2015 M.Tip Easter Phaovibul

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1 205 M.Tip Easter Phaovibul

2 EXTENSIONS OF THE SELBERG-DELANGE METHOD BY M.TIP EASTER PHAOVIBUL DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Dotor of Philosophy in Mathematis in the Graduate College of the University of Illinois at Urbana-Champaign, 205 Urbana, Illinois Dotoral Committee: Professor Kevin B. Ford, Chair Professor Brue C. Berndt, Co-Diretor of Researh Professor Alexandru Zaharesu, Co-Diretor of Researh Professor Adolf J. Hildebrand

3 Abstrat This dissertation involves two topis in analyti number theory. The first topi fouses on extensions of the Selberg-Delange Method, whih are disussed in Chapters 2 and 3. The other topi, whih is disussed in Chapter 4, is a new identity for Multiple Zeta Values. The Selberg-Delange method is a method that is widely use to determined the asymptoti behavior of the sum of arithmeti funtions whose orresponding Dirihlet s series an be written in the term of the Riemann zeta funtion, ζs. In Chapter 2, we first provide a history and reent developments of the Selberg-Delange method. Then, we provide a generalized version of the Selberg-Delange method whih an be applied to a larger lass of arithmeti funtions. We devote Chapter 3 to the proofs of the results stated in Chapter 2. In 96, Matsuoka evaluated ζ2 by means of evaluating the integral π/2 0 x 2 os 2n xdx. The last hapter of this dissertation generalize the idea of Matsuoka and obtains a new identity for Multiple Zeta Values. ii

4 Aknowledgments Render therefore to all their dues:... honour to whom honour. Romans 3:7, King James Version First and for most, I would like to give thanks to God the Father, the Son, and the Holy Spirit for giving me strength, wisdom, and guidane through this proess of written this dissertation and throughout my life. This For sine the reation of the world God s invisible qualities his eternal power and divine nature have been learly seen, being understood from what has been made, so that people are without exuse. Romans :20 NIV I want to thank my advisors, Brue Berndt and Alexandru Zaharesu for their patiene and guidane on my researh and oversee the written proess of this dissertation. I am very graeful for Kevin Ford to served as a hair of my Dotoral Committee, for spending time reading this dissertation very arefully, and provided many valuable omments. And I also want to give thanks to Professor A.J. Hildebrand for his willingness to serve on the ommittee and his input on writing and L A TEX formatting. I also want to express my gratitude to Youness Lamzouri for introdued me to the Selberg- Delange Method, Paul Pollak for the valuable input and suggestions for Chapter 3, and Kenneth Stolarsky for post the problem on the alulation of ζ4 whih lead to the result in Chapter 4. Next, I want to thank University of Illinois Math department, faulty, staffs, and olleague for provide a great atmosphere. I also want to speially thanks Bob Murphy, Jennifer MNeilly, Kathleen Smith, and April iii

5 Hoffmeister for let me invade their offie when I need to get away from my offie and for many wonderful onversations, enouragements, and advies. I want dediate Chapter 4 espeially for my friend, brother, and mentor, Danny Cash and his family Pearl, Isaiah, and Noel, whose I first learn about ζ2 and led me to this wonderful world of Number Theory. iv

6 Table of Contents Chapter Introdution Chapter 2 Selberg-Delange Method Origin of the Method The Development of the Method Modifiations Modifiation on the Dirihlet Series Modifiation on the Representation of F g s Modifiation on the Analyti Region Statement of the Results Chapter 3 Modifiation of the Selberg-Delange Method Preliminary Results for Theorem Proof of Theorem Preliminary Results for Theorem Analyti Continuation of Us The Distribution of Values of Us Proof of Theorem Chapter 4 Partial Multiple Zeta Values Identities History of Speial Values of the Riemann Zeta Funtion Multiple Zeta Funtions Main Result Referenes v

7 Chapter Introdution In this dissertation, we onsider the study of several topis from the field of analyti number theory. Number theory is a field devoted to the study of the integers, espeially prime numbers, and objets made out of integers, suh as rational numbers and algebrai integers. The term analyti refers to the use of analyti tools, suh as tools in real analysis, omplex analysis, and, in reent years, harmoni analysis. One may wonder how tools in the ontinuous world help the study of objets in the disrete world. The first onnetion between these two worlds dates bak to the 8th entury in the time of the great mathematiian Leonhard Euler. In 737, Euler ommuniated a paper entitled Variae observationes ira series infinitas [6]. In the paper, he observed that there is a onnetion between an infinite produt whih runs over the set of prime numbers and an infinite sum that runs over the set of all natural numbers, whih we quote. Theorem.0. Theorem 8. The expression formed from the sequene of prime numbers 2 n 3 n 5 n 7 n n 2 n 3 n 5 n 7 n n has the same value as the sum of the series + 2 n + 3 n + 4 n + 5 n + 6 n + 7 n In this theorem, Euler was only onerned when n is a natural number. Approximately a entury later, the German mathematiian, Bernhard Riemann, extended the study of this onnetion by replaing n in.0. with a omplex value s. The sum.0. is now know as the famous Riemann Zeta Funtion, ζs = n s Rs >.

8 Riemann was able to establish several analyti properties of ζs. Most importantly, he showed that ζs an be extended as a meromorphi funtion to the entire omplex plane with a simple pole at s = by using the funtional equation ζs = 2 s π s sin πs Γ sζ s. 2 This analyti ontinuation enabled Frenh mathematiian, Jaques Hadamard, and Belgian mathematiian, Chales Jean de la Vallée Poussin, to independently omplete the proof of the Prime Number Theorem, whih an be stated as follows. Theorem.0.2 Prime Number Theorem. Let πx denote the number of prime numbers not exeeding x. Then the funtion πx is asymptoti to x log x as x tends to infinity. In other words the funtion πx behaves similarly to the funtion x log x when x is suffiiently large. Many mathematiians onsiders the proof of the Prime Number Theorem as the birth of the analyti number theory. Even after the Prime Number Theorem was proved, many mathematiians ontinued to study analyti properties of ζs. The study brought light and elegane to many problems in the field, in partiular the problems on the asymptoti behavior of arithmeti funtions and the sum of arithmeti funtions. One of the tools used in the study of the asymptoti behavior of arithmeti funtions is a method alled the Selberg-Delange method, whih is the first topi we will study in this dissertation. Before the disovery of the Selberg-Delange method, the main tool in the study of the asymptoti behavior of the summatory funtion of arithmeti funtions was Perron s Formula, whih an also be viewed as a speial ase of an inverse Mellin s transform. fn Theorem.0.3 Perron s Formula. Let fn be an arithmeti funtion, and let F s = n s be the orresponding Dirihlet series. Assume that the Dirihlet series F s is absolutely onvergent 2

