To Maa i
ii
Abstract The Riemann Zeta function is defined as ζ(s), Re(s) >. ns n This thesis presents a new and exceedingly simple method of analytic continuation of ζ(s) to the whole complex plane. This method involves an application of the binomial theorem. As a consequence, we give a simple inductive proof of the well known formula ζ( k) B k /k, where B k is the k-th Bernoulli number. This technique is capable of wider generalization. We will consider the series ζ(s, z) (n + z) s n0 where z is a complex number in a much larger domain: D {z C} \ {z R : z 0} We will derive the analytic continuation for z in the larger complex domain by the method mentioned above and thereby extend the classical result ζ( k, a) B k(a) k for 0 < a < to all a D, where B k (x) denotes the k-th Bernoulli polynomial. More generally, we will consider series of the type (n + z n ) s, {z n } being a sequence in D, n0 iii
iv where D is the cut unit disc D {z C : z < } \ {z R : < z 0}. We will also consider poly-hurwitz zeta functions defined as ζ(s, s 2,..., s r ; x, x 2,..., x r ) (n + x ) s... (n + x r ) sr, for 0 < x i. The above series absolutely converges for Re(s + +s r ) >. Using methods similar to the one mentioned above, we will be able to obtain a meromorphic continuation to C r. n0
Acknowledgements First and foremost, I would like to express my gratitude to my supervisor Professor M. Ram Murty, without whose patient guidance, this study would not have been possible. His expertise and wide ranging knowledge have enabled me to learn and feel comfortable with a subject in which I had no background whatsoever before joining Queen s University. He has encouraged me in my academic endeavours since my undergraduate days and however much I may write, I never will be able to thank him enough. Sir, I am extremely grateful to you for all your help, support and encouragement. I am thankful to you for the time you have so kindly devoted to this project and I also thank you for your guidance at every stage. To me, you have been like a father, friend and a source of inspiration. I would also like to thank Professors Ernst Kani, Roland Speicher and Jamie Mingo, under whom I have taken courses in the past year. I am very grateful to them for their systematic and clear presentations of these courses. I thank my friend Chizuru Sekine, with whom I have studied and discussed many topics in number theory. In the one year that she has spent here, I have learnt a lot from her. I am grateful to my friend and senior Ms. Alina Cojocaru for being a most helpful teaching assistant and a very willing Latex instructor. A very special thanks to my house-mate Arpita didi for her encouragement and friendship. Thank you, didi, for your wonderful coffee and for always making an effort to talk to me about number theory even though you study political science. Last, but not the least, I thank my father for all his love and support. I thank him for always encouraging me to study a subject of my choice and for allowing me to travel abroad to pursue what I wanted to do. I also thank my little brother for his wonderful sense of humour and optimistic attitude which makes me feel better whenever I miss home. v
vi
Contents Abstract Acknowledgements iii v Introduction 2 Preliminaries 5 2. Objectives............................. 5 2.2 The binomial theorem for complex exponents......... 5 2.3 The Euler summation formula.................. 6 2.4 Poisson Summation formula................... 8 2.5 Properties of the Gamma function................ 2.6 Bernoulli numbers and Bernoulli polynomials.......... 2 2.7 Characters mod k and L-Series................. 4 3 The Riemann-zeta function and its generalisations 7 3. Analytic continuation of ζ(s) via partial summation...... 7 3.2 A new method of analytic continuation of ζ(s)......... 9 3.3 A new derivation of ζ( k) B k k............... 24 3.4 An identity involving ζ(2n)................... 28 3.5 Functional equation of the Riemann zeta function....... 30 4 The Hurwitz zeta function 35 4. Classical Hurwitz zeta function................. 35 4.2 Hurwitz zeta function ζ(s, a) for complex values of a..... 36 4.3 Derivation of ζ( k, z) B k(z)................ 37 k 4.4 L-Series and its relation to ζ(s, x)................ 40 vii
5 Generalised Hurwitz zeta series 43 5. Generalization of the Hurwitz Zeta Function.......... 43 5.2 Poly-Hurwitz zeta functions................... 45 Bibliography 47
Chapter Introduction The Riemann Zeta function is defined as ζ(s) n, s where s is any complex variable, satisfying Re(s) >. This function was studied by Riemann in 859. He wrote a paper in which he described the ζ function as the main tool to prove the prime number theorem. He observes that the above series is connected to prime numbers via the Euler product: n n s n ( p s ), where the product is over all primes p. In the same paper, Riemann shows that ζ(s) admits an analytic continuation for all s C apart from s, where it has a simple pole. He also derives the functional equation Γ ( s 2) ζ(s)π s 2 Γ ( ( s) 2 ) ζ( s)π ( s) 2. The proof presented in this thesis is an application of the Poisson Summation formula, and closely follows Riemann. I will follow the proof as treated in [M2]. In the third chapter of this thesis, we will study the analytic continuation of the Riemann zeta function. We will use a simple and elegant summation technique to analytically continue the ζ function. I have followed the treatment as outlined in [MR].
2 CHAPTER. INTRODUCTION One of the main contributions of this thesis is a new and exceedingly simple method for the analytic continuation of ζ(s). This method involves an application of the binomial theorem. For this proof, we will also need the Hurwitz Zeta function, which is defined as ζ(s, x) n0 (n + x) s, where x is a real number such that 0 < x. This series converges absolutely for Re(s) >. In this thesis, we present a new method of analytic continuation for ζ(s, x). We will also give a new proof of the identity ζ( k) B k k, where the Bernoulli numbers B n are defined by the power series t e t t n B n n! In the fourth chapter, we will introduce the Hurwitz zeta function n0 ζ(s, x), x R, 0 < x, first studied by Hurwitz in 882. In his paper, he obtained an analytic continuation of ζ(s, x) to the whole complex plane apart from s, where it has a simple pole. Hurwitz thus unified the treatment of ζ(s) and the study of the classical Dirichlet L-functions L(s, χ) by observing that L(s, χ) k s k χ(r)ζ(s, r k ), r where χ is any Dirichlet character mod k and χ(n) L(s, χ), Re(s) >. ns n In the fourth chapter, we will study ζ(s, z) for a complex number z in domain D, where D is the cut unit disc D {z : z < } \ {z R : < z 0}.
