Sums of Generalized Harmonic Series for Kids from Five to Fifteen

Total Page:16

File Type:pdf, Size:1020Kb

Sums of Generalized Harmonic Series for Kids from Five to Fifteen GENERAL ⎜ ARTICLE Sums of Generalized Harmonic Series For Kids from Five to Fifteen Zurab Silagadze We examine the remarkable connection, first dis- covered by Beukers, Kolk and Calabi, between ζ(2n), the value of the Riemann Zeta-function at an even positive integer, and the volume of some 2n-dimensional polytope. It can be shown that this volume is equal to the trace of a compact Zurab Silagadze gradu- self-adjoint operator. We provide an explicit ex- ated from the Tbilisi State pression for the kernel of this operator in terms University in 1979. He got of Euler polynomials. This explicit expression his PhD from Novosibirsk in theoretical and makes it easy to calculate the volume of the poly- mathematical physics. tope and hence ζ(2n). In the case of odd positive Currently, he works as integers, we rediscover an integral representa- senior researcher at tion for ζ(2n +1), obtained by a different method Budker Institute of by Cvijovi´c and Klinowski. Finally, we indicate Nuclear Physics and as assistant professor at that the origin of the miraculous Beukers–Kolk– Novosibirsk State Calabi change of variables in the multidimen- University. sional integral, which is at the heart of this circle of ideas, can be traced to the amoeba associated with the certain Laurent polynomial. The article is dedicated to the memory of Vladimir Arnold (1937–2010). 1. Introduction In a nice little book [1], Vladimir Arnold has collected 77 mathematical problems for kids from 5 to 15 to stimulate critical thinking in them. Problem 51 in this book asks the reader to calculate the sum of the inverse squares and prove Euler’s celebrated formula ∞ 1 π2 2 = . (1) Keywords n 6 n=1 Riemann zeta function, integral representation, Basel problem. Well, there are many waystodothis(see, for example, 822 RESONANCE ⎜ September 2015 GENERAL ⎜ ARTICLE [2, 3, 4, 5, 6, 7, 8, 9, 10] and references therein), some maybe even accessible for kids under fifteen1. However, 1 Editor’s note: This perhaps ap- in this note we concentrate on the approach of Beuk- plies to Russian kids! ers, Kolk and Calabi [11], further elaborated by Elkies in [12]. This approach incorporates pleasant features which all the kids (and even some adults) adore: sim- plicity, magic and the depth that allows one to go be- yond the particular case (1). The simplicity, however, is not everywhere explicit in [11] and [12], while the magic longs for explanation after the first admiration fades away. Below we will try to enhance the simplicity of the approach and somewhat uncover the secret of the magic. The article is organized as follows. In the first two sec- tions, we reconsider the evaluation of ζ(2) and ζ(3) so that technical details of the general case do not obscure the simple underlying ideas. Then, we elaborate the general case and give the main result of this work, the formula for the kernel which allows us to simplify consid- erably the evaluation of ζ(2n) from [11, 12] and re-derive Cvijovi´c and Klinowski’s integral representation [13] for ζ(2n + 1). Finally, we ponder over the mysterious re- lations between the sums of generalized harmonic series and amoebas, first indicated by Passare in [10]. This relation enables us to uncover somewhat the origin of the