9 for Rs > σ a. Then Ax := n<x fn + 2 fx = 2πi +i i F s xs s ds for > σ a and x > 0. By using Perron s Formula and analyti properties of ζs, espeially the zero-free region of ζs, we are able to estimate the summatory funtion of arithmeti funtions whih an be simply expressed in term of ζs. For example, by using Perron s Formula on the funtions ζs, and ζs2, we an obtain estimations Mx : = µn = O xe log x,.0.2 n x Dx : = n x dn = x log x + 2γ x + O x,.0.3 where, if n =, µn = k, if n = p p 2 p k, 0, otherwise, where p,..., p k are distint prime numbers. The funtion dn is the number of divisors of n, γ is Euler s onstant, and the notation gx = Ofx means that there exists a positive onstant M suh that gx M fx for all x suffiiently large. One limitation to the above method is that all the singularities of the orresponding Dirihlet series must be at most poles. For example, the method fails to give an asymptoti of the summatory funtion for the orresponding Dirihlet Series F s = ζs. The Selberg-Delange Method, essentially developed by Atle Selberg [52] and Hubert Delange [9] [], is an extension of the method that is used to prove.0.2 and.0.3. The most important part of the Selberg-Delange method is that the method enables one to work with a Dirihlet series whose singularities are not poles. In partiular, if the orresponding Dirihlet series admits a 3

10 representation of the type F s = Hsζs z for Rs >, for a ertain omplex number z, and for an analyti funtion Hs that satisfies a ertain rate of growth. In reent years, several mathematiians suh as Naimi and Smida [40], Lau and Wu [36], and Ben Saïd and Niolas [2], extended the methods in several diretions and in various settings. In the work of Lau and Wu, they adapted the method to give an estimate of summatory funtion sof the form gn<x fn, where gn is a positive real-valued multipliative funtion under ertain onditions, and the orresponding Dirihlet series admits a representation of the type F g s = H g sζs κ α s for s and some fixed parameters κ and α. In Chapter 2 and Chapter 3 of this dissertation, we extend the work of Naimi and Smida, Lau and Wu, and Ben Saïd and Niolas to a lass of summatory funtions that have a orresponding Dirihlet series representation of the type F g s = H g s p P χp hps p s for Rs >, a Dirihlet harater χ, a set of prime numbers P under ertain onditions, and analyti funtions H g s and h p s under ertain onditions of growth rate. In Chapter 2, we first provide a more detailed history and the most reent developments of the Selberg-Delange method. After that, we disuss in detail how we will generalize the method further. Lastly, we lose Chapter 2 with statements of the two generalizations of the method. Chapter 3 is 4

11 devoted entirely to the proof of these two main theorems. Due to omplexity of the two theorems, we refrain from stating the full statements of these theorems at the present time. In the same spirit as Chapter 2 and Chapter 3, the seond topi of this dissertation is the asymptoti behavior of arithmeti funtions in residue lasses. A general question in this area is as follows: Given a positive integer N and integral-valued arithmeti funtion fn, how often does fn a mod N for some integer a? Many mathematiians fous on a more speifi question in the area, namely: What are the neessary and suffiient onditions suh that fn will fall into every residue lass modulo N equally often, or in more tehnial terms, fn is uniformly distributed modulo N? In 96, Uhiyama gave suh a riterion. Theorem.0.4 Uhiyama. The sequene of the integral-valued arithmeti funtion {fn} is uniformly distributed modulo N if and only if for r =, 2,..., N, lim x x exp2πifnr/n = 0. n x However, Uhimaya s riterion is somewhat diffiult to apply in pratie. In 969, Delange gave a simpler riterions when fn is an integral-valued additive funtion, suh as ωn, the number of distint prime divisors of n [0]. This an be stated as follows. Theorem.0.5 Delange 969. Let f be an integral-valued additive funtion, and let N be an integer greater than. The sequene {fn} is uniformly distributed modulo N if and only if it satisfies one of the following onditions: d fp p diverges for every divisor d > of N. 2 2f2r d is an odd integer for every divisor d > of N and every r. At the end of Chapter 4, we extended the result of Delange to integral-valued additive funtions with argument in arithmeti progression. Theorem.0.6. For every m, q, a N suh that m, ϕq =, and 0 < k < m, let fn be an 5

12 integral-valued additive funtion suh that kfp is not a multiple of m for all primes p. Then #{n x : n a mod q, fn k mod m} = x qm + ox. Equivalently if kfp is not a multiple of m for all prime p, then the funtion fqn+a is uniformly distributed modulo m. Many arithmeti funtions may not be uniformly distributed in all residue lasses, but uniformly distributed in the residue lasses that are relatively prime to N. This phenomenon is known as weakly uniform distribution modulo N. Many important number theoretial funtions are weakly uniformly distributed modulo N, for ertain values of N. For example, J.P-Serre [53] gave neessary and suffiient onditions on N suh that the sequene of Ramanujan s τ-funtion, τn is weakly uniformly distributed modulo N. Another example is the following theorem due to Delange. Theorem.0.7. The sequene {dn}, where dn is the number of divisors of n, is weakly uniformly distributed modulo N if and only if the least prime not dividing N is a primitive root modulo N. Another important multipliative funtion is the Euler-totient funtion ϕn, the number of positive integers less than n that are relatively prime to n, whih we will be the main fous of our study in Chapter 4. Theorem.0.8 Narkiewiz. The sequene {ϕn} n is weakly uniformly distributed modulo N if and only if N is relatively prime to 6. However, the method employed by Delange for dn and Narkiewiz for ϕn does not give an asymptoti of the number n suh that dn respetively ϕn a mod N. Moreover, how are the values of ϕn distributed if N is not relatively prime to 6? In Chapter 4, we mainly study the above questions. First, we onsider the ase when N is a power of 2. Sine ϕn is always even exept when n =, 2, we an disregard all the odd residue lasses. We onduted a numerial experiment on how ϕn is distributed modulo 2 k. We find that there is a strong orrelation between the number of ϕn ongruent to a modulo 2 k and the highest power of 2 dividing a. This data led us to the proof of the following theorem. 6

