THE DISTRIBUTION OF THE DIVISOR FUNCTION IN ARITHMETIC PROGRESSIONS

Transcription

1 The Pennsylvania Sae Univesiy The Gaduae School Ebely College of Science THE DISTRIBUTION OF THE DIVISOR FUNCTION IN ARITHMETIC PROGRESSIONS A Disseaion in Mahemeics by Papanpong Pongsiiam c 202 Papanpong Pongsiiam Submied in Paial Fulfillmen of he Reuiemens fo he Degee of Doco of Philosophy May 202

2 The disseaion of Papanpong Pongsiiam was eviewed and appoved by he following: ii Robe C. Vaughan Pofesso of Mahemaics Disseaion Advise Chai of Commiee Wen-Ching Winnie Li Pofesso of Mahemaics W. Dale Bownawell Disinguished Pofesso of Mahemaics Jogesh Babu Pofesso of Saisics Svelana Kaok Pofesso of Mahemaics Chai of Gaduae Pogam Signaues ae on file in he Gaduae School.

3 iii ABSTRACT We sudy he disibuion of he diviso funcion in aihmeic pogessions. Fis, we give an inoducion o elaed ideas and esuls in Chape. In Chape 2, we give a lis of basic esuls and lemmas. Then we calculae an asympoic fomula and wo kind of second momens associaed wih he diviso funcion in aihmeic pogessions in Chape 3, 4, and 5, especively. Ou esuls impove hose of Moohashi and ae bee han hose of Blome on aveage. We also emak ha he mehod used in his disseaion should have wide ange of applicabiliy o ohe seuences.

4 iv Table of Conens Lis of Figues Acknowledgmens v vi Chape. Inoducion Lis of definiions and noaions Pimes in aihmeic pogessions Aveage esul on pimes in aihmeic pogessions Diviso funcions in aihmeic pogessions Aveage esuls on he diviso funcion in aihmeic pogessions and he conen of his hesis Chape 2. Lemma Chape 3. An asympoic fomula of he sum of he diviso funcion in aihmeic pogessions Chape 4. Main Resuls Ouline of he poof An uppe bound of V, The calculaion of S The calculaion of S The calculaion of J, Se up he sum of J, The calculaion of he inegals The calculaion of V, Chape 5. Vaughan s mehod Definiion and noaion Lemma Main esuls Bibliogaphy

5 v Lis of Figues 4. The conou γ of inegaion

6 vi Acknowledgmens I would like o epess my deep gaiude o my adviso, Pofesso Robe C. Vaughan. Fo many yeas, I knew I like he subjec of analysis bu I didn find he eacly igh one unil I me him. Afe aending his classes, I chose o do eseach on analyic numbe heoy. Since hen he has played a majo ole in my gowing enhusiasm fo he subjec. The moe I wok wih him, he moe I see he beauy of analysis and numbe heoy. Also by being an ecellen ole model, he moivaes me o wok hade. Though his guidance all hese yeas, I have leaned a lo fom him and have impoved myself o be a bee mahemaician. Thanks also go o Pofesso Wen-Ching Winnie Li, Pofesso W. Dale Bownawell, and Pofesso Jogesh Babu, he commiee membes, fo hei suppo. Thanks o all of my eaches who augh me well duing my sudy in Thailand. To name jus a few, hey ae Associae Pofesso Imchi Temwuipong, Associae Pofesso Naanad Tiphop, Pofesso Robeo Coni, Associae Pofesso Wichan Lewkeeaiyukul, Associae Pofesso Ajchaa Hanchuwong, and Associae Pofesso Naapan Kiisin. I am happy o acknowledge my indebedness fo he financial suppo fom my family bon o pesen, fom he Depamen of Mahemaics, Penn Sae Univesiy , fom Depamen of Mahemaics, Chulalongkon Univesiy and fom he pojec DPST conduced joinly by he Royal Thai Govenmen Agencies and he Insiue fo he Pomoion of Teaching Science and TechnologyIPST. My life in USA is happy and comfoable wih he help fom many fiends. I have obained a lo of suppo fom Saawu Cheunka, Siapa Rujkiegumjon, Bian Tule, Yi-Ting Huang, Shishuo Fu, Pichkii Bannangkoon, Sudahip Saean, Uhompon Domhong, Thap Panianaak, Tani Jaisa-ad, Chayanis Jaisa-ad, Paveena Siangsan, Wachaeepan Aipona, and Kosin Panya-a-isin. Paiculaly, Saawu Cheunka is a suppoe and a pane in able ennis. Playing able ennis wih him make my life a Penn Sae enjoyable. Fuhemoe, Tipaluck Kiyakiene, Sahoko Ichikawa, Lida Ing, and Sam have given me a place o say like a home duing long holidays fo many imes. Visiing hem helps me egain a lo of enegy. I also would like o hank Panupong Vichikunakon and Naawa Klamsakul fo giving a comfoable place o say duing my long visi a UIUC. Spending ime leaning numbe heoy hee is a pleasue. The convesaion wih Jingjing Huang, Boonod Yuanan, and Voapan Chandee is fun and encouaging. Thanks o hem, I have leaned a lo of ineesing soies in he wold of analyic numbe heoy. Las bu no leas, I am vey gaeful o my wife, Tammaada Pongsiiam, fo he suppo and sacifice all hese yeas. Wihou he, mahemaics would means nohing o me. No amoun of hanks and paise would be enough o do he jusice.

7 Chape Inoducion. Lis of definiions and noaions The lees a, b, k, m, n, ae posiive ineges; p, p, p 2,... ae pimes; α is a eal numbe; is a lage eal numbe; s = σ + i is a comple numbe; s is he comple conjugae of s; ε is abiaily small posiive consan. Below is he lis of definiions and noaions we use. Mos of hem ae simila o he one given in Dav and MV. f = Og, o euivalenly f g means ha hee is a consan c > 0 such ha f c g fo all in a specify domain, usually [,, f g means ha f g conveges o as appoach o a specify limi, usually. σ c = he abscissa of convegence σ a = he abscissa of absolue convegence [] = he lages inege less han o eual o, called he inegal pa of {} = [], = min n n Z called he facional pa of dn = m n, called he diviso funcion, µn = σn = m n m Λn = if n = if n = p p 2 p and p, p 2,...,p ae disinc, 0 if hee is p 2 n called he Mobius funcion { log p if n = p k, called he von Mongold funcion 0 ohewise a, is he geaes common diviso of a and π = p π,, a = p p amod

8 ϑ = log p p ϑ,, a = log p p p amod ψ = n Λn ψ,, a = li = Γs = ζs = n n amod 2 0 n= eα = e 2πiα c a = ϕn = log d Λn e s d fo σ > 0, he gamma funcion n s, b= b,= e k n k,n= he Riemann zea-funcion ba, he Ramanujan sum, he Eule s phi funcion γ = lim n log, he Eule consan n A,, a = dn, he sum of he diviso funcion in aihmeic pogession M,, a = ρ R,, a = n n amod c a n R n amod log 2 + 2γ, an esimaion of A,, a c n log n 2 + 2γ, anohe esimaion of A,, a. Below is he lis of noaions and definiions in linea algeba. Mos of hem ae simila o he one given in Lan. Fo vecos u, v C n, a mai H M n C, and a linea opeao T : C n C n define 2

9 3 u, v = n ū i v i, i= u = u, u 2, he inne poduc of u and v he nom of u T = inf{m T M fo all C n }, he opeao nom of T H = H, he conjugae anspose of H H is said o be Hemiian if H = H T is said o be self-adjoin if T, y =, Ty fo all, y C n λ C is an eigenvalue of T if hee eiss v C n {0} such ha Tv = λv v C n {0} is an eigenveco of T if hee eiss λ C such ha Tv = λv..2 Pimes in aihmeic pogessions The pime numbe heoem fo aihmeic pogessions asses ha each educed esidue class amod conains he same popoion of pimes i.e. fo fied a, wih a, = we have π,, a π ψ ϕ, ψ,, a ϕ, o in uaniaive fom Dav, Chape 20 ψ,, a = ϕ + O e c log. whee c is an absolue consan. An impoan hing concening he fomula. is he unifomiy of. Then we should emphasize ha we saed. fo fied. If is no fied, wihou Siegel s heoem,. holds unifomly in only in he ange log δ fo some δ > 0 Dav, p.23. Wih Siegel s heoem MV, p. 372, we obain Siegel-Walfisz heoem MV, p. 38 which asses ha hee is an absolue consan c > 0 such ha fo a given A > 0 ψ,, a = ϕ + O A e c log.2 holds unifomly in log A and he implici consan depend only on A. The analogous esul MV, p.382 of.2 fo ϑ,, a and π,, a can also be obained. Assuming he genealized Riemann Hypohesis, we have an impovemen of.2 as ψ,, a = ϕ + O 2 log 2.3 holds unifomly in. Anyway, wihou he genealized Riemann hypohesis, he fomula.2 is he bes we have Dav, Chape 22. Alhough he genealized Riemann Hypohesis is sill unpoved, we aleady have a subsiue of.3 in he fom of an aveage esul due o Bombiei Bo.