Beukers–Kolk–Calabi’s highly non-trivial change of variables. The approach of Evaluation of ζ(2) Beukers, Kolk and Calabi has the Recall the definition of the Riemann Zeta function: features of simplicity, ∞ 1 magic and depth. ζ(s)= . (2) But, the simplicity is ns n=1 not explicit and the The sum (1) is just ζ(2) which we will now evaluate magic longs for following the method of Beukers, Kolk and Calabi [11]. explanation after the first admiration fades away. RESONANCE ⎜ September 2015 823 GENERAL ⎜ ARTICLE This change of Our starting point will be the dilogarithm function variables ∞ xn transforms the unit Li (x)= . (3) 2 n2 square to an n=1 isosceles triangle. Clearly, Li2(0) = 0 and Li2(1) = ζ(2). Differentiating (3), we get ∞ d xn x Li (x)= = − ln (1 − x), dx 2 n n=1 and, therefore, 1 ln (1 − x) dx dy ζ(2) = Li2(1) = − dx = , (4) x 1 − xy 0 ✷ where ✷ = {(x, y):0≤ x ≤ 1, 0 ≤ y ≤ 1} is the unit square. Let us note dx dy dx dy dx dy + =2 , (5) 1 − xy 1+xy 1 − x2y2 ✷ ✷ ✷ and dx dy dx dy 1 dx dy − = , (6) 1 − xy 1+xy 2 1 − xy ✷ ✷ ✷ where the last equation follows from 2xy 1 d(x2)d(y2) 1 dx dy dx dy = = . 1 − x2y2 2 1 − x2y2 2 1 − xy ✷ ✷ ✷ It follows from equations (5) and (6) that 4 dx dy ζ(2) = . (7) 3 1 − x2y2 ✷ Now let us make the magic Beukers–Kolk–Calabi change of variables in this two-dimensional integral [11] sin u sin v x = ,y= , (8) cos v cos u 824 RESONANCE ⎜ September 2015 GENERAL ⎜ ARTICLE with Jacobian determinant cos u sin v sin u cos v cos2 u 2 2 ∂(x, y) sin u sin v 2 2 = =1− 2 2 =1−x y . ∂(u, v) sin u sin v cos v cos v cos u 2 cos v cos u Then miraculously Editor's note: For a change of variables, the integral gets 4 4 scaled by the Jacobian determi- ζ(2) = du dv = Area(∆), (9) nant. 3 3 ∆ where ∆ is the image of the unit square ✷ under the transformation (x, y) → (u, v). It is easy to show that ∆ is the isosceles right triangle ∆ = {(u, v):u ≥ 0,v ≥ 0,u+ v ≤ π/2} and, therefore, 4 1 π 2 π2 ζ(2) = = . (10) 3 2 2 6 “Beautiful – even more so, as the same method of proof extends to the computation of ζ(2k)intermsofa2k- dimensional integral, for all k ≥ 1” [14]. However, before considering the general case, we check whether the trick works for ζ(3). 3. Evaluation of ζ(3) Inthecaseofζ(3), we begin with trilogarithm ∞ xn Li (x)= , (11) 3 n3 n=1 and using x d ln (1 − y) x Li3(x)=Li2(x)=− dy, dx y 0 we get 1 x dx ln (1 − y) ζ(3) = Li3(1) = − dy. (12) x y 0 0 RESONANCE ⎜ September 2015 825 GENERAL ⎜ ARTICLE A change of variables But of the unit cube x 1 1 ln (1 − y) ln (1 − xz) similar to that of the − dy = − dz square unfortunately x y xz 0 0 leads to a 1 1 complicated integral. dy = dz , A hyperbolic version 1 − xyz gives an interesting 0 0 integral which too is and finally not possible to dx dy dz evaluate; hence ζ(3) = Li3(1) = , (13) 1 − xyz zeta(3) is more ✷3 complicated than ✷ zeta(2). where 3 = {(x,y,z):0≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1} is the unit cube. By a similar trick as before, we can transform (13) into the integral 8 dx dy dz ζ(3) = , (14) 7 1 − x2y2z2 ✷3 and here the analogy with the previous case ends, un- fortunately, because the generalization of the Beukers– Kolk–Calabi change of variables does not lead in this case to a simple integral. However, it is interesting to note that the hyperbolic version of this change of vari- ables sinh u sinh v sinh w x = ,y= ,z= (15) cosh v cosh w cosh u does indeed produce an interesting result 