13 Theorem.0.9. Let k and r be positive integers and let b 2 r a mod 2 k, where a is odd. Then #{n : n < x, ϕn b mod 2 k } 3 2 k xlog log x r r! log x as x tends to infinity. The third and final topi in this dissertation are identities for multiple zeta values. The multiple zeta funtion is a generalization of ζs, whih an be defined as follows. Definition.0.0. Let s, s 2,..., s k be omplex values suh that Rs + Rs Rs m > m for all m k. We define the multiple zeta funtion by ζs, s 2,..., s k = k n >n 2 > >n k >0 i= n s i..0.4 Similar to ζs, multiple zeta funtions have analyti ontinuations to C k with possible simple poles at s k = and s j + + s k = k j + 2 l for positive integers l and j < k [67]. If s, s 2,..., s k are all positive integers greater than, then.0.4 are alled multiple zeta values. The study of relation between multiple zeta values dates bak to the the time of Euler [8, pp ]. Theorem.0.. If a, b >, then ζa, b + ζb, a = ζaζb ζa + b. In partiular, ζa, a = 2 ζ 2 a ζ2a. Multiple zeta values also satisfy many other interesting relations. One example is the relation ζ2, = ζ3, whih an be generalized in the following theorem. Theorem.0.2 Sum Theorem. Let n and k be natural numbers suh that n > k. Then, for n,..., n k N, ζn = n + +n k =n n > ζn, n 2,..., n k. 7

14 This theorem was proved for the ase k = 2 by Euler, for k = 3 by Hoffman and Moen [27], and for the general ase by Granville [22]. In Chapter 5, we derive a new identity for multiple zeta values using a similar idea to that used to evaluate ζ2 by Matsuoka [39]. In partiular, we prove the following theorem. Theorem.0.3. For any positive integer m, we have m,0 + m l m,l l= i= r + +r i =l ζ2r,..., 2r i = 0. where m,l = l π2m l 2 2m 2m! 2m l +!. 8

15 Chapter 2 Selberg-Delange Method 2. Origin of the Method At the turn of the 9th entury, when Jaques Hadamard and Charles Jean de la Vallée-Poussin independently proved the Prime Number Theorem, Edmund Landau, a German mathematiian, published an influential book Handbuh der Lehre von der Verteilung der Primzahlen [35], also know as Landau s Primzaheln, for short. In Primzahlen, Landau disussed the tehniques that Hadamard and Vallée-Poussin used in their appliations to the Prime Number Theorem. One of the appliations onerns the behavior of the ardinality of the set of the natural numbers whih have exatly v distint prime fators, whih an be stated as follows [35, p. 2]. Theorem 2... Landau Let ρ k x be the number of integers x that are divisible by exatly k distint primes, eah ourring in any multipliity. Then ρ k x k! xlog log x k. log x In this theorem, Landau established the asymptoti of ρ k x for a fixed value of k. But, what will happen if the value of k is growing as a funtion of x? A similar question also appeared in Ramanujan s Lost Notebook, whih we quote. [45, p. 337]. Entry. ϕx is the number of numbers not exeeding x whose number of prime divisors doesn t exeed k. ϕx x log log x log log x log x!! log log x[k]. [k]! This is true when k is infinite. Is this true when k is a funtion of x? This question was first answered by L.G. Sathe in the series of papers [48], [49], [50], [5]. Sathe s 9

16 proof was rather ompliated and very involved. Later in the same year, Atle Selberg [52] gave a muh simpler proof of the theorem. Theorem If k log log x and < 2, then ρ k x f p + x p + p log x log log x k, k! where f = Γ + p e /p + k e k/p. p p p Even though Selberg s goal was to give an alternative approah to estimating ρ k x, his method also applies to more general arithmeti funtions, suh as the sum of divisors funtion σn. The main idea of Selberg was to reate a Dirihlet series assoiated with the relevant arithmetial funtion and study the behavior of the series around the pole s =. This idea was later extended by Hubert Delange, [9] []. This method is now know as the Selberg-Delange Method, whih is the main fous of our study. 2.2 The Development of the Method Throughout this hapter, we will adopt the following notation. Let s = σ + it be a omplex number with real part σ and imaginary part t. Let fn be a omplex-valued funtion, not neessary multipliative, whih we want to study, and f + n be a positive real-valued funtion. Finally denote a Dirihlet series orresponding to fn as F s = fn n s. The ore of Selberg s method relies on the following theorem, whih we paraphrase in order to fit our future definitions. Theorem Let F s = Hs, zζs z, 0

17 where and let Hs, z = b z n b z n n s σ >, log 2nB+δ n 2.2. be uniformly bounded for z B. Then Ax := n x fn = H, z Γz x log x z + O x log x 2 z x, uniformly for z B, as x tend to infinity. In addition, Selberg proved the following theorem. Theorem Under the assumptions of Theorem 2.2., let a z n = k nz k, z A, be an arithmeti funtion depending on a parameter z. Moreover, if the seond derivative of H,z Γ+z is uniformly bounded for z A, we have C k x := n x k n = H Γ, k log log x + k log log x xlog log xk k! log x + O x log x k 2 log log x k 3 k!, uniformly for k < 2 δ log log x. By letting fn = z ωn, where ωn is the number of distint prime fators of n, and applying Theorem 2.2. and Theorem 2.2.2, Selberg obtained Theorem Early appliations of Selberg s theorems were used mostly for lasses of funtions fn = z αn, where αn is an additive funtion. In 97 Delange [] extended the result to the lass of funtions fn = χnz αn, where χn is a Dirihlet harater. Before disussing the next development, we denote D := { } s : σ, log3 + t

18 the regionon whih we will fous. Due to omplexity and various parameters, Gerald Tenenbaum [57] has formulated the following terminology in order to apply Theorem 2.2. more effetively. Definition Let z C, 0 > 0, 0 δ <, and M > 0. We say that a Dirihlet series F s has property Pz; 0, δ, M if the following onditions hold. F s admits a representation of the type F s = Hs, zζs z for σ. 2 The funtion Hs, z in equation is a omplex-valued analyti funtion on the region D 0, and satisfiies the inequality Hs, z M3 + t δ Definition If the Dirihlet series F s has property Pz; 0, δ, M, then we say F s has property P + z, w; 0, δ, M if there exists a positive real-valued funtion f + n, suh that fn f + n for all n N and F + f + n s = n s has property Pw; 0, δ, M. Tenenbaum replaed the assumption on the onvergene ondition of the funtion 2.2. and it s derivatives by the analyti ontinuation of Hs, z. The new theorem an be stated as follows. Theorem Let F s be a Dirihlet series that has property P + z, w; 0, δ, M. For x 3, N 0, A > 0, and z, w A, we have Ax := n x fn = N x λ k z log x z log x k + O 0,δ,A MR N x, k=0 with where λ k := Γz k l+j=k l!j! γ jz dl H; z, dsl γ j z = dj ds j s s ζs z 2