We will be able to generalise our new method for analytically continuing ζ(s) to obtain an analytic continuation of ζ(s, z) by applying the binomial theorem. This will enable us to prove that ζ( k, z) B k(z), (.) k where B k (z), the k-th Bernoulli polynomial is defined as k ( ) k B k (z) B j z k j. j j0 This idea originates in the paper[mr], albeit for real values of z, 0 < z. By a slight modification to our arguments, we can also define ζ(s, z) for z belonging to an even larger domain D {z C} \ {z R : z 0}. It will be shown that ζ(s, z), z D can be analytically continued to the whole complex plane except s where it has a simple pole. Furthermore, equation (.) holds for all z D. In the last chapter, we will define the Generalised Hurwitz zeta series ζ(s, Λ) n0 (n + z n ) s, where Λ is a sequence {z n }, z n D, and study some applications to regularized products. We also consider poly-hurwitz zeta functions defined as ζ(s, s 2,..., s r ; x, x 2,..., x r ) (n + x ) s... (n + x r ) sr, for 0 < x i. By the same methods, we will obtain analytic continuation for these functions and prove as a consequence that n0 ζ( k, k 2,..., k r ; x,..., x r ) is a polynomial in x, x 2,..., x r with rational coefficients (given by Bernoulli numbers). We may also consider these zeta functions for x i D and thereby get a generalization of equation (.). 3
4 CHAPTER. INTRODUCTION
Chapter 2 Preliminaries 2. Objectives The aim of this chapter is to introduce some fundamental definitions and results which will be required to prove the main results presented in this thesis. 2.2 The binomial theorem for complex exponents For the sake of completeness, we first review the binomial theorem, which states that: [LA], page 59 Lemma 2.2. If z < and α is any complex number, then Here, ( ) α r ( + z) α r0 ( ) α z r. r α(α )... (α r + ), α C r! This lemma will be applied to prove the analytic continuation of the zeta function to the whole complex plane in chapter 3. 5
6 CHAPTER 2. PRELIMINARIES 2.3 The Euler summation formula The following result is very useful, and is referred to as the technique of partial summation. Theorem 2.3. Suppose {a n } n is a sequence of complex numbers and f(t) is a continuously differentiable function on [, x]. Set A(t) n t a n. Then, a n f(n) A(x)f(x) n x x A(t)f (t)dt. Proof. Suppose first that x is a natural number. Then, A(n )f(n) a n f(n) A(n)f(n) n x n x n x A(n)f(n) A(n)f(n + ) n x n x A(x)f(x) n+ A(n) f (t)dt A(x)f(x) n (x ) n n+ n (x ) n A(t)f (t)dt, since A(t) is a step function. We observe n+ A(t)f (t)dt x n (x ) n A(t)f (t)dt. Thus, the result has been proved if x in an integer. If x is not an integer, an observation that A(x){f(x) f([x])} completes the proof. x [x] A(t)f (t)dt 0
2.3. THE EULER SUMMATION FORMULA 7 Corollary 2.3.2 Suppose A(x) O(x δ ). Then, for Re(s) > δ, a n n s A(t) dt. s ts+ n Hence, the Dirichlet series converges for Re(s) > δ. Proof. By Theorem 2.3., taking f(n) n s, we get, a n n A(x) x A(t) + s dt. s x s ts+ n x For s fixed, Re(s) > δ, we have, A(x) lim 0 x x s as we know that A(x) O(x δ ). Thus, n a n n s s In the special case a n, it follows that Corollary 2.3.3 For Re(s) > 0, where {x} x [x]. Proof. From above, Thus, ζ(s) n x ζ(s) s s s s s A(t) dt. ts+ {x} dx, xs+ x n s [t] dt. s ts+ s s s [x] dx xs+ x dx s s {x} dx, xs+ and the latter integral converges for Re(s) > 0. {x} dx xs+
8 CHAPTER 2. PRELIMINARIES 2.4 Poisson Summation formula Theorem 2.4. Let F L (R). Suppose that the series n Z F (n + v) converges absolutely and uniformly in v, and that m Z F (m) <, where F (m) F (x)e 2πimx. Then, F (n)e 2πinv. n Z F (n + v) n Z The following proof closely follows the treatment in [M2]. Proof. The function G(v) n Z F (n + v) is bounded, measurable and periodic with period. The Fourier coefficients of G are Since c m 0 0 n Z n Z G(v)e 2πimv dv ( ) F (n + v) e 2πimv dv n Z 0 n+ n F (m). F (n + v)e 2πimv dv F (x)e 2πimx dx F (x)e 2πimx dx F (m) <, n Z we can represent G by its Fourier series n Z F (n + v) n Z F (n)e 2πinv.
2.4. POISSON SUMMATION FORMULA 9 Corollary 2.4.2 With F as above, n Z F (n) n Z F (n). Proof. Set v 0 in Theorem (2.4.). We observe that since therefore, f(t) f(t + a) f(x)e 2πixt dx, f(x + a)e 2πixt dx e 2πiat f(y)e 2πiyt dy e 2πiat f(t). Similarly, f( t a ) a f a f(at). ( x a) e 2πixt dx f(y)e 2πiy(at) dy Thus, we have shown that Proposition 2.4.3 It can be shown that ([M2], page 308) f(t + a) e 2πiat f(t) and f( t a ) a f(at). Proposition 2.4.4 For F (x) e πx2, F (u) e πu2. Proof. We have to show that e πx2 e 2πixu dx e πu2.