8 8 ζ(3) = du dv dw = Vol(U3), (16) 7 7 U3 where U3 is a complicated 3-dimensional shape defined by the inequalities u ≥ 0,v≥ 0,w≥ 0, sinh u ≤ cosh v, sinh v ≤ cosh w, sinh w ≤ cosh u. 826 RESONANCE ⎜ September 2015 GENERAL ⎜ ARTICLE Unfortunately, there is no obvious simple way to calcu- late the volume of U3. However, there is a second way to convert the integral (12) for ζ(3) in which the Beukers–Kolk–Calabi change of variables still plays a helpful role. We begin with the identity 1 x dx ln (1 − y) ln (1 − y) ζ(3) = − dy = − dx dy, x y xy 0 0 D (17) where the domain of the 2-dimensional integration is the triangle D = {(x, y):x ≥ 0,y ≥ 0,y ≤ x}. Interchang- ing the order of integration in (17), we get 1 1 ln (1 − y) dx ζ(3) = − dy , y x 0 y which can be transformed further as follows 1 1 1 ln (1 − y)lny dx ζ(3) = dy = − ln y dy y 1 − xy 0 0 0 ln y = − dx dy, 1 − xy ✷ or in a more symmetrical form 1 ln (xy) ζ(3) = − dx dy. (18) 2 1 − xy ✷ Note that 2xy ln (xy) 1 ln (x2y2) dx dy = d(x2)d(y2) 1 − x2y2 4 1 − x2y2 ✷ ✷ 1 ln (xy) = dx dy. 4 1 − xy ✷ RESONANCE ⎜ September 2015 827 GENERAL ⎜ ARTICLE Therefore, we can modify (5) and (6) accordingly and using them transform (18) into 4 ln (xy) ζ(3) = − dx dy. (19) 7 1 − x2y2 ✷ At this point we can use the Beukers–Kolk–Calabi change of variables (8) in (19) and as a result we get 4 ζ(3) = − ln (tan u tan v)du dv 7 ∆ 8 = − ln (tan u)du dv. (20) 7 ∆ But this equation indicates that π/2 π/2−u 8 ζ(3) = − du ln (tan u) dv 7 0 0 π/2 8 π = − − u ln (tan u)du, 7 2 0 π which after the substitution x = 2 − u becomes π/2 π/2 8 8 ζ(3) = − x ln (cot x)dx = x ln (tan x)dx.
Recommended publications
  • The Lerch Zeta Function and Related Functions
    The Lerch Zeta Function and Related Functions Je↵ Lagarias, University of Michigan Ann Arbor, MI, USA (September 20, 2013) Conference on Stark’s Conjecture and Related Topics , (UCSD, Sept. 20-22, 2013) (UCSD Number Theory Group, organizers) 1 Credits (Joint project with W. C. Winnie Li) J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function I. Zeta Integrals, Forum Math, 24 (2012), 1–48. J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function II. Analytic Continuation, Forum Math, 24 (2012), 49–84. J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function III. Polylogarithms and Special Values, preprint. J. C. Lagarias and W.-C. Winnie Li , The Lerch Zeta Function IV. Two-variable Hecke operators, in preparation. Work of J. C. Lagarias is partially supported by NSF grants DMS-0801029 and DMS-1101373. 2 Topics Covered Part I. History: Lerch Zeta and Lerch Transcendent • Part II. Basic Properties • Part III. Multi-valued Analytic Continuation • Part IV. Consequences • Part V. Lerch Transcendent • Part VI. Two variable Hecke operators • 3 Part I. Lerch Zeta Function: History The Lerch zeta function is: • e2⇡ina ⇣(s, a, c):= 1 (n + c)s nX=0 The Lerch transcendent is: • zn Φ(s, z, c)= 1 (n + c)s nX=0 Thus ⇣(s, a, c)=Φ(s, e2⇡ia,c). 4 Special Cases-1 Hurwitz zeta function (1882) • 1 ⇣(s, 0,c)=⇣(s, c):= 1 . (n + c)s nX=0 Periodic zeta function (Apostol (1951)) • e2⇡ina e2⇡ia⇣(s, a, 1) = F (a, s):= 1 . ns nX=1 5 Special Cases-2 Fractional Polylogarithm • n 1 z z Φ(s, z, 1) = Lis(z)= ns nX=1 Riemann zeta function • 1 ⇣(s, 0, 1) = ⇣(s)= 1 ns nX=1 6 History-1 Lipschitz (1857) studies general Euler integrals including • the Lerch zeta function Hurwitz (1882) studied Hurwitz zeta function.