10 4.3 Aveage esul on pimes in aihmeic pogessions In 965 Bombiei Bo impoved he lage sieve of Linnik and Renyi, by which he obained an impoan esul concening he eo em in pime numbe heoem fo aihmeic pogessions in he following fom: Fo given A > 0, hee is B = BA > 0 such ha if 2 log B, hen ma y ma a,= ψy,, a y ϕ log A..4 This esul was lae impoved by A.I. Vinogadov in Vi. Davenpo Dav, Gallaghe Ga2, Mongomey Mon3 and Huley Hu gave a poof wih B = 4A + 40, 6A+03, A+3 and A+0, especively. The bes consan o dae dues o Vaughan Va5 whee he shows ha B can be aken o be A Anohe kind of aveage esuls is he second momen of he eo em. Resuls in his diecion saed wih he wok of Baban Ba, Davenpo and Halbesam DH, and hen impoved by Gallaghe Ga in he following fom: Le A > 0 be given and log A. Then V, = a= a,= ψ,, a ϕ 2 2 log A.5 The impoan poin of his esul is ha he ange of in.5 is wide han he one in he esimaion.4. Then Mongomey Mon succeeded in eplacing.5 by an asympoic fomula in he following fom. Le A > 0 and. Then V, = ψ,, a 2 ϕ = log + O + 2 log A..6 a= a,= Mongomey s mehod depended on a esul of Lavik Lav in addiive numbe heoy and was, heefoe dependen upon Vinogadov heoy of eponenial sums ove pimes Mon, p.34. Replacing ψ,, a in.6 by ϑ,, a, Hooley Ho and Ho2 inoduced and developed a simple mehod which led o he asympoic fomula V, = log + c + O log A fo.7 and by assuming genealized Riemann hypohesis, he fomula is V, = log + c + O ε 2 fo.8 whee c is an absolue consan. Some of he vaiaion of hese esuls ae also obained by Hooley Ho4.

11 The fomula.7 and.8 wee fuhe impoved especively by Fiedlande and Goldson in FG and by Goldson and Vaughan in GV. Cuenly, eseaches ae sill ineesed o invesigae moe on his kind of esuls no only in he seuence of pimes bu also in moe geneal seuences. Fo eample, Hooley Ho3, Ho5, Ho6, Ho7, Ho8, Ho9 and Vaughan Va3, Va4 sudy he second momen V, associaed wih a seuence saisfying ceain condiions moe geneal han he seuence of pimes. Bu hose condiions ae no saisfied by he diviso funcion dn and he funcion n which couns he numbe of epesenaion of n as a sum of wo suaes. The lae case was sudied by Dancs in his hesis Dan unde he guidance of Vaughan. The fis case is lef o be sudied. Theefoe, in his hesis, we ae ineesed o find he second momen associaed wih he diviso funcion in aihmeic pogessions. An obvious condiion which make ou siuaion diffeen fom ha in pimes is ha we sudy he disibuion of he diviso funcion on any esidue class no jus on a educed esidue class. Also he easons ha make ou siuaion moe complicaed is ha he local faco and global faco canno be sepaaed see Theoem 3.3 in his hesis and compae wih Va3, Va4. In he following secions, we will biefly discuss an analogue of Secion.2 and.3 fo he case of diviso funcions and hen we will say somehing abou he conen of his hesis..4 Diviso funcions in aihmeic pogessions In his secion, we suppose ha a, =, and we le A,, a = d n n n amod A,, a = M,, a + E,, a.9 whee M,, a is an appopiae main em and E,, a is he eo em in he esimaion of A,, a by M,, a and d n is he numbe of soluions in a, a 2,...,a N of he euaion n = a a 2 a. Thee ae many aicles in he lieaue concening he size of E,, a and he unifomiy of in he asympoic fomula.9. Resuls in his diecion ae given by, fo eample, Lavik Lav2, Lavik and Edgoov LE, Smih Sm, Nowak No, Masumoo Ma, Fiedlande and Iwaniec FI, FI2, Heah-Bown He, Chace Ch, Nakoya Nak and so on. These aicles ae usually called he diviso poblem fo aihmeic pogessions, which give analogous esuls fo pimes in aihmeic pogessions as in Secion.2. We do no go in his diecion. Insead we focus on he esul of he same kind as in Secion.3, which will be called aveage esuls on he diviso funcion in aihmeic pogession. In he ne secion we conside only he case = 2. We wie d 2 n = dn, he classical diviso funcion. We also wie A 2,, a, M 2,, a and E 2,, a as A,, a, M,, a and E,, a, especively. 5

12 .5 Aveage esuls on he diviso funcion in aihmeic pogessions and he conen of his hesis Thee ae a few aicles which give aveage esuls fo he diviso funcion in aihmeic pogessions analogous o Secion.3. Fo eample, he wok of Fouvy and Iwaniec in FoI, and Banks, Heah-Bown, and Shpalinski in BHS. Howeve he esuls menioned above assume he condiion a, =. Fo insance, Banks e.al BHS defined fo a, =, S,, a = dn. n n amod The auhos BHS menioned ha in an unpublished wok, A. Selbeg and C. Hooley independenly discoveed see, e.g., he discussion in Ho0 ha he Weil bound fo Klooseman sums implies ha fo evey ε > 0, hee eiss δ > 0 such ha S,, a = P log ϕ δ + O,.0 ϕ povided ha < 2/3 ε, whee P is he linea polynomial given by P log = ϕ2 2 log + 2γ + 2ϕ d µdlog d. d Then hey BHS show ha.0 holds on aveage fo all moduli up o ε, moe pecisely, hey show ha fo evey ε > 0, hee eiss δ > 0 such ha S,, a P log ϕ = O δ, < ε.. a Z Fom his i follows ha fo all moduli < ε, he diviso sum S,, a lies close o is epeced value fo almos all a Z in a suiable sense. Thei esuls BHS and he esuls in his hesis do no imply each ohe. The fis is a kind of fis momen while he lae is a kind of second momen. Similaly, he wok of Fouvy and Iwaniec in FoI is anohe kind of fis momen whee hey use a diffeen main em. In fac, Banks e.al BHS use P log while Fouvy and Iwaniec FoI use dn ϕ ϕ n n,= as he main em. Blome s esul Bl is pobably moe elaed o ous. Fis of all, he does no assume he condiion a, =. Secondly his main em is he same as ous. His esul is ha a= n n amod dn c a 2 log 2 + 2γ +ε..2 6

13 Blome s esul and ou esul do no imply each ohe. Bu if we conside hem on aveage, hen ou esul is bee. I basically says ha he uppe bound +ε in.2 can be eplaced by log 3 on aveage. And i is menioned by Blome Blsee p.277 ha his esul Bl is songe han ha of Banks e.al BHS. So on aveage, ou esul is also bee han ha of Banks e.al BHS. Fom his poin on, we dop he assumpion a, = and we le V, = A,, a M,, a 2. a= Then he only esul concening V, in he lieaue so fa is he one given by Moohashi Mo. The main em M,, a in Mo looks diffeen fom ous bu acually is he same compae Mo, p.77 and he definiion in Secion 4.. Bu Moohashi gave he esul only fo = = N N. His main esul Mo is ha hee ae consans g, g 2, g 3, g 4 such ha V, = g 2 log 3 + g 2 2 log 2 + g 3 2 log + g 4 + O 5 8 log 2..3 This is he only aveage esul in he lieaue which is of he eac same kind as ous. In his hesis, we give moe esuls of his kind as follows: We give he big-o esul fo V, fo in Secion 4.2. This is an analogous esul fo he wok of Baban Ba, Davenpo and Halbesam DH, and Gallaghe Ga. Like hei wok, he essenial ool in ou poof is he lage sieve ineualiy in he fom given in Lemma 2.3. Noe ha he lage sieve was invened by Linnik. Then i was developed by many eseaches such as Renyi, Roh, Davenpo, Halbesam, Bombiei, Gallaghe, Mongomey and Vaughan, Cohen, and Selbeg. The fom given in Lemma 2.3 is poved by Selbeg Mon2, p.559 and by Mongomey and Vaughan MV3 wih an ea winkle by Cohen Mon2, p.559. The compehensive accoun of he lage sieve can be found in Mon2 and Bo We give he asympoic fomula fo V, fo in Secion 4.6. This is an analogous esul fo he wok of Mongomey Mon and Hooley Ho. I also eends he esul of Moohashi Mo fom = o he case wih a bee eo em. Thee ae iple inegals of funcions of hee comple vaiables and 36 conou inegals of complicaed funcions appea in he calculaion of V,. When we move he conou o he lef, we will pick up he esidues a s = and s = 0 and some of he inegands have poles of ode 3 o 4 a hose poins. So we see ha hee ae a lo of calculaion involved. The advanage of his is ha i gives a smalle eo em, saving by 8, compae o ha of Moohashi Mo.

14 8 3 We give an asympoic fomula fo he second momen M 2,, ρ R defined by M 2,, ρ R = A,, a ρ R,, a 2 a= c whee ρ R,, a = n log n 2 + 2γ. n R n amod This gives an analogous esul fo he wok of Vaughan Va, and Va2. Alhough he eo em appea in M 2,, ρ R is lage han ha of V,, i is lage only by log. Bu he advanage of M 2,, ρ R ove V, is ha is calculaion is much shoe. Because of his, i has moe poenial o give us he hid momen fo he diviso funcion in aihmeic pogessions in he fuue.