0 CHAPTER 2. PRELIMINARIES In other words, we have to show that e π(x+iu)2 dx (2.) We will first prove equation (2.) for u 0 and then show that the value of the integral e π(x+iu)2 dx is independent of u. This will prove equation (2.). When u 0, we have the integral We note that I 2 I 0 e πx2 dx. e πx2 dx re πr2 dr e πy2 dy e π(x2 +y 2) dxdy 2π 0 dθ, by making the polar substitution x r cos θ, y r sin θ. Since I 2 and I > 0, we deduce I. Also note that u 0 e π(x+iu)2 dx 2πre πr2 dr, i 0. ( u e π(x+iu)2 ( x e π(x+iu)2 [ie π(x+iu)2] x x ) dx ) dx Thus, the value of the integral is independent of u. Since I, equation (2.) is proved. Thus, from Propositions 2.4.3 and 2.4.4, we get
2.5. PROPERTIES OF THE GAMMA FUNCTION Proposition 2.4.5 For F (x) e π t2 x, F (x) x /2 e πt2x. Thus, by Poisson summation formula, Corollary 2.4.6 n Z e n2 π x x 2 n Z e n2 πx. 2.5 Properties of the Gamma function While proving the functional equation of the Riemann zeta function, we will require some properties of the Gamma function Γ(s), as outlined in [Ap]. For Re(s) > 0, we have the following integral representation: Γ(s) 0 x s e x dx. Γ(s) can be continued beyond the line σ 0, and is analytic everywhere in the s-plane except for simple poles at the points s 0,, 2,... with residue ( )n at s n. n! Γ(s) also has the representation Γ(s) lim n n s n! s(s + ) (s + n). for s 0,, 2,... The Gamma function satisfies two functional equations and Γ(s + ) sγ(s) (2.2) Γ(s)Γ( s) It also satisfies a multiplication formula π sin(πs) (2.3) ( Γ(s)Γ s + ) (... Γ s + m ) (2π) m 2 m 2 ms Γ(ms) (2.4) m m for all s and integers m.
2 CHAPTER 2. PRELIMINARIES 2.6 Bernoulli numbers and Bernoulli polynomials Summation of the first n natural numbers, or squares, or cubes, is an elementary problem in number theory. It leads to the following well known formulae, n S (n) i i0 (n )n, 2 n S 2 (n) i 2 i0 (n )n(2n ), 6 and n S 3 (n) i 3 n2 (n ) 2. 4 i0 These formulae go back to Aryabhata in the 5th century A.D. who derived the above formulae for k, 2, 3 in his book Aryabhatiya. (see [?], page 38) The generalization of above formulae to find the sum n S k (n) i0 i k is connected with the work Ars Conjectandi of Jacob Bernoulli. We define the Bernoulli numbers B n by the series t e t t k B k k!. k0
2.6. BERNOULLI NUMBERS AND BERNOULLI POLYNOMIALS 3 We consider the power series: k0 S k (n) tk k! k0 n t k n k! j0 k0 n e tj j0 i0 j0 j k (tj) k k! ent t t e t n i t i i! i j0 n i+ t i (i + )! j0 B j t j j! B j t j j! Comparing coefficients of tk, we get k! S k (n) k i0 ( k n )B i+ k i i i +. Thus S k (n) is a polynomial in n of degree k + involving the Bernoulli numbers as its coefficients. We also observe that t 2 + t e t is an even function of t. Thus, we deduce that B k 0 for k odd, k 3. We also define the n-th Bernoulli polynomial as B n (x) n k0 ( ) n B k x n k. k
4 CHAPTER 2. PRELIMINARIES Note that S k (n) k i0 k + k + ( k n )B i+ k i i i + k ( k + k i + i k ( k + i0 i0 k+ k + i i + ( k + i ) B k i n i+ ) B k i n i+ ) B k+ i n i k + (B k+(n) B k+ ). Bernoulli polynomials will appear later as special values of Hurwitz zeta functions.(see Chapter 4) 2.7 Characters mod k and L-Series Consider the group (Z/kZ) of coprime residue classes mod k. A homomorphism χ : (Z/kZ) (C) into the multiplicative group of complex numbers is called a character mod k. We extend the definition of χ to all natural numbers by setting { χ(n(mod k)) if (n, k), χ(n) (2.5) 0 otherwise. χ, as defined above, is a completely multiplicative function. The character χ 0 : (Z/kZ) (C) satisfying χ 0 (a) for all (a, k) is called the principal character. The set of characters (mod k) forms a group of order ϕ(k), where ϕ is the Euler ϕ function. This has been explained in [M2].
2.7. CHARACTERS MOD K AND L-SERIES 5 We observe that r(mod k) χ 0 (r) ϕ(k). Lemma 2.7. If χ χ 0, then r(mod k) χ(r) 0. Proof. Since χ χ 0, there is a b(mod k) such that (b, k) and χ(b). Then br runs through coprime residue classes mod k as r does. Thus, Thus, s : Since, χ(b), therefore s 0. Lemma 2.7.2 χ(mod k) χ(n) Proof. If n (mod k), then χ(mod k) r(mod k) r(mod k) χ(b)s. χ(r) χ(br) s(χ(b) ) 0. { ϕ(k) if n (mod k), 0 otherwise. χ(n) χ(mod k) ϕ(k). (2.6) Otherwise, suppose n (mod k) and (n, k). Then, there is a character α such that α(n). (Refer to page 224 of [M2]) Note that αχ ranges over all the characters mod k as χ does. Thus, S : χ(n) χ(mod k) χ(mod k) α(n)s. αχ(n)
6 CHAPTER 2. PRELIMINARIES Since α(n), therefore S 0. We now define the L-Series L(s, χ) for Re(s) > as follows. L(s, χ) n χ(n) n s. Notice that when χ is not principal, χ(n) O(). n x So, by corollary 2.3.2, we obtain an analytic continuation of L(s, χ) for Re(s) > 0.