19 and R N x = e log x 2 N + + log x N+ for some positive onstants and 2. In 996, M. Naimi and H. Smida [40] were able to replae the onstant z in equation by an analyti omplex-valued funtion satisfying a ertain rate of growth. We an state their theorem as follows. Theorem Let F s = assume that for some 0 < α < fn n s be a Dirihlet series that has property Phs; 0, δ, M, and hs M log3 + t α t R and log3+ t σ 2. Then there exists a polynomial P k x with degree at most k suh that, uniformly for N and x 3, with fn = n x N x P k log log x log x h log k + O R N x k=0 R N = Me log x log log x + M 2 N + 2N log x N+ for some onstants and 2 whih depend only on, α, δ and M. Several years later Yuk-kam Lua and Jie Wu [36] gave another variation of Theorem Their main purpose was to obtain the asymptoti behavior of the general sum fn gn<x where both fn and gn are multipliative funtions. Their main theorem an be stated as follows. Theorem Let F g s = fn gn. Suppose that f : N C and g : N R + are two s multipliative funtions suh that for all primes p: 3

20 fp κ < p η, for η > 0, C 0 and κ < η, 2 gp = αp or gp = αp + α p + tp, where t l u C 2 l + l u l for > 0, 3 C 0, α > 0, α 0 and > >, ν=2 fp ν gp ν / C 3 p ψ, where C 3 > 0, ψ >, and >. Then for any positive integer N, we have A g x = fn = gn<x x log x κ/α/ N k= P k log log x log x k + O R N,λ x, where P k x = k λ k,l x l and the oeffiients λ k,l are given by l= λ k,l := κ/α l! k m m l m=l n=l i=0 λ m,n,i, where λ m,n,i := log αm κ/α / n log n l i a k,l b n,m n n l i!i!γ i κ/α / k where a n is the n-th oeffiient of the Laurent series expansion of the funtion s F g ss κ/αs at s =, and where b m,n := n +n 2 + +n m=n n +! n m +!. The error term R N,λ x is given by R N,λ x := N + λ log log x + N+ 2 + e 3 log x log x3/5 log log x /5, with 0 < λ < and for some onstants, 2, and 3. In the same year, Fethi Ben Saïd and Jean-Louis Niolas [2] introdued the use of a Dirihlet harater to obtain an asymptoti of ertain arithmeti funtions, with a restrition on the primes 4

21 dividing n are in ertain sets of arithmeti progressions. Their theorem an be stated as follows. Theorem Let b be a positive integer, ξ be a Dirihlet harater modulo b and J Z/kZ. Let gn be a multipliative funtion suh that gn > 0 and gn. Let a J,ξ n b J,ξ n, respetively be a omplex-valued multipliative funtion respetively, real-valued. Suppose that for all n, a J,ξ n b J,ξ n and the series F g,j,ξ s = a J,ξ n gn s and F + g,j,ξ s = b J,ξ n gn s are analyti on the half plane σ >. Also suppose that there exist three real onstants B > 0, 0 < < 2, 0 δ < and funtions f js j J and f + s, analyti on the domain D suh that max{ f j s, f + s} Blog3 + t δ for j J and s D. Also suppose that in a half-plane σ > the series F g,j,ξ s admits a representation of the type F g,j,ξ s = H g,j,ξ s j J p j mod k ξp fj s p s, where H g,j,ξ s is analyti on D, and satisfies the inequality H g,j,ξ s B3 + t δ. Similarly, in a half plane σ > the series F + g,j,ξ s admits a representation of the type F + g,j,ξ s = H+ g,j,ξ sζsf + s, 5

22 where H + g,j,ξ s is analyti on D and satisfies the inequality H + g,j,ξ s B3 + t δ. Let fs = ϕk f j s. j J Then for a non-prinipal harater ξ, A g,j,ξ x := gn<x log log x a J,ξ n = O x log x 2, and for a prinipal harater ξ 0 A g,j,ξ0 x = x Hg,J,ξ0 C J,k log x f + O Γf log log x log x, where the onstant C J,k is defined by C J,k = j J p j mod k p fj p fj /ϕk fj. p p p b,p j mod k The proof of Theorem relies on another result of Naimi and Smida whih an be stated as follows. Theorem Theorem A, [2]. Let F g s = hs and h + s are analyti on D and satisfy the inequality fn gn s have property P+ h, h +,, δ, M, where Also, let A g x = n,gn x max{ hs, h + s } log3 + t δ. fn. Then A g x = xlog x h Hh Γh + O log log x log x. 6

23 We are unable to loate the paper of Naimi and Smida, so we will give a proof of a stronger version of this theorem in the next hapter. 2.3 Modifiations In this setion, we will ombine the ideas of Lau and Wu [36], Naimi and Smida [40], and Ben Saïd and Niolas [2] with some of our modifiations to obtain a more general version of Selberg s Theorem. We will use Definition 2.2.3, Definition 2.2.4, and Theorem of Tenenbaum as the base of our modifiations. As usual, we let s = σ + it Modifiation on the Dirihlet Series The first modifiation we make is on the summatory funtion Ax = n<x fn. We define an analogue, A g x := fn n gn<x for some funtion gn. Following Selberg s idea, we need to reate an assoiated Dirihlet series for A g x, namely, F g s := fn gn s. It is evident that one must put onditions on F g s and gn. In Selberg s proof, there is a part where he applies Perron s formula to the orresponding Dirhlet series. Thus, we need to find onditions suh that we are able to apply analogues of Perron s formula. For this reason we need F g s and gn to satisfy the following onditions. The Dirihlet series F g s have a finite absissa of onvergene. 2 The funtion gn is a real-valued multipliative funtion suh that g : N [,, and gn 2.3. as n tends to infinity. 7

24 3 The limit superior of the ratio lim sup n log n log gn exists and is non-negative. In the paper of Ben Saïd and Niolas [2], they only assume ondition 2. We believe that this ondition is not suffiient to arry out the proof without assuming that the Dirihlet series is absolutely onvergent at some real number σ > Modifiation on the Representation of F g s The next modifiation we make is to the equation We divide these modifiations into two stages. Later on, we will state a result orresponding to eah stage. For the urrent disussion, we assume that all funtions have an analyti ontinuation and do not vanish on some region D. We will disuss this region in more detail in the next subsetion. First Modifiation In the first stage, similar to Lau and Wu [36], we introdue a parameter > 0. We replae ζs in equation with ζs. Next, similar to Naimi and Smida [40], we replae a omplex onstant z by an analyti funtion hs suh that hs Mlog3 + t α for some positive onstant M and 0 α < in the region D. With these modifiations, the analogue of equation an be written as F g s = H g s, h; ζs hs where H g s, h; has an analyti ontinuation to region D and satisfies the inequality H g s, h; M3 + t δ for some 0 δ < and M > 0 in the region D. 8