15 9 Chape 2 Lemma In his chape, we give some well-known esuls which will be used in he calculaion lae. Lemma 2.. The ohogonaliy elaion of addiive chaaces { if n amod n a e = = 0 ohewise. 2. Poof. If n amod, hen each summand is and he esul is clea. If no, i is he sum of geomeic pogession which can be compued diecly. Lemma 2.2. Le sα = eαn. If α Z, hen sα = b. If α / Z, hen sα sinπ α 2 α. a<n a+b Poof. If α Z, hen each summand is and he esul follows. Fo he ohe case, see he poof of Lemma 4.7 in Na, p.04. Lemma 2.3. The lage sieve ineualiy. Le a n be a seuence of comple numbes. Then 2 an a n e N + 2 a n 2. a= M<n M+N a,= Poof. See Te, p M<n M+N Lemma 2.4. Cauchy inegal fomula If f is holomophic in an open se Ω, hen f has infiniely many comple deivaives in Ω. Moeove if C Ω is a cicle whose ineio is also conained in Ω, hen f n z 0 = n! 2πi C fz z z 0 n+dz fo all z 0 in he ineio of C, whee he conou inegal is aken coune-clockwise. Poof. See e.g.ss, p.47.

16 Lemma 2.5. Peon fomula Le αs = 0 n= a n n s be a Diichle seies which conveges fo σ > σ c. If σ 0 > ma{0, σ c } and > 0, hen n a n = 2πi σ0 +i σ 0 i αs s s ds whee indicae ha if Z, hen he las em is 2 a. Poof. See e.g. MV, p.38 o Ap, p.245. Lemma 2.6. Eule poduc fomula Assume ha fnn s conveges absoluely fo n= σ > σ a. If f is muliplicaive, hen fn n s = + fp p s + fp2 p 2s + p n= Poof. See e.g. Ap, p.23. d, d fo σ > σ a. Lemma 2.7. Diichle hypebola mehod If u, v R + such ha uv =, hen fdg = fd g + g fd fd g. d u v d d u v d Poof. See e.g. Ap, p.69. Lemma 2.8. Paial summaion Le a be aihmeic funcion and A = n an. Assume ha f has a coninuous deivaive on [y, ], whee 0 < y <. Then we have y<n anfn = Af Ayfy Poof. See e.g. Ap, p.77. { if n = Lemma 2.9. µd = 0 ohewise. d n y Af d. Poof. See e.g. Ap, p.25. Lemma 2.0. d n Poof. See e.g. Ap, p. 26. ϕd = n and ϕn n = d n µd d.

17 Lemma 2.. n= ϕn ζs n s = ζs Poof. See e.g. Ap, p. 23. Lemma 2.2. d n log d = dn 2 fo σ > 2, log n. n= µn n s = ζs fo σ >. Poof. This is euivalen o d = n dn 2. Fo each u n, u < n hee is v n and d n v > n such ha vu = n. We goup such u, v ogehe o ge he desied esul. See also Ap, p.47, poblem 2. Lemma 2.3. dn = o n ε. Poof. See e.g. MV, p. 56. Lemma 2.4. Fo each σ, le µσ = inf{c [0, ζσ + i c }. Then µ is a nonnegaive, deceasing, coninuous, conve downwad funcion such ha Poof. See e.g. Iv, p.25. µσ = σ, 2 fo σ 0 µσ σ, 2 fo 0 σ µσ = 0, fo σ. Moeove ζs µσ+δ fo any δ > 0 and if ζs µσ+δ, hen ζ s µσ+δ 2.2 Lemma 2.5. Le δ > 0, R = {s C : s δ and ag s < π δ}. Then Γ s = log s + O unifomly fo s R. Γ s Poof. See e.g. MV, p.523. Lemma 2.6. ζs is holomophic on C ecep fo a simple pole a s = wih esidue and he Lauen epansion of ζs a s = is ζs = s + γ + γ s + γ 2 s whee γ is Eule s consan and γ k = k k! lim n m n log m k m log nk+. k + Poof. See e.g. Iv, Theoem.3.

18 Lemma 2.7. If Res > 0, hen Γs + = sγs. Moeove, Γ has an analyic coninuaion o a meomophic funcion on C whose only singulaiies ae simple poles a s = n n = 0,, 2,... wih esidue n. n! Poof. See e.g. SS, p. 6. Lemma 2.8. Γ is an enie funcion wih simple zeos a s = 0,, 2,... and i vanishes nowhee else. Poof. See e.g. SS, p. 65. Lemma 2.9. Fo Res > 0 Rez > 0, Poof. See e.g. AAR, p s z d = ΓsΓz Γs + z. Lemma Le g be analyic a z = z 0 and a fz = N z z 0 N + a + a z z 0 + a z z 0 + a 2 z z be he Lauen seies 0 of f, whee he case N = coesponds o fz = a n z z 0 n. n= Then Res z=z0 fzgz = gn z 0 N! a N + gz 0 a. In paicula, we will use his in he case N =, 2, 3, 4. When N =, Res z=z0 fzgz = gz 0 a When N = 2, Res z=z0 fzgz = g z 0 a 2 + gz 0 a When N = 3, Res z=z0 fzgz = g z 0 2 a 3 + gz 0 a When N = 4, Res z=z0 fzgz = g3 z 0 6 a 4 + gz 0 a. Poof. This can be poved by diec calculaion. Lemma 2.2. log log 2 log 2 log 3 d = + C, d = + C, 2 3 log 2 d = log log 2 + C, log 2 2 d = log 3 log 3 3log 2 2 d = 6 log 6 + C 2 log Poof. This can be compued by subsiuion echniue o inegaion by pa. Lemma Le U = f...f n g...g m, V = f f + + f n f n g g g m g m. 2 + C, 2

19 3 Then d d U = UV, d 2 d 2U = UV 2 + UV, d 3 d 3U = UV 3 + 3UV V + UV. Poof. We wie log U = n log f i i= m log g i. Then diffeeniae o ge U = UV as desied. The ohe wo cases follow fom he fomula of U. Lemma Fo σ >, n= d 2 n n s i= = ζ4 s ζ2s. Poof. See HW, Theoem 304. Lemma d 2 n = a 3 log 3 + a 2 log 2 + a log + a 0 + O whee n a 3 = π 2, a 2 = 34γ π2 36ζ 2 π 4, 3 4 a = 6π4 4γ + 6γ 2 4γ ζ 2 2 2π 2 ζ 2 + 4γζ 2 ζ 2 π 6 a 0 = 24 π 4 3 2γ ζ 2 + 2ζ 3 2 3ζ 2 + 8γ 2 ζ 2 2γζ 2 ζ 2 + π 6 4γ + γ4 2γ + 2γ 2 + 4γ 3 6γ 2 728ζ π 2 ζ 22ζ 2 + 4γζ 2 ζ 2. Poof. By Lemma 2.23 and he Peon fomula, we see ha o appoimae n d 2 n one needs o compue he esidue a s =. Since ζ 4 s has pole of ode 4 a s =, he calculaion is vey long. Founaely, he esidue is compued by R.Baillie Bai using Mahemaica in Bai, p.42. O In fac, we can eplace he eo em O ep Alog 5 log log 5 SSi, bu O in he fomula by O ε Wi o is good enough in ou siuaion. The ne lemma is pobably a sandad esul bu we did no find a efeence fo i. So we give a poof hee. Lemma Fo η C, Re η > 0, and σ 0 > ma0, σ a, we have a η = 2πi σ0 +i σ 0 i As s+η ΓsΓη + ds. Γs + η +

20 whee As = a s = Poof. I is easy o see ha η = η u η du. Muliply boh sides by a and sum ove and change he ode of summaion we obain a η = η u η a 0 du 2.3 u We can divide he limi of inegaion [0, ] o 0,, 2... [], [] [],. This gives he same inegal and since u / Z, we have u Subsiue 2.4 in 2.3, we obain a η η = 2πi a = 2πi = η 2πi σ0 +i σ 0 i σ0 +i 0 σ 0 i σ0 +i σ 0 i As s By subsiue v = u, he inne inegal above is Theefoe 2.5 become s+ 0 a η = 2πi 4 As us ds. 2.4 s η Asu s dsdu s u η u s du ds 2.5 u v η v s dv = s+ Γs + Γη. Γs + η + = 2πi σ0 +i σ 0 i σ0 +i σ 0 i 0 As s Muliply boh sides by η, we ge he desied esul. s Γs + ηγη Γs + η + ds As s ΓsΓη + Γs + η + ds.

21 5 Chape 3 An asympoic fomula of he sum of he diviso funcion in aihmeic pogessions In his chape, we begin wih he calculaion of n dne o find an asympoic fomula fo n n amod an. Then we poceed dn, by applying Lemma 2. o capue he condiion n amod and using Lemma 2.2 in he esimaion of is eo em. Definiion 3.. Le A,, a = n n amod dn Gα, = n dnenα When hee is no confusion, we will wie Gα fo Gα,. Lemma 3.2. Le a, =. Then a G = log 2 + 2γ + O + log 2. Poof. Wie dn = u n mehod, we have G, eaange he ode of summaion and use Diichle hypebola a = 2 u v u a e uv u v a e uv.