    [Show full text]
  • A Short and Simple Proof of the Riemann's Hypothesis
    A Short and Simple Proof of the Riemann’s Hypothesis Charaf Ech-Chatbi To cite this version: Charaf Ech-Chatbi. A Short and Simple Proof of the Riemann’s Hypothesis. 2021. hal-03091429v10 HAL Id: hal-03091429 https://hal.archives-ouvertes.fr/hal-03091429v10 Preprint submitted on 5 Mar 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A Short and Simple Proof of the Riemann’s Hypothesis Charaf ECH-CHATBI ∗ Sunday 21 February 2021 Abstract We present a short and simple proof of the Riemann’s Hypothesis (RH) where only undergraduate mathematics is needed. Keywords: Riemann Hypothesis; Zeta function; Prime Numbers; Millennium Problems. MSC2020 Classification: 11Mxx, 11-XX, 26-XX, 30-xx. 1 The Riemann Hypothesis 1.1 The importance of the Riemann Hypothesis The prime number theorem gives us the average distribution of the primes. The Riemann hypothesis tells us about the deviation from the average. Formulated in Riemann’s 1859 paper[1], it asserts that all the ’non-trivial’ zeros of the zeta function are complex numbers with real part 1/2. 1.2 Riemann Zeta Function For a complex number s where ℜ(s) > 1, the Zeta function is defined as the sum of the following series: +∞ 1 ζ(s)= (1) ns n=1 X In his 1859 paper[1], Riemann went further and extended the zeta function ζ(s), by analytical continuation, to an absolutely convergent function in the half plane ℜ(s) > 0, minus a simple pole at s = 1: s +∞ {x} ζ(s)= − s dx (2) s − 1 xs+1 Z1 ∗One Raffles Quay, North Tower Level 35.
    [Show full text]
  • The Riemann and Hurwitz Zeta Functions, Apery's Constant and New
    The Riemann and Hurwitz zeta functions, Apery’s constant and new rational series representations involving ζ(2k) Cezar Lupu1 1Department of Mathematics University of Pittsburgh Pittsburgh, PA, USA Algebra, Combinatorics and Geometry Graduate Student Research Seminar, February 2, 2017, Pittsburgh, PA A quick overview of the Riemann zeta function. The Riemann zeta function is defined by 1 X 1 ζ(s) = ; Re s > 1: ns n=1 Originally, Riemann zeta function was defined for real arguments. Also, Euler found another formula which relates the Riemann zeta function with prime numbrs, namely Y 1 ζ(s) = ; 1 p 1 − ps where p runs through all primes p = 2; 3; 5;:::. A quick overview of the Riemann zeta function. Moreover, Riemann proved that the following ζ(s) satisfies the following integral representation formula: 1 Z 1 us−1 ζ(s) = u du; Re s > 1; Γ(s) 0 e − 1 Z 1 where Γ(s) = ts−1e−t dt, Re s > 0 is the Euler gamma 0 function. Also, another important fact is that one can extend ζ(s) from Re s > 1 to Re s > 0. By an easy computation one has 1 X 1 (1 − 21−s )ζ(s) = (−1)n−1 ; ns n=1 and therefore we have A quick overview of the Riemann function. 1 1 X 1 ζ(s) = (−1)n−1 ; Re s > 0; s 6= 1: 1 − 21−s ns n=1 It is well-known that ζ is analytic and it has an analytic continuation at s = 1. At s = 1 it has a simple pole with residue 1.
    [Show full text]
  • 2 Values of the Riemann Zeta Function at Integers
    MATerials MATem`atics Volum 2009, treball no. 6, 26 pp. ISSN: 1887-1097 2 Publicaci´oelectr`onicade divulgaci´odel Departament de Matem`atiques MAT de la Universitat Aut`onomade Barcelona www.mat.uab.cat/matmat Values of the Riemann zeta function at integers Roman J. Dwilewicz, J´anMin´aˇc 1 Introduction The Riemann zeta function is one of the most important and fascinating functions in mathematics. It is very natural as it deals with the series of powers of natural numbers: 1 1 1 X 1 X 1 X 1 ; ; ; etc. (1) n2 n3 n4 n=1 n=1 n=1 Originally the function was defined for real argu- ments as Leonhard Euler 1 X 1 ζ(x) = for x > 1: (2) nx n=1 It connects by a continuous parameter all series from (1). In 1734 Leon- hard Euler (1707 - 1783) found something amazing; namely he determined all values ζ(2); ζ(4); ζ(6);::: { a truly remarkable discovery. He also found a beautiful relationship between prime numbers and ζ(x) whose significance for current mathematics cannot be overestimated. It was Bernhard Riemann (1826 - 1866), however, who recognized the importance of viewing ζ(s) as 2 Values of the Riemann zeta function at integers. a function of a complex variable s = x + iy rather than a real variable x. Moreover, in 1859 Riemann gave a formula for a unique (the so-called holo- morphic) extension of the function onto the entire complex plane C except s = 1. However, the formula (2) cannot be applied anymore if the real part of s, Re s = x is ≤ 1.