25 Seond Modifiation In the seond modifiation, we will modify the equation We will first adopt the idea of Ben Saïd and Niolas [2] and generalize it further. In the paper of Ben Saïd and Niolas, they replaed ζs z in the equation by j J p j mod k ξp p s h j s, where J Z/kZ, χ is a Dirihlet harater modulo q, and h j s are analyti funtions satisfying a ertain rate of growth. In the light of Ben Saïd and Niolas, we wish to generalize to a produt on a ertain set of primes P, more preisely p P χp hps p s. But some sets of prime numbers or some hoies of h p s do not possess ertain analyti properties that we need. Thus, we need to put onditions on P and h p s. For the onditions on P, first, let χ be a Dirihlet harater modulo q and let q be a multiple of q. Let πa, q, x denote the number of primes less than x and ongruent to a modulo q. Next, let λ : Z/ qz [0, ]. For all η > log 3 and for all a Z/ qz, # {p : p < x, p P, p a mod q} = λaπa, q, x + Ox η We will see later that the error term x η in is the best possible. Next, for the ondition h p s, we first introdue a funtion hs analyti in the region D,. Also for all s D, max{ h p s, hs } Mlog3 + t α for 0 α < and uniformly for all p P. Next, we define a region K T := K,,T = {s : s D,, σ 2, t T }. 9

26 For all η > log 3, h p h,k x η, p P p<x where f,k = sup s K T fs. In other words, on average the funtion h p s should behave similarly to λpχphs on ompat set K T for all T Modifiation on the Analyti Region The next modifiation is on the analyti region. The first neessary ondition is that ζs needs to have an analyti ontinuation to the region with an exeption of a simple pole at s =. We also need ζs to not vanish in the region. Moreover, in order to ompromise with equation 2.3.4, we also need Ls, ξ, where ξ is a Dirihlet harater modulo q, to have an analyti ontinuation to the region with an exeption of a simple pole at s =, and not vanish in the region. The analytiity of ζs and Ls, ξ an be showed by their funtional equations whih were proved by Riemann [46] and Hurwitz [29, pp 72-88], respetively. All that remains is to onsider the zero-free regions of ζs and Ls, ξ. The first result along this line was first proved by de la Vellée Poussin [7]. In 899, de la Vellée Poussin showed that there exists a onstant > 0 suh that the Riemann Zeta funtion ζs does not vanish in the region σ > log t for suffiiently large t. This was improved by several people. First, Littlewood [37] showed that there exists a onstant 2 suh that ζs dose not vanish in the region σ > 2 log log t. log t 20

27 Later Chudakov [4] extended the region to 3 σ > log t 3 4 +ɛ for some onstant 3 > 0. The most reent result was given by Korobov [3] and Vinogradov [6], who independently showed that there exists a onstant 4 > 0 suh that ζs does not vanish in the region 4 σ > log t 2 3 log log t 3 for suffiiently large t. These results also hold for Ls, χ for a fixed χ. For our purpose, we will only use the lassial zero-free region of de la Vallée Poussin. For > 0, we define a region D, to be the lassial zero-free region on of ζs, more preisely, D, := { s : σ > } logmax{3, t } One may ask, how large is the onstant in the zero-free region of de la Vallée Poussin? de la Vallée Poussin showed that one an take = This result was reently improved by Kaidiri [30] to = For our purposes, we will assume that < 2. One of the onsequenes of working in the larger analyti and zero-free region is the improvement on the error terms in the asymptoti formula for A g x. We define Errx to be an inreasing funtion suh that πx Lix MxErrx. The funtion Errx depends on the zero-free region of ζs; in our ase Errx = e 5 log x for some onstant 5 > 0. Remark: If one uses the region of Korobov and Vinogradov, then we an effet an improvement, log x 3/5 Errx = exp log log x /5. 2

28 Moreover, under the assumption of the Riemann Hypothesis, von Koh [62] showed that Errx = x log x. 2.4 Statement of the Results First, we will introdue a notation. For z 0 C and any positive integer k, we define dk := Γ k z 0 dz k = Γz k! z=z 0 2πi γ dz Γzz z 0 k+ We are now introdue definitions analogue to Definition and Defintion Definition Let 0 < < 2, > 0, M 0, 0 δ <, 0 α <, and κ 0, and let hs be an analyti funtion in D, where D, = { s : σ > }. logmax{3, t } fn We say that a Dirihlet series F g s = with a finite absissa of onvergene has property gn s Bh;,, M, δ, α, κ if the following onditions hold. gn is a real-valued multipliative funtion suh that g : N [,, gn tends to infinity as n tends to infinity, and lim sup n log n log gn = κ for some onstant κ 0. 2 F g s admits a representation of the type F g s = H g s, h; ζs hs for σ >. 3 The funtion H g s, h; has analyti ontinuation to the region D, and satisfies the inequal- 22

29 ity H g s, h; M3 + t δ for all s D,. 4 The funtion hs is analyti in the region D, and satisfies the inequality hs M log3 + t α for all s D,. Definition We say that a funtion F g s = fn gn has property B + h, h + ;,, M, δ, α, κ s if F g s have property Bh;,, M, δ, α, κ and there exists a positive real-valued funtion f + n, suh that fn f + n for all n N and F g + f + n s = gn s has property Bh+ ;,, M, δ, α, κ. Now, we are ready to state a stronger version of Theorem Theorem Let 0 < < 2, > 0, M 0, 0 δ <, 0 α <, and κ 0, and let hs and h + s be analyti funtions in D, where D, = { s : σ > }. logmax{3, t } Let a Dirihlet series F g s = fn gn s have property B+ h, h + ;,, M, δ, α, κ. Also, let As := s H g s, h; s ζs hs. Then, uniformly for N and x 3, we have A g x := fn = gn<x x log x h N m=0 P m log log x log x m + O R N, where P m x := m m j=0 n=j e m,n h n j i=0 n n i log n i j j+i i j Γ i h x j, m 23

30 e m,n := n n! a k,n := m A m k ak,n, m k! k=n k +k 2 + +k n=k k i n i= h k i, k i! and R N := N + N+ log x 2 + log log x + MErrx 3 where Errx = e 4 log x for some positive onstants, 2, 3 and 4 depend on, M, δ,, and α. In partiular, for N =, A g x = x A log x h log log x h Γ h + O log x By letting = in 2.4.2, we obtain Theorem Now, for the next theorem, we will introdue another definitions analogue to Definition and Defintion Definition Let 0 < < 2, > 0, M 0, 0 δ <, 0 α <, and κ 0. Let P be a set of prime numbers, χ be a Dirihlet harater modulo q, q be a positive integer divisible by q and define a funtion λ : Z/ qz [0, ]. And lastly, let hs and for all primes p P let h p s a omplex-valued funtions analyti in the region D, where D, = We say that a Dirihlet series F g s = the following onditions hold. { s : σ > }. logmax{3, t } fn gn s has property Ah p, h; P, χ, λ, q,,, M, δ, α, κ if The funtion gn is a multipliative funtion suh that g : N [,, gn tends to infinity as n tends to infinity, and lim sup n log n log gn = κ 24