22 6 Then apply Lemma 2.2 o ge G a = 2 u u [ u] u u [ ] + O u u au = log 2 + 2γ + O + O If =, he esul is clea, so we assume ha 2. Conside he second eo em. We have Theefoe G u u Theoem 3.3. au = = = u u mod au + n= n + n 2 = a u u + au + 2 n log. n. a = log 2 + 2γ + O + log 2, as euied. A,, a = c a n log 2 + 2γ + O + log 2 Poof. We apply Lemma 2. o capue he condiion n amod, and eaange he ode of summaion o ge A,, a = n = dn = m= m,= e n a m e = m a G a e = G

23 7 Applying Lemma 3.2, we see ha he main em is and he eo em is c a log 2 + 2γ, φ + log 2 φ + log 2 = + log 2. So A,, a = c a log 2 + 2γ + O + log 2, as euied.

24 8 Chape 4 Main Resuls 4. Ouline of he poof Fis we give definiion of uaniies ha we will discuss abou in he ouline of he poof. Fo, R, a, N, a, we define Fo R <, we define M,, a = V, = c a log 2 + 2γ A,, a M,, a 2 a= and V, R, = S = S, R, = S 2 = S 2, R, = S 3 = S 3, R, = J, = A,, a M,, a 2, R< a= R< a= R< a= R< a= A,, a 2, A,, am,, a, M,, a 2, < m<n m nmod and dmdn. Fo he main esuls, we fis give an uppe bound of V, in Secion 4.2. Then we follow Hooley s mehod o calculae an asympoic fomula of V,. We wie V, = V, R + V, R, whee V, R, is defined above. Then we esimae V, R by applying he uppe bound we obain in Secion 4.2 so ha V, = V, R, + ORlog 3.

25 So i emains o calculae only V, R,. By suaing ou he ems A,, a M,, a 2, we can wie V, R, = S 2S 2 + S 3 whee S, S 2 and S 3 ae defined above. Then V, = S 2S 2 + S 3 + ORlog 3. We calculae S 3 and S 2 in Secion 4.3 and 4.4, especively. Fo S, we wie S = R< a= m,n m n amod = R< m,n m nmod = R< n = [] [R] n dmdn d 2 n + 2 dmdn R< m<n m nmod dmdn d 2 n + 2 J, R J, 9 whee J, is defined above. We calculae J, in Secion 4.5. Afe ha we combine all he esuls we obain o calculae V, in Secion 4.6. This will give he asympoic fomula fo V, as desied. 4.2 An uppe bound of V, In his secion, we give an uppe bound fo V,. I will be seen lae ha his uppe bound is shap. Theoem 4.. Le, R,. If, hen V, log 2 2. If >, hen V, log 3. Poof. Fo, we wie c a = b= b,= e ba = b= b,= e ba,

26 20 hen we have M,, a = = = e b= b= b= b,= ba e ba log 2 + 2γ b,= log 2 + 2γ ba, b e log, b γ. Then wie A,, a = dn ba n e n b= = ba e dne bn b= n = ba e G b b= Then we have V, = a= = 2 A,, a M,, a 2 e a= b= ba H b, whee H b, = G b, b log, b γ. Then by he ohogonaliy elaion, we have V, = b= H b, 2 = d d b= b,d= 2 Hb, d 2, whee Hb, d = G b log d d d 2 + 2γ. Then changing he ode of summaion o ge V, = d d d b= b,d= Hb, d 2 l d l.

27 Le D be a paamee o be deemined lae. We divide he sum d of V, ino wo pas : d D and D < d and wie V, = V, D + V 2, D, 4. whee V, D and V 2, D, coespond o d D and D < d, especively. Then 2 V, D d D d D d d b= b,d= ϕd d log 2D 2 Hb, d 2 log 2 d + dlog 2d 2 log 2 d d D + d 2 log 2 d, by Lemma 3.2 and he definiion of Hb, d log 2D 2 D + D 2 + D 3 log 2 D log 2D 2 D + D 3 log 2 D, as D2 ma{d, D 3 } 4.2 In paicula when, we le D = and obain V, = V, log log 2 2. To obain a shap esimae when >, we will caefully calculae V 2, D,. We have V 2, D, log 2 d d Hb, d 2. d Noe ha Hb, d 2 2 b G d V 2, D, D<d + D<d D<d D<d b= b,d= d 2 log d 2 + 2γ 2, so log 2 d d log 2 d d log 2 d d d b= b,d= b 2 G d ϕd 2 d 2 log d 2 + 2γ 2 d b= b,d= b 2 G + log d 2 log 2 D. 4.3

28 We will esimae he fis em above by he lage sieve ineualiy. d Le G 0 d = b G d n b= b,d= d 2 n log 3, we have D<d Then by he lage sieve ineualiy and he bound G 0 d 2 + n d 2 n 2 + log D<d log 2 d d = log 2 = log 2 G 0 d = D log 2 A AD + A AD + da AD, whee A = d log 2 + d D D A AD 2 log 2 A ADd + D G 0 d + log 2A AD 2 = log 2 A AD + a + a 2, say 4.5 d By 4.4, we have A AD 2 + log 3. So log 2 A AD + a 2 log log 3 + D 2 d log log 3 D 4.6 a D log 2 + log 3 d = log 3 log 2 D + 2d log 3 D + 2 log d 4.7 We have + D 2d = D + D and + D 2 log d = log D log D + D + log + log D D + D = Dlog + D log D + D + D log D + D log.

29 Subsiue above inegals in 4.7, hee is some cancellaion going on and we obain a log 3 + log 3 log D Hence fom 4.5, 4.6 and 4.8 we obain 23 D D. 4.8 D<d log 2 d d G 0 d log 3 + log 3 log D D D + 2 log 3. D This gives he esimae fo he fis em in 4.3 and hus V 2,, D log 3 + log 3 log D D D + 2 log D Recall ha >. We le D =, hen fom 4., 4.2 and 4.9 we obain V, log 3. To summaise when >, V, log 3 and when, V, log The calculaion of S 3 In his secion we will fis compue S 3, defined by S 3, = M,, a 2. Afe ha, we will obain S 3, R, = S 3, S 3, R as desied. Theoem 4.2. Le, R,. Then hee ae consans C, C 2, C 3, C 4 such ha a= S 3, = C 2 log 2 log + γc C 2 2 log 2 + 4γ 2C 4C 2 2 log log + 2γ2γ C 24γ C 2 + 4C 3 2 log + 2γ 2 C 8γ 4C 2 + 4C 3 2 log + 2γ 2 γc 2γ 2 + 4γ2γ C 2 + 2γ 4C 3 4C O log 2 log

30 Poof. We pu c a = We see ha 2 2 n= n,= an e M,, a 2 is eual o a= log 2 + 2γ log 2 + 2γ a= 24, suae ou and change he ode of summaion. n= m= a= n,= m, = an e By he ohogonaliy elaion, he innemos sum is a n e m { if n = m 0mod 0 ohewise am e Since 0 < n, m and n, = m, =, we have n m 0mod n = m = and n = m. So he innemos sum is if = and n = m, o ohewise i is 0. Thus a= M,, a 2 = 2 ϕ 2 log 2 + 2γ Pu 4.0 in he definiion of S 3, and change he ode of summaion. Then we ge Puing k k and heefoe S 3, = 2 S 3, = 2 ϕ 3 log 2 + 2γ 2 k k. 4. = log +γ log +O in 4. gives he eo O 2 log 2 log, ϕ 3 log γ log + γ log + O Ne we will calculae he main em eplicily. We wie he main em as 2 log 2 log 2 ϕ 3 log + 2γ 2 log 2 log + γ log = 2 a 2 bf a 2 + 4abg + 4a + 4bh 4I whee a = a = log + 2γ, b = b = log + γ,.

31 25 f = h = ϕ 3, ϕlog 2 g = 3, I = ϕlog 3 ϕlog 3 3. Ne le C = C 3 = ϕ 3, C 2 = ϕlog 2 3, C 4 = = = ϕlog 3, ϕlog 3 = = Then we have log f = C + O, g = C 2 + O, log 2 log 3 h = C 3 + O, I = C 4 + O, whee he eo ems ae esimaed by using Lemma 2.2. Puing eveyhing ogehe we ge S 3, = 2 C a 2 b C 2 a ab + C 3 4a + 4b 4C 4 + O log 2 log = C 2 log 2 log + γc C 2 2 log 2 + 4γ 2C 4C 2 2 log log + 2γ2γ C 24γ C 2 + 4C 3 2 log + 2γ 2 C 8γ 4C 2 + 4C 3 2 log + 2γ 2 γc 2γ 2 + 4γ2γ C 2 + 2γ 4C 3 4C O log 2 log This complees he poof. Coollay 4.3. Le R <. Then S 3, R, = C 2 log 2 log + 4γ 2C R 4C 2 2 log log R + 2γ 2 C 42γ C 2 + 4C 3 2 log R + O 2 R log 2 log R Poof. S 3, R, = S 3, S 3, R, apply Theoem 4.2 o ge he euied esul.