Chapter 3 The Riemann-zeta function and its generalisations 3. Analytic continuation of ζ(s) via partial summation We recall that the Riemann ζ function was originally defined by the Dirichlet series ζ(s), Re(s) >. ns n In this section, we will prove the following theorem, using partial summation. This method has also been explained in [MR] and [Ap]. Theorem 3.. ζ(s) extends to an analytic function for s C, apart from s, where it has a simple pole. Proof. From corollary 2.3.3, by partial summation, we have ζ(s) s [x] dx xs+ + s s {x} dx, xs+ where {x} denotes the fractional part of x. This gives us the analytic continuation of ζ(s) for Re(s) > 0. 7
8CHAPTER 3. THE RIEMANN-ZETA FUNCTION AND ITS GENERALISATIONS Next, we write ζ(s) + s s + s s {x} x dx s+ n+ n n + s s n Integrating the last integral by parts, we get n 0 u (u + n) s+ du 2 n 0 x n dx xs+ u du. (u + n) s+ (n + ) + s + s+ 2 and the second integral converges for Re(s) >. Thus ζ(s) + s s s(s + ) (ζ(s + ) ) 2 2 {x} 2 dx xs+2 {x} 2 dx. xs+2 In this way, ζ(s) has been analytically continued in the half-plane Re(s) >. From above, we also infer that ζ(0). Integration by parts leads to 2 similar representations in successively larger half-planes, as indicated below. ζ(s) + s m s(s + )... (s + r ) (ζ(s + r) ) (r + )! r s(s + )... (s + m) u m+ du. (m + )! (u + n) s+m+ n 0 (3.) and the infinite sum on the right hand side converges for Re(s) > m.
3.2. A NEW METHOD OF ANALYTIC CONTINUATION OF ζ(s) 9 3.2 A new method of analytic continuation of ζ(s) The aim of this section is to derive the analytic continuation of the Riemann zeta function by an even simpler observation than the one in the previous section. The idea is to use the Hurwitz zeta function for x /2. As a consequence, we give a new inductive proof of the well known formula ζ( k) B k /k, where B k is the k-th Bernoulli number. The technique is capable of wider generalization. In the last section of this thesis, we will consider generalizations of the classical Hurwitz zeta function and obtain analytic continuations for them. To derive the analytic continuation of ζ(s), we will make use of the Hurwitz zeta function. Now, fix 0 < x <. Recall that the Hurwitz zeta function ζ(s, x) is defined by the series ζ(s, x) n0, which converges absolutely for Re(s) >. (n + x) s We notice that x s + ζ(s, x) ζ(s) ( n (n + x) s n s ). The summand on the right hand side can be written as n s ( ( + x ) ) s. n Using the binomial theorem, we obtain Proposition 3.2. If 0 < x <, then x s + ζ(s, x) ζ(s) ( s r r ) ζ(s + r)x r
20CHAPTER 3. THE RIEMANN-ZETA FUNCTION AND ITS GENERALISATIONS Proof. Since 0 < x < and n, using binomial theorem, we get Thus, we write ( + x ) s ( ) s (x ) r. n r n r0 x s + ζ(s, x) ζ(s) n n s ( ) s (x ) r. r n We note that by ratio test([ru], page 66), the series of positive terms n r r [ ( ) ] s ( x ) r n s r n converges for Re(s) >. So, we can interchange the order of summation to get x + ζ(s, x) ζ(s) ( ) s ζ(s + r)x r s r The beauty of this proposition is that the right hand side is analytic for Re(s) > 0. This is because for Re(s) > 0, ( ) s x r ζ(s + r) r ( ) s x r r, (3.2) r where σ Re(s). Notice Then, we note that ( ) s x r r r r r σ + r > r. n σ+r n n σ+r n ( ) s x r r n +σ r n ( ) C s x r, r r
3.2. A NEW METHOD OF ANALYTIC CONTINUATION OF ζ(s) 2 where Once again, by ratio test, r C n n +σ. ( ) C s x r r r is convergent. Thus, by Weierstrass M-test, ([Ru], page 48), ( ) s x r r n σ+r n is uniformly convergent for Re(s) > 0. It follows that ( ) s ζ(s + r)x r r r converges uniformly in every compact subset of the half plane Thus, {s C : Re(s) > 0}. ( ) s ζ(s + r)x r r r is analytic for Re(s) > 0. We will exploit this fact below. However, before we do this, let us note that In this way, we get ζ(s, /2) 2 s (2n + ) s n0 2 s (ζ(s) (2n) ) s n 2 s (ζ(s) 2 s ζ(s)) (2 s )ζ(s).
22CHAPTER 3. THE RIEMANN-ZETA FUNCTION AND ITS GENERALISATIONS Lemma 3.2.2 ζ(s, /2) (2 s )ζ(s). This is the key observation that we will use to analytically continue ζ(s). In Proposition 3.2., for x /2, we get ( ) s 2 s + ζ(s, /2) ζ(s) ( r 2 )r ζ(s + r). Using Lemma 3.2.2, we deduce Theorem 3.2.3 (2 s 2)ζ(s) 2 s + r r ( ) ( ) r s ζ(s + r). r 2 Since the right hand side converges for Re(s) > 0, we obtain an analytic continuation of (2 s 2)ζ(s) in this region. It is immediate that this gives a meromorphic continuation of ζ(s) for Re(s) > 0 with possible poles only at s + 2πim, where m is an integer. It is also interesting that we can use log 2 Theorem 3.2.3 to derive (by induction) a meromorphic continuation of ζ(s) to the entire complex plane. We do this below. Note first that So, ( r r ) ( 2 )r ζ( + r) ( ) r ( 2 )r r n n r 2 n ( 2n )r n 2n(2n + ). n r+ lim s (2s 2)ζ(s) 2 2 2n(2n + ) n ( 2 2 2n ) 2n + 2 log 2. n So we obtain, as a corollary to Theorem 3.2.3, the following
3.2. A NEW METHOD OF ANALYTIC CONTINUATION OF ζ(s) 23 Corollary 3.2.4 (2 s 2)ζ(s) extends to an analytic function for Re(s) > 0 and lim s (2s 2)ζ(s) 2 log 2. Thus so that lim s 2 s 2 (s )ζ(s) 2 log 2 s lim(s )ζ(s). s There is nothing special about ζ(s, ). Lemma 3.2.2 can be generalized as 2 follows : Lemma 3.2.5 Proof. (m s m)ζ(s) m a [ζ(s, a m ) ζ(s) ]. Thus, Thus, m s m a ζ(s, a m ) m m s ζ(s) (m s m)ζ(s) a n0 (mn + a) s ζ(s) m ζ(s, a m ) a m a [ζ(s, a m ) ζ(s) ]. The interesting feature of Lemma 3.2.5 is that the right hand side is analytic for Re(s) > 0 by Proposition 3.2. and the remark afterwards. Hence, by the same logic as used in the remarks following Theorem 3.2.3, we deduce that ζ(s) extends to a meromorphic function for Re(s) > 0 with possible poles at s + 2πik, k Z. Choosing m 3, we find that the only possible poles of log m ζ(s) lie in { + 2πik } { log 3 : k Z + 2πil } log 2 : l Z {}.