31 for some onstant κ 0. 2 F g s admits a representation of the type F g s = H g s p P χp hps p s for σ >. 3 For all η > log 3 and for eah a Z/ qz, P a, q x := # {p : p < x, p P, p a mod q} = λ q aπa, q, x + Ox η where πa, q, x = # {p : p < x, p a mod q}. 4 The funtion H g s has an analyti ontinuation to the region D, and satisfies the inequality H g s M3 + t δ for all s D,. 5 The funtion h p s and hs are analyti in the region D, and satisfies the inequality max{ hs, h p s } M log3 + t α for all s D,, for all p P. 6 For all suffiiently large T, and for all η > log 3, h p h,kt Cx η, p P p<x where onstant C depends on P and η and K T := K,,T = {s : s D,, σ 2, t T }. 25

32 and Definition We say a funtion F g s = f,k = sup s K T fs. fn has property gn s A + h p, h, h + ; P, χ, λ, q,,, M, δ, α, κ if F g s has property Ah p, h; P, χ, λ, q,,, M, δ, α, κ and there exists a positive real-valued funtion f + n, suh that fn f + n for all n N, with the following properties holds. F g + f + n s = gn s admits a representation of the type F + g s = H + g sζs h+ s, for σ >. 2 The funtion H + g s has an analyti ontinuation to the region D, and satisfies the inequality H + g s M3 + t δ for all s D,. 3 h + s is analyti in the region D, and satisfies the inequality h + s M log3 + t α. for all s D,. We are now ready to state our main result. fn Theorem Let F g s = gn s have property A+ h p, h, h + ; P, χ, λ, q,,, M, δ, α, κ. Define T s := χp hps p s ζs λhs, p P where λ = a Z/ qz λaχa, 26

33 and As := s H g s s ζs λhs. Then, for x 3 and N 0, A g x := fn = gn<x x log x λh N m=0 P m log log x log x m + O R N where P m x := m m j=0 n=j e m,n λh e m,n = n n! n j i=0 m k=n l=0 n n i log n i j j+i i j Γ i λh x j, m m k a k,n = λ n m k l T l k +k 2 + +k n=k k i n i= A m k l h k i, k i! ak,n, m k! and R N = N + N+ log x 2 + log log x + MErrx 3, where Errx = e 4 log x for some positive onstants, 2, 3 and 4. For our onveniene, we define Γ i m = 0 for integers m 0 and all i. In most known appliations, h p s = z, where z is a omplex number for all primes p. In this setting, by hoosing hs = z we an show that, n = 0, a k,n = 0, otherwise. 27

34 We an redue from equation to A g x = x log x λh N m=0 β m z log x m + O R N where β m z := z Γ λz m m k=0 m k l=0 m k l T l z Am k l z m k! and RN N + N+ = + MErrx log x In some appliations, suh as [20], interested in the seondary term of the asymptoti expansion of A g s of speifi funtion fs. By letting N =, an be written as x A g x = log x λh A T Γ λh λh + A T logxγ λh A T λh log log x + + A T + A T + logxγ λh + O R Moreover, if hs = z is a omplex onstant, we an simplify further to obtain A g x = x A T + log x λz Γ λz A T + A T + A T + O R. logxγ λz

35 Chapter 3 Modifiation of the Selberg-Delange Method 3. Preliminary Results for Theorem In this setion, we will give the proofs of several lemmas whih are neessary to prove Theorem The first lemma, whih we are proving, is onerned about the existene of the absissa of absolute onvergene of the Dirihlet series F g s. Lemma 3... Let gn be a real-valued funtion suh that g : N [,, gn tends to infinity as n tends to infinity, and lim sup n log n log gn = κ, fn where κ is a non-negative real number. We also let F g s = gn s have a finite absissa of onvergene σ. Then the absissa of absolute onvergene, σ a, exists and satisfies σ σ a σ + κ. Proof. Fix gn as in the theorem and assume that F g s = onvergene σ. Let ɛ > 0. By definition of σ, the series fn gn s has a finite absissa of fn gn σ+ɛ onverges. It follows that lim n fn = 0. gn σ+ɛ 29

36 Hene, there exists an N ɛ suh that, for all n N ɛ, fn gn σ+ɛ <. 3.. Next, sine lim sup n log n log gn = κ, for some κ 0, we see that for δ > 0, there exists N δ suh that for all n N δ, log n κ + δ log gn. Exponentiating both sides, we find that n gn κ+δ, 3..2 for all n N δ. Now let N = max{n ɛ, N δ }. By 3.. and 3..2, we see that, for n > N, fn gn σ+ɛ+κ+δ+ɛκ+δ = fn gn σ+ɛ gn κ+δ+ɛ < gn κ+δ +ɛ n +ɛ. It follows that for σ σ + ɛ + κ + δ + ɛκ + δ, fn gn s = N fn gn s + n=n fn gn s N fn gn s + n=n n +ɛ. Hene fn gn s is absolutely onvergene for σ σ + ɛ + κ + δ + ɛκ + δ. Sine this is true for any ɛ > 0 and δ > 0, then σ a σ + κ. This ompletes the proof. Another essential omponent of the proof is the behavior of F g s when s D,. 30

37 Lemma Let F g s = fn gn s have property Bh;,, M, δ, α, κ and let B >. Then for s D,, σ < B and s > F g s M,M,δ,,α,B 3 + t δ+ 2 log 3. Proof. Let F g s have property Bh;,, M, δ, α, κ. Then there exists H g s, h; suh that F g s = H g s, h; ζs hs 3..3 and H g s, h; M3 + t δ Next, we need to obtain an upper bound for ζs hs. First, reall bounds of the Riemann Zeta funtion [58, p. 49, Theorem 3.5]. Uniformly for log3+ t σ 2, and t > t for some t > 0 ζs C log3 + t, where C is a positive onstant depending on. Sine ζs onverges for σ 2, thus uniformly for log3+ t σ B, and t > t,b ζs C 2 log3 + t, where C is a positive onstant depending on and B. It follows that ζs C 3 log3 + t, 3..5 where C is a positive onstant depending on, B, and, uniformly for and t > t,,b. Next, by property 4 of Bh;,, M, δ, α, κ, we see that log3+ t σ B, hs M log3 + t α,