32 Recall he definiion of C, C 2, C 3 in euaion 4.2. We can calculae hem eplicily as follows : Fom Lemma 2. we have = ϕ s = ζs ζs 26 fo σ > Le s = 3, we ge C = = ϕ 3 Diffeeniae 4.3 and pu s = 3 o ge = ζ2 ζ3 = π2 6ζ C 2 = π2 6 Diffeeniae 4.3 wice and pu s = 3 o ge ζ 3 ζ 2 3 ζ 2 ζ C 3 = ζ 2 ζ3 2ζ 2ζ 3 ζ 2 π2 ζ ζ 2 3 π2 ζ ζ If we pu he value of C, C 2, C 3 in Coollay 4.3, we obain S 3, R, wih eplici consan. We ecod i in he ne coollay. Coollay 4.4. Le. Then hee ae consans C, C 2, C such ha 3 S 3, R, = C 2 log 2 log +C R 2 2 log log +C 3 R 2 log R 2 +O R log 2 log whee C = C 2 π2 6ζ3 = 2γ π2 3ζ3 2π2 ζ 3 3ζ ζ 2 ζ3 C 3 = 2γ 2 π 2 42γ 6ζ3 + 4 π 2 ζ 3 6ζ 2 3 ζ 2 ζ3 ζ 2 ζ3 2ζ 2ζ 3 ζ 2 π2 ζ ζ 2 3 π2 3 ζ 3 2 ζ 2 3

33 The calculaion of S 2 As in he pevious secion, we fis compue S 2,, defined by Recall fom Theoem 3.3 ha A,, a = n n amod If we pu a = 0, we ge A,,0 = n n S 2, = dn = dn = ϕ A,, am,, a a= c a In he ne lemma, we find a diffeen eo em of A,,0. log 2 + 2γ + O + log 2. log 2 + 2γ + O + log 2. Lemma 4.5. Suppose ha. Then A,,0 = ϕ log 2 + 2γ + O log k Poof. A,,0 = n n = v = v = v dn = kl k,k=v j v j, v= j v j, v= kl = k 2 l k kl l kl 2 l k kl 2 l = v jv v l 2 j l v l v + O v l kl j v j, v= j v j, v= [ ] 2 j. dk. k [ ] v

34 The eo em is v k k,=v = k. Apply Lemma 2.9, and change he ode of summaion, we ge A,,0 = µw 2 v lw + O 4.7 v l vw l = log w v w v Noe ha he condiion w vw vw + γ + O A,,0 = v w v w v v l vw in 4.7 o ge 28 is supefluous bu no afe we subsiue [ 2 vw µw log w vw + γ + O ] v w + O +O. The eo em is v w v v = k µw 2 vw + w v v + w v dk k + k v µw w v dk k. + Thus A,,0 = The main em above is log v w v µw w v w v w v µw w vw 2 + 2γ Since, he second em above is log vw 2 + 2γ v w v v <w µw w v log log v k w v + O k dk. k log vw 2 + 2γ dk k

35 29 So A,,0 = v w v µw w log vw 2 + 2γ + O log k dk. k Le = vw. Then, w and heefoe A,,0 = µw log w 2 + 2γ + O log dk k w k = ϕ log 2 + 2γ + O log dk. k k Theefoe Fom Lemma 3.2, fo b, N, b, =, we have b G = log 2 + 2γ + O + log 2. b= b,= G b = ϕ In he ne lemma, we find a diffeen eo em fo b= b,= log 2 + 2γ + Oϕ + log 2. b= b,= G b. Lemma 4.6. b G = dnc n = ϕ log +O 2 + 2γ dkσ n k Poof. Apply he ideniy c n = apply Lemma 4.5, we ge,n µ k log., change he ode of summaion, and

36 30 n dnc n = k µ dn n n = µ A,,0 = ϕk µ log k k 2 + 2γ + O k k = ϕk µ log k k 2 + 2γ + O k k The eo em is dk k log = k k The main em is = k µ k ϕk k dk k kl log = dkσ k l k log k 2 + 2γ µ = l k µ kl = k { if k = 0 ohewise. Theefoe he mainem is = ϕ log 2 + 2γ. Hence dnc n = ϕ log 2 + 2γ + O dkσ n k ha b= b,= G b This complees he poof. = n b= b,= dne bn = n dnc n. dk log k dk log. k log. k log. Noe k

37 3 Theoem 4.7. S 2, = S 3, + O 2log 3 2 log 2 Poof. The innesum A,, am,, a of S 2, is eual o a= A,, a c a log 2 + 2γ a= = log 2 + 2γ c aa,, a a= = log 2 + 2γ dnc n = + O n log ϕ 2 + 2γ log 2 + 2γ log 2 + 2γ dkσ k log. k Fom euaion 4.0 in he calculaion of S 3,, we see ha he main em above is M,, a 2. a= Theefoe S 2, = S 3, + O 3 2 log Le E be he eo em above. Then E 3 2log 2 k k k k k log 2 + 2γ dkσ dkσ. k Change he ode of summaion as follows : = =. k Wie = kl and = k. Then k k = k l k = k k k. l. k

38 32 Then E 3 2log 2 k l dk k k = 2log 3 2 dk k l k = 2log 3 2 dk k 2 k = 3 2log 2 k dk k 2 k l k 3 2log 2 log k σl l 2 k = l dk k 2 d l k= l σl l 2 σ k kkl σ σl l u kl dk k 2 l d l 2 = d d c d dk k 2 u σl l 2. kl k c 2 log = ζ 2 2 = O. Hence E 3 2log 2 log 2. Coollay 4.8. Fo R <, we have S 2, R, = S 3, R, + O 3 2log 2 log The calculaion of J, In his secion, we will calculae J,. We divide he calculaion ino wo subsecions. In Secion 4.5., we se up he sum J, using Hooley s ick Ho, p.20. Then we poceed o wie J, in he fom of comple inegaion. Thee ae 36 ems appea and afe gouping hem, hee ae 5 inegals o evaluae. In Secion we poceed o he calculaion of he 5 inegals we obain in Secion This will give us an asympoic fomula fo J,.

39 Theoem 4.9. Le, R, and 0 < Θ <. Then hee ae consans C, C 2,..., C 22 such ha J, = C 2 log log 2 + C 2 2 log log log + C 3 2 log 2 + C 4 2 log log + C 5 2 log log + C 6 2 log log + C 7 2 log + C 8 2 log + C 9 2 log +C C log 3 + C 2 log log 2 + C 3 log log 2 + C 4 log log log + C 5 log 2 + C 6 log log + C 7 log log + C 8 log log + C 9 log + C 20log + C 2 log + C O log log + 3 2log log + 2log 3 log 3 + +Θ Θ log 2 As menioned above, we divide he poof ino wo subsecions Se up he sum of J, Recall ha Fis we wie A,, a = = c a c a log 2 + 2γ + O + log 2 0 J, = 33 log y 2 + 2γ dy + O + log < m<n n mmod dndm. We eaange he ode of summaion following Hooley s mehod Ho, p.20. Wie n m =. Then n mmod, <, and < n m, and we ge J, = dmdn = dm dn < m<n < m n mmod < n m m+<n n mmod

40 The innemos sum of J, is now in a fom ha can be calculaed by he applicaion of 4.8, which gives J, = dm c m log y < m m γ dy + O dm + log 2. < m 2 The eo em above is O 5 log log + 3 bm em, we wie c m = e < = < b= b,= b= m m+ b,= b= b,= 2 log 34 log. Fo he main and change he ode of summaion, which yields bm e m y dm log y 2 + 2γ dy bm e dm log y 2 + 2γ dy. The sum in he above inegal is G b, y, which is, by Lemma 3.2, eual o y log y 2 + 2γ + O + log 2. Theefoe J, = < b= b,= + O < O log log y 2 + 2γ y log y 2 + 2γ dy φ + log 2 log y 2 + 2γ dy log + 3 2log log.