    [Show full text]
  • Special Values of Riemann's Zeta Function
    The divergence of ζ(1) The identity ζ(2) = π2=6 The identity ζ(−1) = −1=12 Special values of Riemann's zeta function Cameron Franc UC Santa Cruz March 6, 2013 Cameron Franc Special values of Riemann's zeta function The divergence of ζ(1) The identity ζ(2) = π2=6 The identity ζ(−1) = −1=12 Riemann's zeta function If s > 1 is a real number, then the series X 1 ζ(s) = ns n≥1 converges. Proof: Compare the partial sum to an integral, N X 1 Z N dx 1 1 1 ≤ 1 + = 1 + 1 − ≤ 1 + : ns xs s − 1 Ns−1 s − 1 n=1 1 Cameron Franc Special values of Riemann's zeta function The divergence of ζ(1) The identity ζ(2) = π2=6 The identity ζ(−1) = −1=12 The resulting function ζ(s) is called Riemann's zeta function. Was studied in depth by Euler and others before Riemann. ζ(s) is named after Riemann for two reasons: 1 He was the first to consider allowing the s in ζ(s) to be a complex number 6= 1. 2 His deep 1859 paper \Ueber die Anzahl der Primzahlen unter einer gegebenen Gr¨osse" (\On the number of primes less than a given quantity") made remarkable connections between ζ(s) and prime numbers. Cameron Franc Special values of Riemann's zeta function The divergence of ζ(1) The identity ζ(2) = π2=6 The identity ζ(−1) = −1=12 In this talk we will discuss certain special values of ζ(s) for integer values of s.
    [Show full text]
  • The Riemann Zeta Function and Its Functional Equation (And a Review of the Gamma Function and Poisson Summation)
    Math 259: Introduction to Analytic Number Theory The Riemann zeta function and its functional equation (and a review of the Gamma function and Poisson summation) Recall Euler's identity: 1 1 1 s 0 cps1 [ζ(s) :=] n− = p− = s : (1) X Y X Y 1 p− n=1 p prime @cp=1 A p prime − We showed that this holds as an identity between absolutely convergent sums and products for real s > 1. Riemann's insight was to consider (1) as an identity between functions of a complex variable s. We follow the curious but nearly universal convention of writing the real and imaginary parts of s as σ and t, so s = σ + it: s σ We already observed that for all real n > 0 we have n− = n− , because j j s σ it log n n− = exp( s log n) = n− e − and eit log n has absolute value 1; and that both sides of (1) converge absolutely in the half-plane σ > 1, and are equal there either by analytic continuation from the real ray t = 0 or by the same proof we used for the real case. Riemann showed that the function ζ(s) extends from that half-plane to a meromorphic function on all of C (the \Riemann zeta function"), analytic except for a simple pole at s = 1. The continuation to σ > 0 is readily obtained from our formula n+1 n+1 1 1 s s 1 s s ζ(s) = n− Z x− dx = Z (n− x− ) dx; − s 1 X − X − − n=1 n n=1 n since for x [n; n + 1] (n 1) and σ > 0 we have 2 ≥ x s s 1 s 1 σ n− x− = s Z y− − dy s n− − j − j ≤ j j n so the formula for ζ(s) (1=(s 1)) is a sum of analytic functions converging absolutely in compact subsets− of− σ + it : σ > 0 and thus gives an analytic function there.