38 for 0 α <. By 3..5 and 3..6, for t > t,,b, ζs hs = ζs Rhs e Argζs Ihs exp { hs logζs + π} exp {M log 3 + t α log log3 + t + logc 3 + π} exp {C 4 log 3 + t α log log3 + t }, where C is a positive onstant depending on, B, and. Sine α <, then there exists t,,α,b suh that for t > t,,α,b, C 4 log 3 + t α log log3 + t δ 2 log3 + t. Thus, for t > t,,α,b, { } ζs hs δ exp log3 + t log3 + t δ By ombining 3..3, 3..4, and 3..7, for t > t,,α,b, we obtain F g s = H g s, h; ζs hs M log3 + t δ log3 + t δ 2 M log3 + t +δ 2. Hene, for t > 0, F g s M,M,δ,,α,B 3 + t δ+ 2, as desired. This ompletes the proof. One ruial part of the proof of Theorem is to establish an analogue of the effetive form of Perron s formula. We now introdue an analogue of a normalized summatory funtion, A gx = n,gn<x fn + 2 fn n,gn=x 32

39 and the funtion, x >, αx = 2 x =, 0, 0 < x <. Lemma For any positive, T, and T, we have αx 2πi +it it x s ds s x 2π log x T + T x 3..8 and α 2πi +it it x s ds s T + x = The proof of this lemma an be found in Tenenbaum [57, p. 3]. Theorem 3..4 Analogue of Perron s Formula. Let F g s = has absissa of onvergene σ and absissa of absolute onvergene σ a suh that fn gn s σ σ a σ + κ for some κ 0. Let ξ > max{0, σ }. Assume that for σ ξ, F g s Mt δ for some 0 δ < and positive onstant M depending on ξ. Denote the set gn as the image of the funtion gn. Then A gx := n,gn<x fn + 2 n,gn=x fn = 2πi ξ+i ξ i F g sx s s ds, 3..0 where the integral is onditionally onvergent for x R\gN and onvergent in the sense of Cauhy s Priniple Value for x gn. Proof. Fix F g s, σ, σ a, κ, δ, and M as in the statement of the theorem. First, we will prove the theorem for σ > σ a. Suppose ξ > σ a. Sine F g s is absolutely and uniformly onvergent for 33

40 σ σ a + ɛ for a fixed ɛ > 0, we are able to interhange the sum and the integral of the right hand side of the equation We obtain ξ+it F g s xs 2πi ξ it s ds = ξ+it 2πi ξ it Then by 3..8, for x R + \gn, we see that A gx 2πi ξ+it ξ it fn x s gn s s ds = 2πi F g s xs s ds = fn n,gn<x x = fn α gn fn x α gn x fn ξ log 2πgn ξ T + T xξ 2π ξ+it x s ds fn ξ it gn s. fn ξ+it x 2πi ξ it gn 2πi 2πi x gn fn log gn ξ ξ+it ξ it ξ+it ξ it T + T x gn gn s ds s x s ds gn s x s ds s. 3.. Now, sine x R + \gn, there exists a onstant C x,g > 0 suh that log x gn for all n. Therefore, from equation 3.., A gx 2πi ξ+it ξ it for x R + \gn. Next, sine ξ > σ a, the series C x,g F g s xs s ds xξ 2π C x,g fn gn ξ T + fn T gn ξ. onverges. Thus A gx 2πi ξ+it ξ it F g s xs s ds M x ξ T + T 3..2 for some onstant M > 0. By letting T and T tend to infinity independently, we omplete the proof the first assertion of the theorem for ξ > σ a. 34

41 For the seond assertion, it is suffiient to take T = T. By proeeding in the same manner as in the proof of the first assertion, and using 3..9 as the upper bound when gn = x, we an show that A gx 2πi ξ+it ξ it F g s xs s ds = fn + 2 n gn x x = fn α gn fn α n gn x + n fn gn=x n gn=x fn 2πi x gn fn ξ+it x s ds 2πi ξ it gn s x s ds gn s ξ+it 2πi α 2πi x fn ξ log n 2πgn ξ gn x x gn ξ it ξ+it ξ+it ξ it 2 T + ξ it x gn s ds s x s ds gn s n gn=x + ξ fn. ξ + T Next, note that sine the domain of gn is the set of nature numbers and gn tends to infinity as n tends to infinity, then for any x there are only finitely many values of n suh that gn = x. Thus, n gn=x fn is a finite sum. Therefore, by 3..2, A gx 2πi ξ+it ξ it F g s xs s ds M x ξ 2 T + M 2 ξ ξ + T for some positive onstants M and M 2. Thus by letting T tend to infinity, we omplete the proof of the theorem for ξ > σ a. Now, suppose that σ < ξ σ a. By Lemma 3.., ξ + κ > σ a. Consider a retangular ontour integral F g s xs 2πi R s ds, 35

42 where R is the positively oriented retangle with verties ξ±it and ξ+κ±it. Sine F g s t δ for 0 δ <. Thus the ontribution of a horizontal segment is 2πi ξ+κ±it ξ±it F g x xs s ds T δ xξ+κ T x T ɛ. for some ɛ > 0, as T tends to infinity. Thus by the residue theorem, ξ+it F g x xs 2πi ξ it s ds = ξ+κ+it F g x xs 2πi ξ+κ it s ds + O T ɛ. Letting T tends to infinity, we find that 2πi ξ+i ξ i F g x xs s ds = 2πi ξ+κ+i ξ+κ i F g x xs s ds. Hene A gx 2πi ξ+it ξ it F g s xs s ds = A gx 2πi ξ+κ+it ξ+κ it F g s xs s ds. Sine ξ + κ > σ a, by proeeding in the same way as in the proof for the ase ξ > σ a, the theorem follows for the ase σ < ξ σ a. The equation 3..0 of the analogue of the Perron s Formula is insuffiient to prove Theorem We need a more effetive version of Theorem 3..4, whih an be stated as follows. fn Theorem Let F g s = gn s have absissa of onvergene σ and absissa of absolute onvergene σ a suh that σ σ a σ + κ for some κ 0. Assume that for σ ξ, F g s Mt δ for some 0 δ < and positive onstant M depending on ξ. Define A g x := fn. n gn<x 36

43 Then x A g tdt = 2πi ξ+i ξ i F g sx s+ ds ss Proof. Let w 0 and x R + \gn. First note that F g s = By Theorem 3..4 for s = s + w, we see that n gn<x fngn w = 2πi fn gn s = ξ+i ξ i fngn w gn s+w. F g s xs+w ds s + w Also note that n gn<x fnx w = 2πi ξ+i ξ i F g s xs+w ds s Therefore by subtrating 3..4 from 3..5, we obtain n gn<x fn x w gn w = 2πi = 2πi ξ+i ξ i ξ+i ξ i F g s xs+w ds s 2πi ξ+i ξ i F g s xs+w s + w ds F g s xs+w w ds ss + w The equation 3..6 still holds if x gn. Thus letting w =, we find that x A g tdt = fn x gn = 2πi n gn<x ξ+i ξ i x s+ F g s ss + ds. This ompletes the proof. Another important estimation we will need is an estimation of a trunated Hankel ontour integral. The Hankel ontour Ha, r is a path formed by joining the irle of radius r and enter at a, exluding the point s = a r, and the segment, a r] traed twie as shown in Figure