41 Le M and E be he main em and eo em of J,, especively. The fis eo em is log ϕ + log 2 < = log ϕlog + ϕlog. 4.9 < < Now ϕlog log d log, by Lemma 2.2, ϕlog log by Lemma 2.0. Theefoe 4.9 is O 32 3 log log + 2 log log. Combine he second eo em, we see ha E 5 2 log log + 3 2log log + 2log 3 log Ne, we compue he main em. M = < = < = < = 0 φ y log y 2 + 2γ y log y 2 + 2γ dy φ log y y 2 + 2γ 2 log φ log y y 2 + 2γ 2 log φ log y y 2 + 2γ 2 log y 2 + log + 2γ dy y 2 + log + 2γ dy y 2 + log + 2γ dy Le α R, α > 0, and le C 0 be a small cicle abou 0 in posiive oienaion. By he Cauchy inegal fomula, we have 2πi z 2 + A dz = αlog α + A 4.22 z 2πi C 0 α z+ C 0 α z z 2 + B dz = log α + B 4.23 z

42 Using 4.23 wih α = y 2 and 4.22 wih α = y 2 log y 2, we ge log y 2 + 2γ = y w 2πi C 0 2 w 2 + 2γ w y 2 + log + 2γ = 2πi C 0 y 2 36 dw 4.24 z+ log + 2γ z2 + dz z 4.25 Subsiue 4.24 and 4.25 in 4.2 and le gw = w 2 + 2γ w, hz = log + 2γ z2 + z φ y z+ fy, z, w = 2z+2w+2 y w, we obain y 4.26 M = = = = 2πi 2 2πi 2 2πi 2 2πi C 0 y C 0 C 0 gwhzy φ C 0 C0 w y z+ 2z+2w+2 dzdwdy φ y z+ 2z+2w+2 y w gwhzdzdwdy y fy, z, wgwhzdzdwdy C 0 C 0 gwhz fy, z, wdy dzdw C Apply Lemma 2.25 fo a = fy, z, w = 2πi σ0 +i σ 0 i φ 2z+2w+2, = y, η = z +. We obain Ds, z, w y s+z+ y w ΓsΓz + 2 ds Γs + z + 2 whee Ds, z, w = 2.. = +s φ ζs + ζs z + 2w 2z+2w+2 =, by Lemma ζs z + 2w

43 37 Then 0 fy, z, wdy = 2πi = 2πi σ0 +i σ 0 i σ0 +i σ 0 i 0 Ds, z, wy s+z+w+ s z ΓsΓz + 2 dyds Γs + z + 2 Ds, z, w s z ΓsΓz + 2 Γs + z + 2 s+z+w+2 s + z + w + 2 ds. Subsiue in 4.27, we obain σ0 +i Ds, z, w s+z+w+2 ΓsΓz + 2 M = 2πi 3 gwhz C 0 C 0 σ 0 i s + z + w + 2 s+z+ Γs + z + 2 ds dzdw σ0 +i s+z+w+2 Ds, z, w ΓsΓz + 2 = 2πi 3 σ 0 i C 0 C 0 s + z + w + 2 s+z Γs + z + 2 gwhzdzdwds σ0 +i = 2πi 3 ζs + Γs hzgw σ 0 i C 0 ζs z + 2w s+z+w+2 Γz + 2 ζs z + 2w s + z + w + 2 s+z Γs + z + 2 dzdwds. C 0 Fo his poin on, i will be a long calculaion of M bu i is saighfowad. We will also subsiue gw and hz defined in 4.26 wihou menioning i, and in wha follows f, g, h will be edefined bu hey ae no he same as in Le ζs z + 2w s+z+w+2 Γz + 2 fz, w, s = gw ζs z + 2w s + z + w + 2 s+z Γs + z + 2. By he Cauchy inegal fomula he innemos inegal is log + 2γ = fz, w, s C 0 z2 + dz z f0, w, s = = 2πi f z z=0 +2πilog + 2γ f0, w, s. w 2 + 2γ w = g w, s, say. ζs w ζs w To evaluae f z, we use Lemma 2.22 and wie fz, w, s = w 2 + 2γ s+w+2 ζs z + 2w w s ζs z + 2w s+w+2 s + w + 2 s Γs + 2 z Γz + 2 s + z + w + 2 Γs + z + 2

44 38 f z = w 2 + 2γ z s+w+2 ζs z + 2w Γz + 2 w s ζs z + 2w s + z + w + 2 Γs + z + 2 [ 2ζ s z + 2w ζs z + 2w + log + Γ z + 2 Γz + 2 2ζ s z + 2w ζs z + 2w s + z + w + 2 Γ s + z + 2 ] Γs + z + 2 f z z=0 = 2 w 2 + 2γ s+w+2 ζ s w w s ζs w s + w + 2Γs log w 2 + 2γ s+w+2 ζs w w s ζs w s + w + 2Γs Γ 2 w 2 + 2γ s+w+2 ζs w w s ζs w s + w + 2Γs w 2 + 2γ s+w+2 ζs wζ s w w s ζ 2 s w s + w + 2Γs + 2 w 2 + 2γ s+w+2 ζs w w s ζs w s + w Γs + 2 w 2 + 2γ s+w+2 ζs w Γ s + 2 w s ζs w s + w + 2Γ 2 s + 2 = g 2 w, s + g 3 w, s + g 4 w, s + g 5 w, s + g 6 w, s + g 7 w, s, say. So he innemos inegal is Thus M = 2πi 2 Ne, we evaluae σ0 +i σ 0 i = 2πi 7 g i w, s + 2πi log + 2γ g w, s. i=2 ζs + Γs C 0 C 0 i=2 7 g i w, s + log + 2γ g w, sdwds. i=2 7 g i w, s + log + 2γ g w, sdw Le f i s = C 0 g i w, sdw fo i {, 2,...,7}. 4.29

45 f s = C 0 w 2 + 2γ ζs w s+w+2 w ζs w s + w + 2 s Γs + 2 = C 0 w 2 + 2γ hw, sdw, say w h = 2πi w w=0 +2γh0, s h0, s = ζs + 2 s+2 ζs + 3s + 2 s Γs + 2 = h s, say 39 To evaluae h w w=0, wie hw, s = 2 s ζs w w Γs + 2 ζs w s + w + 2 h s w = ζs w w 2 Γs + 2 ζs w s + w + 2 2ζ s w ζs w + log 2ζ s w ζs w s + w + 2. s s h w w=0 = 2 2 ζ s + 2 ζs + 3 s + 2Γs ζs log ζs + 3 s + 2Γs + 2 s s 2 2 ζ s + 3ζs + 2 ζ 2 s + 3 s + 2Γs + 2 ζs ζs + 3 s Γs + 2 = h 2 s + h 3 s + h 4 s + h 5 s, say. Then Ne, 5 f s = 2πi h i s + 2γh s i=2 f 2 s = g 2 w, sdw C 0 = 2 C 0 w 2 + 2γ s+w+2 w = 2 C 0 w 2 + 2γ Aw, sdw, w = 2πi 2 A w w=0 +4γA0, s. s ζ s w ζs w say s + w + 2Γs + 2 dw

46 40 A0, s = s+2 ζ s + 2 s ζs + 3 s + 2 Γs + 2 = h 6s, Aw, s = s+2 ζ s w s Γs + 2 ζs w s + w + 2 A w = s+2 ζ s w s Γs + 2 ζs w s + w + 2 2ζ s w ζ s w + log 2ζ s w ζs w s + w + 2 w w say So s s A w w=0 = 22 ζ s + 2 ζs + 3 s + 2Γs log ζ s + 2 ζs + 3 s + 2Γs + 2 s s 22 ζ s + 3ζ s + 2 ζ 2 s + 3 s + 2Γs ζ s + 2 ζs + 3 s Γs + 2 = h 7 s + h 8 s + h 9 s + h 0 s, say. Ne, f 3 s = g 3 w, sdw C 0 = C 0 w 2 + 2γ w = w 2 + 2γ w C 0 = 2πi f 2 s = 2πi 2 0 i=7 log s+w+2 Bw, sdw B w w=0 +2γB0, s. h i s + 4γh 6 s. 4.3 ζs w s ζs w s + w + 2Γs + 2 dw B0, s = log s+2 ζs + 2 s ζs + 3 s + 2Γs + 2 = h s, say Bw, s = log s+2 ζs w w s Γs + 2 ζs w s + w + 2 B w = log s+2 ζs w w s Γs + 2 ζs w s + w + 2 2ζ s w ζs w + log 2ζ s w ζs w s + w + 2

47 4 So B w w=0 = 2 2 log s ζ s + 2 ζs + 3 s + 2Γs log s ζs + 2 log ζs + 3 s + 2Γs log s ζ s + 3ζs + 2 ζ 2 s + 3 s + 2Γs log s ζs + 2 ζs + 3 s Γs + 2 = h 2 s + h 3 s + h 4 s + h 5 s, say. f 3 s = 2πi 5 i=2 h i s + 2γh s In wha follows, Cw, s, Dw, s, Ew, s, Fw, s ae defined in he same way as Aw, s and Bw, s. Ne, f 4 s = g 4 w, sdw C 0 = Γ 2 w 2 + 2γ w C 0 = 2πiΓ 2 Cw, sdw C w w=0 +2γC0, s C0, s = s+2 ζs + 2 s ζs + 3 s + 2Γs + 2 = h 6s, say Cw, s = s+2 s Γs + 2 ζs w ζs w s + w + 2 C w = s+2 ζs w s Γs + 2 ζs w s + w + 2 2ζ s w ζs w + log 2ζ s w ζs w s + w + 2 w w s s C w w=0 = 2 2 ζ s + 2 ζs + 3 s + 2Γs ζs log ζs + 3 s + 2Γs + 2 s s 2 2 ζ s + 3ζs + 2 ζ 2 s + 3 s + 2Γs + 2 ζs ζs + 3 s Γs + 2 = h 7 s + h 8 s + h 9 s + h 20 s, say.