24CHAPTER 3. THE RIEMANN-ZETA FUNCTION AND ITS GENERALISATIONS This shows that ζ(s) extends to an analytic function for Re(s) > 0 except for a simple pole at s. From proposition 3.2. and Lemma 3.2.2, we have ( ) s (2 s 2)ζ(s) 2 s + ζ(s + r)2 r. (3.3) r Let us now assume that ζ(s) is analytic for Re(s) > (m ), m N. Equation (3.3) can be rewritten as m ( ) s ( ) s (2 s 2)ζ(s) 2 s + 2 r ζ(s + r) + 2 r ζ(s + r). r r If Re(s) > m, then r r r m ( ) s 2 r ζ(s + r), r rm+ being a finite sum of analytic functions is analytic. From arguments following Proposition 3.2., we deduce that ( ) s 2 r ζ(s + r) r rm+ is analytic for Re(s) > m. So, assuming that ζ(s) has an analytic continuation in the half plane Re(s) > (m ), m N we have shown that ζ(s) can be analytically continued in the region Re(s) > m. Thus, by induction, we see that ζ(s) extends analytically for all s C, except for a simple pole of residue at s. This gives us an elementary proof of Theorem 3... 3.3 A new derivation of ζ( k) B k k We now use the derivation in the previous section to obtain a new proof of the formula ζ( k) B k k, k 2.
3.3. A NEW DERIVATION OF ζ( K) B K K 25 Since t exp(t) + t 2 is an even function, the Bernoulli numbers for odd subscripts 3 vanish, and we can write the above formula as Theorem 3.3. ζ( k) ( ) k B k /k, k. Proof. Observe that equation (3.3) can be written as (2 s 2)ζ(s) 2 s ζ(s + ) ( ) s s + ζ(s + r)2 r 2 r where we see that the right hand side is analytic for Re(s) >. We may substitute s 0 in the above formula and deduce that ζ(0) 2. Since B, the above theorem holds for k. 2 We proceed by induction on k. Suppose the result holds for all m N, m < k. We note that for r k, ζ( k + r) has a simple pole of residue. So, Then, lim r k Also, if r > k, the term r2 lim(r k)ζ( k + r). r k ( k r ) ζ( k + r) 2r 2 k k. ( ) k ζ( k + r) r 2r vanishes, since ζ( k + r) is analytic. Thus, if we put s k in Theorem (3.2.3), we obtain the recurrence k ( ) k (2 k 2)ζ( k) 2 k + ζ( k + r) r 2r 2 k k. (3.4) r
26CHAPTER 3. THE RIEMANN-ZETA FUNCTION AND ITS GENERALISATIONS Then, by induction hypothesis and the fact that ζ(0) B, we get ζ( k + r) ( ) k r B k r, r (k ). k r So, the right hand side of equation (3.5) becomes k ( ) k 2 k + B r 2 r ( )k r k r k r 2 k k r 2 k + k ( ) k k r 2 r ( )k r B k r 2 k k r 2 k + k ( ) k k r 2 r ( )k r B k r k ( )k B k. Thus, we get r0 (3.5) (2 k 2)ζ( k) 2 k ( ) k B k k k Let us define c k s as follows. k0 x k c k k! x k 2 k k! k0 k0 k r0 ( ) k r 2 r ( )k r B k r. ( ) k x k B k k!. (3.6) Then, c k k ( k r0 r) ( ) k r B 2 r k r for every k. Note that c k x k exp( x k! 2 ) ( x) exp( x) k0 ( x ) x x exp + 2 exp( x) exp ( ). x 2 2 Now, coefficient of xk k! in x exp( x 2 ) is k 2 k. We have to determine the coefficient of xk k! in ( x ) x exp + 2 x exp( x 2 ) exp ( x 2 ( x ) ) x exp + ix 2 2 (sin( ix2 ) )
3.3. A NEW DERIVATION OF ζ( K) B K K 27 Recall from [Co](page 88) that (sin(x)) x + m Using this formula, we can deduce that ix 2 Thus, if k is even, As a result, ( sin( ix 2 ) ) c k B 2m ( ) m+ (2 2m 2) x2m (2m)! k ( x ) 2k B 2k (2 2k 2) 2 (2k)! k 2 B 2 k 2 k k 2 k (2 k 2)ζ( k) 2 k ( ) k B k k (2 2 k ) B k k { } k 2 B (2 k 2) k k 2 k Thus that is, ζ( k) B k k If k is odd, c k k 2 k. (2 k 2)ζ( k) ( ) k B k k. In particular, if k, then ζ(0) 2 B If k 3, then ζ( k) 0 B k for k odd. The theorem for m k follows from the different cases considered above. Thus, we have proved our result by induction. Corollary 3.3.2 ζ(s) 0 for s 2n, n N.