44 Is a r a a + r Rs H Figure 3.: Hankel s ontour enter at a with radius r One important fat about a Hankel ontour integral is its onnetion to Γz, whih an be seen in the following theorem. Theorem 3..6 Hankel s Formula. For any omplex number z and positive integer k, we have dk := Γ k z dz k = k s z e s log s k ds. Γz 2πi H0,r The proof of the theorem an be find in [40, pp. 2]. Now, we define a trunated Hankel ontour Ha, r, X to be the part of the ontour Ha, r where σ > x. By using Theorem 3..6, we obtain the following orollary. Corollary Let X >, and let k and m be non-negative integers suh that k < m. For an arbitrary omplex number z, we have k 2πi H0,r,X s m z e s log s k ds = Γ k z m + E k,m,zx, 38

45 where E k,m,z eπ Iz π X ρ m Rz e ρ logρ + π k dρ. Proof. By Thoerem 3..6, we see that k 2πi Then for ρ >, H0,r,X s m z e s log s k ds + k 2πi E k,m,z X : = k 2πi 2π π ρ X X s=ρe ±iπ ρ X s=ρe ±iπ ρ X s m z e s log s k ds = s m z e s log s k ds s m z e s log s k ds ρ m Rz e π Iz ρ logρ + π k dρ. Γ k z m. This ompletes the proof. Lastly before we prove Theorem 2.4.3, we need to establish a series expansion of F g s near s =. We will break up this proess into several lemmas. But, first we will prove a lemma that we frequently use in our estimations. Lemma Suppose a funtion fz is analyti in an open disk γ of radius r and enter at z 0 and fz m for all z γ. Then for eah positive integer k, f k z k!mr r z z 0 k+ for all z in γ. In partiular, if fz is also ontinuous on the boundary of γ then f k z 0 k!r k sup fs. s γ Proof. The proof of the first assertion an be found in [43, p.67]. For the seond assertion, fix fz as in the statement of the theorem. Sine fz is ontinuous on the boundary of γ, then by the Maximum Modulus Priniple, fz attains its maximum on the boundary of γ. Thus, by applying 39

46 3..8 at z = z 0, f k z 0 k!r r z 0 z 0 sup k+ s γ fs = k!r k sup fs, s γ as desired. This ompletes the proof. Lemma Let 0 < 2, > 0, M 0, 0 δ <, 0 α <, and κ 0. Let hs be a omplex-valued funtion and analyti in the region D,. Let F g s have property Bh;,, M, δ, α, κ. Define As := s Hs, h; s ζs hs. Then for integers N, As = N j=0 A j j! j s j + O M where A j = As j! 2πi s = s 2 uniformly for s < 2 and s D,. s j+ ds, N+, Proof. Fix, M, δ, α, κ, hs, F g s and As as in the statement of Lemma The funtion As is analyti in the region s < 2. Thus As = j=0 A j s j = j! j=0 A j j! j s j, where A j = As j! 2πi s = s ds. 2 j+ Let γ be a disk of radius 2 and enter. Sine As is analyti in the region s < by Lemma 3..8, Aj j! 2 log 3, then j sup As s γ Sine s ζs has an analyti ontinuation to C, thus there exists an analyti funtion f s, 40

47 suh that for σ >, s ζs = e f s. Therefore, for s < 2, As = s s Hs, h; ζs hs 2M3 + t e hs fs M Thus, by equations 3..7 and 3..8, for N, As = = = N j=0 N j=0 N j=0 This ompletes the proof. A j j! j s j + A j j! j s j + O j=n+ M A j j! j s j + O M A j j! j s j N +! N+ s N+ N+ N +! N+. s N+. N+ Lemma Let hs be an analyti funtion on the region D, satisfying the inequality hs Mlog t α for some onstant M > 0 and 0 α < in the region D,. For N and s D, \, ], we have N k s hs = s h n where k=0 n=0 n! logs n a k,n k s k + OR N, 3..9 a 0,n =, a k,n = k + +k n=k k i n i= h k i, k i! 4

48 and R N = 2M N+ s N+ logs + N+. Proof. Fix hs as in the statement of the lemma. For any positive integer N, we an write the left hand side of 3..9 as s hs = s h s hs h = s h exp hs h logs = s h n u n n! n=0 N = s h n u n n u n + n! n! n=0 = s h SN + R N, n=n+ where and u := S N := R N := N n u n, n! n u n, n! 3..2 n=0 n=n+ Sine hs is analyti on the disk s < 2, we an write Thus an be written as hs h logs hs = k=0 u = logs h k k! k s k. k= h k k! k s k

49 Next, by raising to the nth power, we see that u n = logs n k= = logs n k=n = logs n k=n h k k! k s k k + +k n=k k i n i= a k,n s k k n h k i k i! s k k with a k,n = k + +k n=k k i n i= h k i k i! Thus, replaing u n in by 3..24, we find that S N = = N n u n n=0 n! N n logs n n! n=0 k=n a k,n s k k N n N = logs n n! n=0 k=n N k n = logs n a k,n n! k=0 n=0 N n + logs n a k,n n! k=n+ n=0 N k n = logs n a k,n n! k=0 n=0 a k,n k s k + s k k k=n+ s k k k s k + K N a k,n s k k where K N := k=n+ N n logs n a k,n n! k s k. n=0 Next, by Lemma 3..8 and property 4 of Bh;,, M, δ, α, κ, h k k! k k sup hs 2k s = k 2 43 M k

50 for some positive onstant. Thus, applying to 3..25, we dedued that a k,n k = k + +k n=k k i n h k i k i! k i i= 2 k + +k n=k k i for some positive onstant 2. Hene, for s < K N = k=n+ N n n=0 logs N n! 2 N+ M N+ s k=n+ N+ k M n k 2 M n k 2 2 k M n k n 2, logs n a k,n k 2 k s k N n=0 logs + N+. s k n M n n! Therefore S N = N k n k=0 n=0 n! logs n a k,n k Lastly, we estimate RN. For s < RN = n=n+ 4, s k +O n u n n! 2M N+ s u N+ N +! n=0 u n. n! N+ logs + N By 3..26, we see that u n 2 n M n s n logs n. Therefore R N 2N+ M N+ s N+ logs N+ N+ N +! n=0 2M n n! n s logs n. Sine x log x attains a loal maximum value of e when x = e for 0 x, we have s log s = s log s + π e + π 2. 44