48 42 So Ne, f 4 s = 2πiΓ 2 f 5 s = g 5 w, sdw C 0 = 2 = 2πi C 0 w 2 + 2γ w 20 i=7 h i s + 2γh 6 s Dw, sdw 2 D w w=0 4γD0, s D0, s = s+2 ζs + 2ζ s + 3 s ζ 2 s + 3 s + 2Γs + 2 = h 2s Dw, s = s+2 s Γs + 2 ζs wζ s w ζ 2 s w w s + w + 2 D w = s+2 ζs wζ s w s Γs + 2 ζ 2 s w s + w + 2 2ζ s w ζs w + 2ζ s w ζ s w + log 4ζ s w ζs w s + w + 2 w s D w w=0 = 2 2 ζ s + 2ζ s + 3 ζ 2 s + 3 s + 2Γs + 2 s ζ s + 3ζs + 2 ζ 2 s + 3 s + 2Γs + 2 s + 2 log ζ s + 3ζs + 2 ζ 2 s + 3 s + 2Γs + 2 ζ 4 2 s s ζs + 2 ζ 3 s + 3 s + 2Γs + 2 s 2 ζ s + 3ζs + 2 ζ 2 s + 3 s Γs + 2 = h 22 s + h 23 s + h 24 s + h 25 s + h 26 s, say. So f 5 s = 2πi 2 26 i=22 h i s 4γh 2 s. 4.34

49 43 Ne, f 6 s = g 6 w, sdw C 0 = w 2 + 2γ w = 2πi C 0 Ew, sdw, E w w=0 2γE0, s say E0, s = s+2 ζs + 2 s ζs + 3 s Γs + 2 = h 27s Ew, s = s+2 s Γs + 2 ζs w ζs w s + w w E w = s+2 ζs w s Γs + 2 ζs w s + w ζ s w ζs w + log 2ζ s w ζs w 2 s + w + 2 w s s E w w=0 = 2 2 ζ s + 2 ζs + 3 s Γs ζs log ζs + 3 s Γs + 2 s s 2 2 ζ s + 3ζs + 2 ζ 2 s + 3 s Γs + 2 ζs ζs + 3 s Γs + 2 = h 28 s + h 29 s + h 30 s + h 3 s, say. So Ne, f 6 s = 2πi 3 i=28 f 7 s = g 7 w, sdw C 0 = w 2 + 2γ w = 2πi C 0 h i s 2γh 27 s Fw, sdw, F w w=0 2γF0, s say

50 44 F0, s = s+2 ζs + 2 Γ s + 2 s ζs + 3 s + 2Γ 2 s + 2 = h 32s, say Fw, s = s+2 Γ s + 2 ζs w s Γ 2 s + 2 ζs w s + w + 2 w So F w = s+2 Γ s + 2 ζs w s Γ 2 s + 2 ζs w s + w + 2 2ζ s w ζs w + log 2ζ s w ζs w s + w + 2 s s F w w=0 = 2 2 ζ s + 2 Γ s + 2 ζs + 3 s + 2Γ 2 s ζs + 2 Γ s log ζs + 3 s + 2Γ 2 s + 2 s s 2 2 ζ s + 3ζs + 2 Γ s + 2 ζ 2 s + 3 s + 2Γ 2 s + 2 ζs Γ s + 2 ζs + 3 s Γ 2 s + 2 w = h 33 s + h 34 s + h 35 s + h 36 s, say. Fo each i {, 2,...,36}, le f 7 s = 2πi H i = 2πi 36 i=33 σ0 +i σ 0 i h i s 2γh 32 s ζs + Γsh i sds Then by 4.28, 4.29, and each euaion fom 4.30 o 4.36, we have σ0 +i 7 M = 2πi 2 ζs + ζs f i s + log + 2γ f s ds = log σ 0 i 5 k i H i + i= 36 i= i=2 l i H i,

51 45 whee k = 2γ, k 2 = k 3 = k 4 = k 5 = l = 2γ 2γ, l 2 = l 3 = l 4 = l 5 = 2γ l 6 = 4γ, l 7 = l 8 = l 9 = l 0 = 2 l = 2γ, l 2 = l 3 = l 4 = l 5 = l 6 = 2γΓ 2, l 7 = l 8 = l 9 = l 20 = Γ 2 l 2 = 4γ, l 22 = l 23 = l 24 = l 25 = l 26 = 2 l 27 = 2γ, l 28 = l 29 = l 30 = l 3 = l 32 = 2γ, l 33 = l 34 = l 35 = l 36 =. Then we have he following inegal o evaluae H = 2πi σ0 +i σ 0 i = σ0 +i 2πi σ 0 i = 2 σ0 +i 2πi σ 0 i ζs + Γsh sds ζs + ζs + 2 ζs + 3 ζs + ζs + 2 ζs + 3 s 2 Γs s + 2Γs + 2 ds s ss + s + 2 ds by using he euaion Γs + = sγs s C. The ohe 35 ems ae s H 2 = 2 2 σ0 +i ζs + ζ s + 2 2πi σ 0 i ζs + 3 ss + s + 2 ds H 3 = 2 log s σ0 +i ζs + ζs + 2 2πi σ 0 i ζs + 3 ss + s + 2 ds s H 4 = 2 2 σ0 +i ζs + ζs + 2ζ s + 3 2πi σ 0 i ζ 2 s + 3 ss + s + 2 ds s H 5 = 2 σ0 +i ζs + ζs + 2 2πi ζs + 3 ss + s + 2 2ds σ 0 i H 6 = 2 σ0 +i 2πi σ 0 i H 7 = 2 2 σ0 +i 2πi σ 0 i ζs + ζ s + 2 ζs + 3 ζs + ζ s + 2 ζs + 3 σ0 +i H 8 = 2 log 2πi H 9 = 2 2 σ0 +i 2πi σ 0 i σ 0 i ζs + ζ s + 2 ζs + 3 s ss + s + 2 ds s ss + s + 2 ds s ζs + ζ s + 2ζ s + 3 ζ 2 s + 3 ss + s + 2 ds s ss + s + 2 ds

52 46 H 0 = 2 σ0 +i 2πi σ 0 i H = 2 log 2πi H 2 = 2 2 log H 3 = 2 log log s ζs + ζ s + 2 ζs + 3 ss + s + 2 2ds s ζs + ζs + 2 ζs + 3 ss + s + 2 ds σ0 +i σ 0 i σ0 +i 2πi 2πi H 4 = 2 2 log 2πi H 5 = 2 log 2πi H 6 = 2 σ0 +i 2πi σ 0 i H 7 = 2 2 σ0 +i 2πi σ 0 i H 8 = 2 log 2πi H 9 = 2 2 2πi σ 0 i σ0 +i ζs + ζ s + 2 ζs + 3 σ 0 i σ0 +i σ 0 i σ0 +i σ 0 i ζs + ζs + 2 ζs + 3 ζs + ζ s + 2 ζs + 3 σ0 +i σ 0 i σ0 +i σ 0 i σ0 +i H 20 = 2 2πi σ 0 i H 2 = 2 σ0 +i 2πi σ 0 i H 22 = 2 2 σ0 +i 2πi σ 0 i H 23 = 2 2 σ0 +i 2πi σ 0 i H 24 = 2 log 2πi ζs + ζs + 2 ζs + 3 s ss + s + 2 ds s ss + s + 2 ds s ζs + ζs + 2ζ s + 3 ζ 2 s + 3 ss + s + 2 ds s ζs + ζs + 2 ζs + 3 ss + s + 2 2ds s ζs + ζs + 2 ζs + 3 ss + s + 2 ds s ss + s + 2 ds s ss + s + 2 ds s ζs + ζs + 2ζ s + 3 ζ 2 s + 3 ss + s + 2 ds s ζs + ζs + 2 ζs + 3 ss + s + 2 2ds s ζs + ζs + 2ζ s + 3 ζ 2 s + 3 ss + s + 2 ds ζs + ζ s + 2ζ s + 3 ζ 2 s + 3 ζs + ζs + 2ζ s + 3 ζ 2 s + 3 σ0 +i σ 0 i σ0 +i H 25 = 4 2 2πi σ 0 i H 26 = 2 σ0 +i 2πi σ 0 i ζs + ζs + 2ζ s + 3 ζ 2 s + 3 s ss + s + 2 ds s ss + s + 2 ds s ss + s + 2 ds ζs + ζs + 2 ζ s + 3 s 2 ζ 3 s + 3 ss + s + 2 ds s ζs + ζs + 2ζ s + 3 ζ 2 s + 3 ss + s + 2 2ds