    [Show full text]
  • Numerous Proofs of Ζ(2) = 6
    π2 Numerous Proofs of ζ(2) = 6 Brendan W. Sullivan April 15, 2013 Abstract In this talk, we will investigate how the late, great Leonhard Euler P1 2 2 originally proved the identity ζ(2) = n=1 1=n = π =6 way back in 1735. This will briefly lead us astray into the bewildering forest of com- plex analysis where we will point to some important theorems and lemmas whose proofs are, alas, too far off the beaten path. On our journey out of said forest, we will visit the temple of the Riemann zeta function and marvel at its significance in number theory and its relation to the prob- lem at hand, and we will bow to the uber-famously-unsolved Riemann hypothesis. From there, we will travel far and wide through the kingdom of analysis, whizzing through a number N of proofs of the same original fact in this talk's title, where N is not to exceed 5 but is no less than 3. Nothing beyond a familiarity with standard calculus and the notion of imaginary numbers will be presumed. Note: These were notes I typed up for myself to give this seminar talk. I only got through a portion of the material written down here in the actual presentation, so I figured I'd just share my notes and let you read through them. Many of these proofs were discovered in a survey article by Robin Chapman (linked below). I chose particular ones to work through based on the intended audience; I also added a section about justifying the sin(x) \factoring" as an infinite product (a fact upon which two of Euler's proofs depend) and one about the Riemann Zeta function and its use in number theory.
    [Show full text]
  • An Integral Representation for the Riemann Zeta Function on Positive Integer Arguments
    An integral representation for the Riemann zeta function on positive integer arguments Sumit Kumar Jha IIIT-Hyderabad, India Email: [email protected] May 23, 2019 Abstract In this brief note, we give an integral representation for the Riemann zeta function for positive integer arguments. To the best of our knowledge, the representation is new. Keywords: Ramanujan’s Master Theorem; Riemann Zeta Function; Polygamma Function; Polylogarithm Function AMS Classification: 33E20 We prove the following Theorem 1. We have, for integers r ≥ 2, Z ¥ −1=2 1 x Lir(−x) z(r) = r dx (1) p(2 − 2 ) 0 1 + x ¥ xk ¥ 1 where Lir(x) = ∑k=1 kr is the polylogarithm function, and z(r) = ∑n=1 nr is the Riemann zeta function. The above can be obtained as a direct consequence of the following result Theorem 2. For integers r ≥ 2 and 0 < n < 1, we have Z ¥ Li (−x) p y − (1 − n) xn−1 r dx = z(r) + (−1)r−1 r 1 .( 2) 0 1 + x sin np (r − 1)! ¥ xk dn G0(x) where Lir(x) = ∑k=1 kr is the polylogarithm function, yn(x) = dxn G(x) is the Polygamma function, G(x) is the Gamma function, and z(r) is the Riemann zeta function. Proof. Let n (r) 1 Hn = ∑ r k=1 k be the generalized Harmonic number. Then, we have the following generat- ing function from [2] Li (x) ¥ r = H(r)xn, − ∑ n 1 x n=1 for jxj< 1. We can also write ¥ Lir(−x) (r) n = ∑ Hn (−x) .( 3) (1 + x) n=1 We have for the following explicit form from [3] y − (n + 1) H(r) = z(r) + (−1)r−1 r 1 .( 4) n (r − 1)! 1 2 (r) r−1 yr−1(1) First note in this form, we have, H0 = z(r) + (−1) (r−1)! = 0.
