Balasubramaia et al. i ay simply coected domai cotaiig Re s > which does ot cotai ay zero or pole of Ls, χ). For ay complex costat z we ca defie Ls, χ)) z as expz log Ls, χ)). Let S 2 cosist of the derivatives of Ls, χ) for all L-series ad let S 3 cosist of the logarithms as defied above for all L-series. Let P s) be ay fiite power product with complex expoets) of fuctios i S. Let P 2 s) be ay fiite power product with o-egative itegral expoets) of fuctios i S 2. Also let P 3 s) be ay fiite power product with o-egative itegral expoets) of fuctios i S 3. Let b =, 2, 3,...) be complex umbers which are O ε explog ) ε )) for every fixed ε > 0 ad suppose that F 0 s) = b s is absolutely coverget i Re s δ where δ 0 < δ < /0) is a positive costat. Fially, put 2) F s) = P s)p 2 s)p 3 s)f 0 s) = a s. The we have 3) Theorem 3. We have F + it) 2 dt = T where C 6 > 0) is a large costat. a O log log T + T C 6 a 2 ) R e m a r k. It is possible to have a more geeral result. For example we ca replace F s) i 2) ad 3) by F s) + d m) + α) s where m is a positive iteger costat ad α is ay costat with 0 < α <. The the right had side of 3) has to be replaced by T a T d m )) 2 + α) 2 + Olog log T ) + O T C 6 a 2 + d m )) 2 ) ). 2. Proof of Theorem 3. We form the mhh ) cotour associated with L-fuctios occurrig i F s)) as i [2]. But we select a small costat δ 0 < δ < ) ad treat the poits δ + iν ν = 0, ±, ±2,...) as though they were zeros associated with L-fuctios occurrig i F s). We recall

5 Hooley Huxley cotour 49 Rw) = expsiw/00)) 2 ). Put s = + it, = C 7 log log T t T, ) X 4) As) = a s where u) ad X are as i 8). The 5) 2πi 2+i 2 i F s + w)x w Rw) dw w = As). We write w = u + iv ad trucate the portio v 2 ad move the w-lie of itegratio so that s + w lies i the portio of the mhh ) cotour pertaiig to v 2. We obtai 6) F s) = As) + Es) where for fixed t i t T ), 7) Es) = 2πi P F s + w)x w Rw) dw w where P is the path cosistig of the mhh ) cotour i u δ, v 2 ) ad the lies coectig it to σ = by lies perpedicular to it at the eds. Notice that to the right of the mhh ) we have by Lemma 5 of [2]) 8) F s + w) explog t) ψ ) with a certai costat ψ satisfyig 0 < ψ < ) for s + w o M, ad M,2 we adopt the otatio of [2]). Also 9) F s + w) explog T ) ψ ) with a small costat ψ 0 < ψ < /5) for s + w o M,3. With these we have the followig cotributios to + /2 Es) dt ad + /2 /2 Es) 2 dt. /2 We hadle the first itegral ad the treatmet of the secod is similar. We have deotig by P the cotour P with the horizotal lies coectig P to σ = omitted) 20) Es) dt log T ) 2 log T ) 3 Q P F s + w) X u dw dt + T 0 F s) X σ ds + T 0 where Q is the portio of the mhh ) i σ δ, /2 t T + /2). Note that s is used as a variable o the mhh ) i the itegral i 20).)

6 50 R. Balasubramaia et al. I the case of Es) 2 dt we majorise it by log T ) 4 ) 2 F s + w) X u dw dt + T 0 P log T ) 5 F s + w) 2 X 2u dw dt + T 0 by Hölder s iequality.) The cotributio to 20) from M, is Olog T ) 20 max Nσ, T )X σ) ) explog T ) ψ )) δ σ τ ad that from M,2 is Olog T ) 20 max Nσ, T )X σ) ) explog T ) ψ )) τ σ τ 2 ad that from M,3 is P Olog T ) D explog T ) ψ )X τ 3 ) where τ ad τ 2 are determied by M,, M,2 ad M,3 ad τ 3 = C 3 log T ) 2/3 log log T ) /3. Here D > 0) is some costat. Note that X is a large positive costat power of T.) Usig the stadard estimates for some details which are very much similar to what we eed, see equatios ) 3) of [3]) we obtai 2) Lemma. Both Es) dt ad Es) 2 dt are Oexp log T ) 0. )). Lemma 2. We have As) = Oexplog T ) ε )). P r o o f. Follows from the fact that ) X As) a. Lemma 3. The itegral As)Es) dt is Oexp 2 log T )0. )). P r o o f. Follows from Lemmas ad 2. Lemma 4. We have F s) 2 dt = As) 2 dt + Oexp 2 log T )0. )). P r o o f. Follows from Lemmas 2 ad 3. Now the itegral o the right had side of 2) is ) X 2 T + O)) a 2 2

7 Hooley Huxley cotour 5 by a well-kow theorem of H. L. Motgomery ad R. C. Vaugha, ad so Theorem 3 follows by a slight further work sice a = O ε ε ) for all ε > 0. Refereces [] R. Balasubramaia, A. Ivić ad K. Ramachadra, The mea square of the Riema zeta-fuctio o the lie σ =, Eseig. Math ), [2] K. Ramachadra, Some problems of aalytic umber theory, I, Acta Arith ), [3] A. Sakaraarayaa ad K. Sriivas, O the papers of Ramachadra ad Kátai, ibid ), MATSCIENCE THARAMANI P.O. MADRAS 600 3, INDIA SCHOOL OF MATHEMATICS TATA INSTITUTE OF FUNDAMENTAL RESEARCH HOMI BHABHA ROAD BOMBAY , INDIA KATEDRA MATEMATIKE RGF-a UNIVERSITET u. BEOGRADU, DJUŠINA 7 BEOGRAD, YUGOSLAVIA Received o )