28CHAPTER 3. THE RIEMANN-ZETA FUNCTION AND ITS GENERALISATIONS Proof. We know that B k 0 for odd k Z, k 3. Thus, ζ( 2n) B 2n+ 2n + 0. 3.4 An identity involving ζ(2n) This section is based on [GW]. In this paper, G.T.Williams presented, among other things, a new method of evaluating ζ(2n). His method is elementary and involves manipulations of the defining series. He proves the following Theorem 3.4. ζ(2)ζ(2n 2) + ζ(4)ζ(2n 4) +... ζ(2n 2)ζ(2) (n + 2 )ζ(2n). This convolution serves to give ζ(2n), recursively, as a rational multiple of (ζ(2)) n. Proof. The left hand side is the limit, as N, of N N ν µ { µ 2 ν 2n 2 + µ 4 ν 2n 4 + + µ 2n 2 ν 2 } (3.7) Here, N is a positive integer. Summing the geometric series within braces, and taking note of the exceptional case when µ ν, this becomes ν 2 2n µ 2 2n ν µ µ 2 ν 2 + (n ) ν. (3.8) ν2n Throughout this proof, all sums run from to N and an accent on the inner sum indicates that the index(in this case, µ) does not take on the value of the index of the outer sum (ν here.) Ignoring the term on the far right, (3.4) is equal to ν µ ν 2 2n µ 2 ν 2 + ν µ µ 2 2n ν 2 µ 2. (3.9)
3.4. AN IDENTITY INVOLVING ζ(2n) 29 Inverting the order of summation in the second double sum, and noting that the condition µ ν is the same as the condition ν µ, (3.5) can be written as 2 ν ν 2n 2 µ µ 2 ν 2 (3.0) Combining the above equations, we find that equation 3.3 can be written as (n ) ν ν 2n + 2 ν ν 2n 2 µ µ 2 ν 2. (3.) Now, 2ν µ µ 2 ν 2 µ N+ν µ 3 2ν µ ν µ µ + ν µ + N ν ν + µ + 2ν µ { N ν + + + N + ν }. When we substitute this into (3.7), we get that (3.3) is equal to (n + 2 ) ν ν 2n ν ( ν 2n N ν + + + ) N + ν (3.2) Finally, N ν + + N ν + 2 + + N + ν < 2ν N ν +
30CHAPTER 3. THE RIEMANN-ZETA FUNCTION AND ITS GENERALISATIONS and so 0 < ν ( ν 2n N ν + + + ) N + ν ν 2n 2. N ν + < 2 ν 2 ν(n ν + ) ν 2 N + < 4 N + ν N ν ( ν + N ν + ν 4 ( + log(n)) N + ) We observe that 4 ( + log(n)) 0 N + as N. The above statement and equation (3.8) taken together give us the theorem. 3.5 Functional equation of the Riemann zeta function We now derive the functional equation of the Riemann Zeta function. ζ(s) satisfies the functional equation ζ(s) 2 s π s sin( πs )Γ( s)ζ( s) 2 The functional equation can be written in a number of ways. Changing s into s and using the functional equation for Γ(s), it becomes ( πs ) ζ( s) 2 s π s cos Γ(s)ζ(s). 2
3.5. FUNCTIONAL EQUATION OF THE RIEMANN ZETA FUNCTION3 It can also be written in the symmetric form ( π s s ) ( ) s 2 ζ(s)γ π ( s)/2 ζ( s)γ 2 2 and we will prove the functional equation in the above form. We define the Θ function as Θ(z) n Z e πin2z, z C, Im(z) > 0. Now, define ω(y) Θ(iy). Thus, ω(x) n Z e πn2 x and ω ( ) e πn2 /x. x n Z We know, from corollary (2.4.6), that Now, Put t n 2 πx. Then, For Re(s) >, ω ( s Γ 2) ( ) x /2 ω(x). x ( s Γ 2) ( s π s/2 n s Γ 2) ( s π s/2 Γ ζ(s) 2) 0 0 e t s/2 dt t t. e n2 πx n s π s/2 x s/2 dx x. 0 0 ( n e n2 πx x s/2 dx x e n2 πx ) x s/2 dx x.
32CHAPTER 3. THE RIEMANN-ZETA FUNCTION AND ITS GENERALISATIONS Note that + 2 n e n2 πx n e n2 πx. So, Let us set W (x) ω(x) 2. Now, n e n2πx ω(x). 2 Now, ( s π s/2 Γ ζ(s) 2) W ( ) x 0 0 ( ) ω(x) dx x s/2 2 x x s/2 W (x) dx x x s/2 W (x) dx x + x s/2 W (x) dx 0 x x s/2 W (x) dx ( x + x s/2 W x ω ( x) 2 x/2 ω(x) 2 x/2 {2W (x) + } 2 2 + 2 x/2 + x /2 W (x). ) dx x. Thus, W ( ) x s/2 dx x x ( 2 + ) 2 x/2 + x /2 W (x) x s/2 dx x s + s + x ( s)/2 W (x) dx x.
3.5. FUNCTIONAL EQUATION OF THE RIEMANN ZETA FUNCTION33 Hence, π s/2 Γ ( s 2) ζ(s) x s/2 W (x) dx x + x ( s)/2 W (x) dx x + s s ( x s/2 + x ( s)/2) W (x) dx x + s s. Thus, it follows from above that ( π s s ) ( ) s 2 ζ(s)γ π ( s)/2 ζ( s)γ 2 2 This is Riemann s derivation of the analytic continuation of ζ(s) and functional equation.