53 47 H 27 = 2 σ0 +i 2πi σ 0 i H 28 = 2 2 σ0 +i 2πi σ 0 i ζs + ζs + 2 ζs + 3 ζs + ζ s + 2 ζs + 3 s ss + s + 2 2ds s ss + s + 2 2ds H 29 = 2 log s σ0 +i ζs + ζs + 2 2πi σ 0 i ζs + 3 ss + s + 2 2ds s H 30 = 2 2 σ0 +i ζs + ζs + 2ζ s + 3 2πi σ 0 i ζ 2 s + 3 ss + s + 2 2ds s H 3 = 2 2 σ0 +i ζs + ζs + 2 2πi σ 0 i ζs + 3 ss + s + 2 3ds s H 32 = 2 σ0 +i ζs + ζs + 2 Γ s + 2 2πi σ 0 i ζs + 3 ss + s + 2Γs + 2 ds s H 33 = 2 2 σ0 +i ζs + ζ s + 2 Γ s + 2 2πi σ 0 i ζs + 3 ss + s + 2Γs + 2 ds H 34 = 2 log s σ0 +i ζs + ζs + 2 Γ s + 2 2πi σ 0 i ζs + 3 ss + s + 2Γs + 2 ds s H 35 = 2 2 σ0 +i ζs + ζs + 2ζ s + 3 Γ s + 2 2πi σ 0 i ζ 2 s + 3 ss + s + 2Γs + 2 ds s H 36 = 2 σ0 +i ζs + ζs + 2 Γ s + 2 2πi σ 0 i ζs + 3 ss + s Γs + 2 ds. Some inegands ae he same, and we have he following 5 inegals o evaluae I = s σ0 +i ζs + ζs + 2 2πi ζs + 3 ss + s + 2 ds σ 0 i I 2 = σ0 +i 2πi σ 0 i I 3 = σ0 +i 2πi σ 0 i I 4 = σ0 +i 2πi σ 0 i I 5 = σ0 +i 2πi σ 0 i I 6 = σ0 +i 2πi σ 0 i ζs + ζ s + 2 ζs + 3 s ss + s + 2 ds s ζs + ζs + 2ζ s + 3 ζ 2 s + 3 ss + s + 2 ds s ζs + ζs + 2 ζs + 3 ss + s + 2 2ds s ζs + ζ s + 2 ζs + 3 ss + s + 2 ds ζs + ζ s + 2ζ s + 3 ζ 2 s + 3 s ss + s + 2 ds

54 48 I 7 = σ0 +i 2πi σ 0 i I 8 = σ0 +i 2πi σ 0 i ζs + ζ s + 2 ζs + 3 s ζs + ζs + 2ζ s + 3 ζ 2 s + 3 ss + s + 2 2ds s I 9 = σ0 +i ζs + ζs + 2 ζ s πi σ 0 i ζ 3 s + 3 I 0 = σ0 +i 2πi σ 0 i I = σ0 +i 2πi σ 0 i I 2 = σ0 +i 2πi σ 0 i I 3 = σ0 +i 2πi σ 0 i I 4 = σ0 +i 2πi σ 0 i I 5 = σ0 +i 2πi σ 0 i ss + s + 2 ds s ss + s + 2 ds s ζs + ζs + 2ζ s + 3 ζ 2 s + 3 ss + s + 2 2ds s ζs + ζs + 2 ζs + 3 ss + s + 2 3ds s ζs + ζs + 2 Γ s + 2 ζs + 3 ss + s + 2Γs + 2 ds s ζs + ζ s + 2 Γ s + 2 ζs + 3 ss + s + 2Γs + 2 ds s ζs + ζs + 2ζ s + 3 Γ s + 2 ζ 2 s + 3 ss + s + 2Γs + 2 ds s ζs + ζs + 2 Γ s + 2 ζs + 3 ss + s Γs + 2 ds The calculaion of he inegals The calculaion of he bound Le 0 < Θ <. Le γ be he conou consising of γ, γ 2, γ 3, γ 4 whee γ is a veical line fom σ 0 it o σ 0 +it, γ 2 is a hoizonal line fom σ 0 +it o Θ+iT, γ 3 is veical line fom Θ+iT o Θ it, γ 4 is hoizonal line fom Θ it o σ 0 it, as in picue

55 49 Fig. 4.. The conou γ of inegaion In his pa, we le µ be he funcion defined in Lemma 2.4, and le i j, s o be he inegand of I j. Conside he inegand i, s of I on γ 2. Since µ is deceasing, µσ is lage when σ is small. So ζs + T µ Θ++δ = T 2 +Θ+δ, ζs + 2 T µ Θ+2+δ T Θ 2 +δ Res + 3 Θ + 3 = 2 Θ, so ζs + 3 s σ σ0 =, s, s +, s + 2 T, so s s + s + 2 T 3. Theefoe he inegand is T 2 +Θ+δ+Θ 2 +δ 3 = T Θ le δ = Θ 4 So he inegal 4T 5 2 +Θ T 0 Because ζs = ζs, we can compue he bound on γ 4 like hose on γ 2, and we will ge he inegand on γ 4 Theefoe I = T Θ+i 0. Θ i i, sds + Res s=0 i, s + Res s= i, s. Fo I 2, we can use he same bound of ζs + 2 fo ζ s + 2, by Lemma 2.4.

56 50 Eveyhing else is he same o he one in I. So we have I 2 = Θ+i Θ i i 2, sds + Res s=0 i 2, s + Res s= i 2, s. Fo I 3, we have Rs + 3 Θ + 3 = 2 Θ, so ζ s + 3, ζ 2 s + 3, and eveyhing else is he same as he one in I 2. So I 3 = Θ+i Θ i i 3, sds + Res s=0 i 3, s + Reg s= i 3, s. The calculaion of he bound fo I i, 4 i ae all he same as he one in I o I 2. Fo I i 2 i 5 we apply Lemma 2.5 o bound Γ Γ. On γ 2, when T is lage, Γ s + 2 Γ log s + 2 log T T δ fo any small value δ > 0. Eveyhing else is he same. So γ 2 i 2, sds on γ 2 conveges o 0 as T. We can also apply Lemma 2.5 fo I 2 on γ 4. Hence I 2 = Θ+i Θ i i 2, sds + Res s=0 i 2, s + Reg s= i 2, s The calculaion of he bound fo I i, 3 i 5 ae he same as he one in I 2. The calculaion of Fo I, we conside Θ+i Θ i i j, sds. Θ+i 2πi Θ i ζs + ζs + 2 ζs + 3 s ss + s + 2 ds. Θ i ζ Θ + iζ Θ + i = 2π ζ2 Θ + i Θ + i Θ + i Θ + i d To see ha he inegal conveges, ζ Θ + i 2 +Θ+δ, ζ Θ + i Θ 2 +δ, i ζ2 Θ + i, =, Θ + i, Θ + i, Θ + i. So he inegand Θ choose δ = Θ 4.

57 2π = ζ Θ + iζ Θ + i ζ2 Θ + i + + i Θ + i Θ + i Θ + i d Θ d which conveges. Similaly, Then Θ+i 2πi Θ i Θ d ζs + ζs + 2 ζs + 3 which conveges. s Theefoe I = Res s=0 + Res s= + O Similaly, fo each i {, 2,...,5}, ss + s + 2 ds = O Θ. I j = Res s=0 i j, s + Res s= i j, s + O Θ. Θ. The calculaion of esidue Now, we will compue he esidue of I i a s = 0 and s =. Fo I i i 5, we show he pocess of calculaion and compue he coefficien eplicily. Fo I i 6 i 5, we show only he esul because he pocess ae simila o hose of I i i 5. We begin wih, I = σ0 +i 2πi σ 0 i ζs + ζs + 2 ζs + 3 s ss + s + 2 ds

58 52 Res a s = 0 : Res a s = : Res s= ζs + = s s 2 + γ s + γ So s ζs + ζs + 2 s ζs + 3s + s + 2 Res s=0 s = d ζs + 2 ds ζs + 3s + s + 2 s=0 + γζ2 2ζ3 = ζ2 ζ ζ 2 2ζ3 ζ ζ 3 + log + γζ2 2 2ζ3 ζs + 2 s + ζs + 2 s + = s γ So s + s ζs + ζs + 3ss + 2 s = d ζs + γζ0 ds ζs + 3ss + 2 s= ζ2 ζ0 = ζ2 ζ ζ ζ 0 ζ 2 + log + γζ0 ζ2 Then Θ I = a + b log + c + d log + O whee a = ζ2 ζ 2ζ3 ζ ζ c = ζ0 ζ2 ζ 2 ζ ζ γ 0 ζ ζ 2 + γ, b = ζ2 2ζ3,, d = ζ0 ζ2. Ne, I 2 = s ζs + ζ s + 2 2πi ζs + 3 ss + s + 2 ds. ζs + Res a s = 0 = s s 2 + γ So s

59 53 Res a s = : Res s= Res s=0 ζ s + 2 s + ζs + s s ζ s + 2 ζs + 3s + s + 2 s = d ζ s + 2 ds ζs + 3s + s + 2 s=0 + γζ 2 2ζ3 = ζ 2 ζ ζ 2ζ3 ζ 2 ζ 3 + log + γζ 2 2 2ζ3 ζ s + 2 s + = s γ So s + s s ζs + = d 2 ζs + γ ζ0 ζs + 3ss ds 2 ζs + 3ss + 2 s= ζ2 s ζs + ζ ζ Le Us =, V s = s + ζs + 3ss + 2 ζ ζ s log s s + 2. s d 2 ζs + Then ds 2 ζs + 3ss + 2 = UsV 2 s + UsV s, and hen s d 2 ζs + ds 2 ζs + 3ss + 2 s= = U V 2 + U V = ζ0 ζ2 a + log 2 ζ0 ζ2 whee a = ζ ζ ζ ζ 2 0 2, b = ζ ζ ζ 0 ζ 0 ζ ζ 2 ζ 2 + ζ Theefoe I 2 = a 2 +b 2 log +c 2 +d 2 log +e 2 whee a 2 = ζ 2 ζ ζ 2ζ3 ζ 2 ζ γ, b 2 = ζ 2 2ζ3, c 2 = ζ0 2ζ2 a2 + b 2γ, d 2 = ζ0 ζ2 a, e 2 = ζ0 2ζ2. log 2 +O b, Θ