34CHAPTER 3. THE RIEMANN-ZETA FUNCTION AND ITS GENERALISATIONS
Chapter 4 The Hurwitz zeta function 4. Classical Hurwitz zeta function In the last chapter, we defined the Hurwitz zeta function : ζ(s, a) (n + a), s n0 where we fix 0 < a. The above series converges absolutely for Re(s) >. For Re(s) >, partial summation gives us ζ(s, a) s 0 [x] dx. (x + a) s+ As in section 3., integration by parts leads to the formula ζ(s, a) a s s m + r s(s + )... (s + m) (m + )! s(s + )... (s + r ) ( ) ζ(s + r, a) a s r (r + )! u m+ du (u + n + a) s+m+ n 0 (4.) The infinite sum on right hand side converges for Re(s) > m. Thus, the method used for Riemann zeta function generalizes to give the analytic continuation of the Hurwitz zeta function ζ(s, a) for real values of a between 0 and. However, we can consider the Hurwitz zeta function for complex values of a, provided we are careful that log(a) is well-defined. We do this below 35
36 CHAPTER 4. THE HURWITZ ZETA FUNCTION and derive an analytic continuation of ζ(s, a). As before, we can inquire into the nature of the special values ζ( k, a). This turns out to be B k (a)/k for appropriate complex values of a. 4.2 Hurwitz zeta function ζ(s, a) for complex values of a Our new method to analytically continue ζ(s), explained in section 3.2, is a versatile method because it enables us to analytically continue ζ(s, a), where a belongs a much larger complex domain D, where We consider the series D {z : z < } \ {z R : < z 0} ζ(s, z) n0 (n + z) s where z is a complex number. The summand is to be interpreted as exp( s log(n + z)) and therefore, for the logarithm to be defined, we need to have z not lie on the negative real axis. We note that z s Writing the summand as + ζ(s, z) ζ(s) ( n s n ( ( + z ) ) s n and using the binomial theorem, we obtain Theorem 4.2. For z D, we have z s + ζ(s, z) ζ(s) ( s r r (n + z) s n s ). ) ζ(s + r)z r.
4.3. DERIVATION OF ζ( K, Z) B K(Z) K 37 Since the right hand side converges for Re(s) > 0, we obtain an analytic continuation of ζ(s, z) in this region. Analogous to remarks made after theorem 3.2.3, we can derive, by induction the following result. Theorem 4.2.2 The function ζ(s, z) for z D is analytic in the whole complex plane except for a simple pole at s. Furthermore, given any non-zero complex number z not lying in the negative real axis, we can find a positive integer m such that z < m. Now, m z + ζ(s, z) ζ(s) s n n s ( ( + z ) ) s n + nm n s ( ( + z ) ) s n Since, z < m, therefore z < n m. n Thus, applying binomial theorem to the second term on the right hand side and by a change in the order of summation, we obtain a modification of Theorem 4.2.2. Theorem 4.2.3 Let D {z C} \ {z R : z 0}. Then, given z D, m N such that m z +ζ(s, z) ζ(s) s n n s ( ( + z ) ) s n ( ( s + )z r ζ(s + r) r Since the right hand side is analytic for Re(s) > 0, we obtain an analytic continuation of ζ(s, z) in this region. Thus, by induction, we can derive that the function ζ(s, z), for z D is analytic in the whole complex plane except for a simple pole at s. 4.3 Derivation of ζ( k, z) B k(z) k We now prove the following result: Theorem 4.3. For z D, r ζ( k, z) B k(z), k Z, k 2. k m n ). n s+r
38 CHAPTER 4. THE HURWITZ ZETA FUNCTION Proof. We first prove the theorem for z D. From Theorem 4.2., we have z s + ζ(s, z) ζ(s) r (s)(s + )... (s + r ) ( ) ζ(s + r)z r (4.2) r! r In equation (4.2), put s k and observe that if r > k, then the sum vanishes, since ζ( k + r) is analytic. At r k, ζ( k + r) has a simple pole of residue. Thus, This implies that lim r k So, lim(r k)ζ( k + r). r k ( k)(2 k)... (r k)(r k) ( )r r! z + ζ( k, z) ζ( k) k k Thus, r0 r ( k r ζ( k + r) ( )k ( ) k (k )! k! k. )ζ( k + r)z r zk k. (4.3) k ( k ζ( k, z) z k + ζ( k) + )ζ( k + r)z r zk r k r k 2 ( ) k z k Bk r + ζ( k) r k r zr + ζ(0)z k zk k r zk + ζ( k) k 2 ( k )B k r z r zk 2 k r k r B z k B k k B z k 0 k k 2 ( ) k B k r z r k r r k ( ) k B k r z r k r B k(z) k
4.3. DERIVATION OF ζ( K, Z) B K(Z) K 39 More generally, let z D. From Theorem 4.2.3, we have m z + ζ(s, z) ζ(s) s ( ( + z ) ) s + n ( ( m ) s )z r. r n s+r n s n r n In the above equation, we put s k where k Z, k 2. Then, we get the following equation : z k + ζ( k, z) ζ( k) m n n k zk k k ( ) s z r ζ(s + r) r r (4.4) ( ( + z ) ) k k ( k + n r r ( m k )z r r n. k+r r n (4.5) ) ζ( k + r)z r Now, the first term on the right hand side in equation 4.5 is m n [ k n k r0 ( k r ) ] ( z ) r n m n k n k ( k r r [ k ( ) ] k ( z ) r r n r m )z r n n k+r. Thus, equation 4.5 reduces to equation 4.3. Since equation 4.3 holds for all z D, our result is proved. The idea of this proof originates in [MR].
40 CHAPTER 4. THE HURWITZ ZETA FUNCTION 4.4 L-Series and its relation to ζ(s, x) If χ is a character mod k, we rearrange the terms in the series for L(s, χ) according to the residue classes mod(k). That is, we write and obtain n qk + r, r k, q 0,, 2,... L(s, χ) n χ(n) n s k q0 r k r k s χ(qk + r) (qk + r) s χ(r) ( q + r k q0 ) s k k s χ(r)ζ(s, r k ). Theorem 4.4. For the principal character χ 0, the L-function L(s, χ 0 ) is analytic everywhere except for a simple pole at s with residue ϕ(k). k If χ χ 0, L(s, χ) is an entire function of s. Furthermore, if we define the generalized Bernoulli polynomial B n,χ, by B n,χ k n r k χ(r)b n (r/k), r then, L( n, χ) B n,χ n. Proof. From Lemma 2.7., we have the the following r mod k χ(r) { 0 if χ χ 0, ϕ(k) if χ χ 0. (4.6)
4.4. L-SERIES AND ITS RELATION TO ζ(s, X) 4 Since ζ(s, r/k) has a simple pole at s with residue, the function χ(r)ζ(s, r/k) has a simple pole at s with residue χ(r). Therefore Moreover, Res s L(s, χ) lim(s )L(s, χ) s k lim s (s )k s r k χ(r) k r { 0 if χ χ 0, ϕ(k) if χ χ k 0. χ(r)ζ(s, r k ) L( n, χ) This proves our theorem. k χ(r)k n ζ( n, r k ) r k r B n,χ n. χ(r)k n B n(r/k) n
42 CHAPTER 4. THE HURWITZ ZETA FUNCTION
Chapter 5 Generalised Hurwitz zeta series 5. Generalization of the Hurwitz Zeta Function In this chapter, we make a few final remarks about how the definition of ζ(s, z) for complex z can be extended more generally to define what we call the Generalized Hurwitz zeta function. This function has applications in a more general theory of regularized products of certain sequences satisfying a few basic axioms[jl]. A basic example has been presented below. Let Λ be a sequence {z n }, z n D. We define the Generalized Hurwitz Zeta function ζ(s, Λ) (n + z n ) s. n0 This series is absolutely convergent for Re(s) >. As a further generalization of Proposition 3.2., we obtain Proposition 5.. ζ(s, Λ) (z 0 ) + ζ(s) + ( s s r r ) n z r n n s+r. Define L r (s) n z r n n s+r 43
44 CHAPTER 5. GENERALISED HURWITZ ZETA SERIES where s C, r N. The above series converges absolutely for Re(s) > r. It immediately follows that ζ(s, Λ) is analytic for Re(s) > 0 except at s where it has a simple pole. Then, we consider the derivative ζ (s, Λ) Putting s 0 formally, we find ζ (0, Λ) n log(z n + n) (z n + n) s log(z n + n) n If we assume that L (s) analytically extends to s 0, then ζ(s, Λ) has an analytic continuation at s 0. Therefore, exp( ζ (0, Λ)) (n + z n ) We thus obtain an interpretation of the infinite product {n + z n }, which is called a regularized product. The notion is useful in certain cases to give meaning to the determinant of a linear operator T with eigenvalues given by a sequence {λ n } n. Thus, we may define det T where the infinite product is defined via the formalism above. Namely, let Z(s) (λ n ) s. n n n Suppose Z(s) admits an analytic continuation for Re(s) 0. Then, as Z (s) (log λ n )λ s n we formally have Z (0) n λ n log λ n. Hence, we may interpret exp( Z (0)) as the determinant of the operator T. n
5.2. POLY-HURWITZ ZETA FUNCTIONS 45 5.2 Poly-Hurwitz zeta functions We define poly-hurwitz zeta functions as follows ζ(s, s 2,..., s r ; x, x 2,..., x r ) (n + x ) s... (n + x r ) sr, for 0 < x i. This series absolutely converges when Re(s +s 2 + +s r ) >. Now, let Then, ζ (s, s 2,..., s r ; x, x 2,..., x r ) n0 (n + x ) s... (n + x r ) sr, ζ (s, s 2,..., s r ; x, x 2,..., x r ) n. s n... ( + x ) s ( s 2... n sr n n ( )( ) ( s s2 sr... j,...,j r n j j 2 j r + x r n ) x j....x jr r A, where A ζ(s + s 2 + + s r + j + + j r ). Using methods similar to those used in previous sections, we can obtain a meromorphic continuation of this function to C k. For example, let r 2. We will try to find the value of when s i k i, k i Z, k i. Then, by Theorem 3.3.2, ζ ( k, k 2 ; x, x 2 ) ζ (s, s 2 ; x, x 2 ) k k 2 j 0 j 2 0 k ( )( k k2 k 2 j 0 j 2 0 where l(j, j 2 ) k + k 2 j j 2. j j 2 ( )( k k2 j j 2 ) sr ) x j x j 2 2 ζ( k + k 2 + j + j 2 ) ) x j x j B 2 l(j,j 2 ) 2 l(j, j 2 ),
46 CHAPTER 5. GENERALISED HURWITZ ZETA SERIES More generally, we can show that ζ( k, k 2,..., k r ; x, x 2,..., x r ) is a polynomial in x, x 2,..., x r with rational coefficients given by Bernoulli numbers. We may also consider these functions for x i D. This gives us a further generalization of (.).
Bibliography [Ap] [Co] Tom Apostol, Introduction to Analytic Number Theory, UTM, Springer-Verlag, 976. Louis Comtet, Advanced Combinatorics, D. Reidel Publishing Company, 974. [GW] G.T.Williams, A new method of evaluating ζ(2n), Amer. Math. Monthly, 60(953) 9-25. [Hu] Hurwitz, A:Mathematische Werke Vol 2, Basel: Birkhauser 932-33. [JL] Jay Jorgenson and Serge Lang, Basic Analysis of Regularized Series and Products, Lecture Notes in Mathematics, 564, Springer Verlag 993. [LA] Serge Lang, Complex Analysis, GTM 03, Springer Verlag, 988 [M] M. Ram Murty, Introduction to p-adic analytic number theory, to appear in International Press. [M2] M. Ram Murty, Problems in Analytic Number Theory, GTM 206, Springer-Verlag, 200. [MR] M.Ram Murty and Marilyn Reece, A simple derivation of ζ( k) B k k, Functiones et Approximatio, Vol28 (2000), 4-54. [Ru] Walter Rudin, Principles of Mathematical Analysis, McGraw Hill International Editions, Third Edition, 976. 47