    [Show full text]
  • Contents 1 Dirichlet Series and the Riemann Zeta Function
    Contents 1 Dirichlet Series and The Riemann Zeta Function 1 2 Primes in Arithmetic Progressions 4 3 Functional equations for ζ(s);L(s; χ) 9 4 Product Formula for ξ(s); ξ(s; χ) 18 5 A Zero-Free Region and the Density of Zeros for ζ(s) 23 6 A Zero-Free Region and the Density of Zeros for L(s; χ) 25 7 Explicit Formula Relating the Primes to Zeros of ζ(s) 25 8 Chebyshev Estimates and the Prime Number Theorem 28 9 Prime Number Theorem in Arithmetic Progressions 33 [Sections 1,2, and 3 are OK - the rest need work/reorganization] 1 Dirichlet Series and The Riemann Zeta Function Throughout, s = σ + it is a complex variable (following Riemann). Definition. The Riemann zeta function, ζ(s), is defined by X 1 Y ζ(s) = = (1 − p−s)−1 ns n p for σ > 1. Lemma (Summation by Parts). We have q q−1 X X anbn = An(bn − bn+1) + Aqbq − Ap−1bp n=p n=p P P where An = k≤n ak. In particular, if n≥1 anbn converges and Anbn ! 0 as n ! 1 then X X anbn = An(bn − bn+1): n≥1 n≥1 P Another formulation: if a(n) is a funciton on the integers, A(x) = n≤x a(n), and f is C1 on [y; x] then X Z x a(n)f(n) = A(x)f(x) − A(y)f(y) − A(t)f 0(t)dt y<n≤x y Proof. q q q q X X X X anbn = (An − An−1)bn = Anbn − An−1bn n=p n=p n=p n=p q q−1 q−1 X X X = Anbn − Anbn+1 = An(bn − bn+1) + Aqbq − Ap−1bp: n=p n=p−1 n=p For the second forumulation, assume y is not an integer and let N = dye;M = bxc.
    [Show full text]
  • 15 the Riemann Zeta Function and Prime Number Theorem
    18.785 Number theory I Fall 2015 Lecture #15 11/3/2015 15 The Riemann zeta function and prime number theorem We now divert our attention from algebraic number theory for the moment to talk about zeta functions and L-functions. These are analytic objects (complex functions) that are intimately related to the global fields we have been studying. We begin with the progenitor of all zeta functions, the Riemann zeta function. For the benefit of those who have not taken complex analysis (which is not a formal prerequisite for this course) the next section briefly recalls some of the basic definitions and facts we we will need; these are elementary results covered in any introductory course on complex analysis, and we state them only at the level of generality we need, which is minimal. Those familiar with this material should feel free to skip to Section 15.2, but may want to look at Section 15.1.2 on convergence, which will be important in what follows. 15.1 A quick recap of some basic complex analysis The complex numbers C are a topological field whose topology is definedp by the distance metric d(x; y) = jx − yj induced by the standard absolute value jzj := zz¯; all implicit references to the topology on C (open, compact, convergence, limits, etc.) are made with this understanding. For the sake of simplicity we restrict our attention to functions f :Ω ! C whose domain Ω is an open subset of C (so Ω denotes an open set throughout this section). 15.1.1 Holomorphic and analytic functions Definition 15.1.
    [Show full text]
  • A Note on the Zeros of Zeta and L-Functions 1
    A NOTE ON THE ZEROS OF ZETA AND L-FUNCTIONS EMANUEL CARNEIRO, VORRAPAN CHANDEE AND MICAH B. MILINOVICH 1 Abstract. Let πS(t) denote the argument of the Riemann zeta-function at the point s = 2 + it. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for S(t). We discuss a generalization of this bound (and prove some related results) for a large class of L-functions including those which arise from cuspidal automorphic representations of GL(m) over a number field. 1. Introduction Let ζ(s) denote the Riemann zeta-function and, if t does not correspond to an ordinate of a zero of ζ(s), let 1 S(t) = arg ζ( 1 +it); π 2 where the argument is obtained by continuous variation along straight line segments joining the points 1 2; 2 + it, and 2 + it, starting with the value zero. If t does correspond to an ordinate of a zero of ζ(s), set 1 S(t) = lim S(t+") + S(t−") : 2 "!0 The function S(t) arises naturally when studying the distribution of the nontrivial zeros of the Riemann zeta-function. For t > 0, let N(t) denote the number of zeros ρ = β + iγ of ζ(s) with ordinates satisfying 1 0 < γ ≤ t where any zeros with γ = t are counted with weight 2 . Then, for t ≥ 1, it is known that t t t 7 1 N(t) = log − + + S(t) + O : 2π 2π 2π 8 t In 1924, assuming the Riemann hypothesis, Littlewood [12] proved that log t S(t) = O log log t as t ! 1.
    [Show full text]
  • The Computation of Polylogarithms
    The Computation of Polylogarithms David C. Wood ABSTRACT The polylogarithm function, Lip(z), is defined, and a number of algorithms are derived for its computation, valid in different ranges of its real parameter p and complex argument z. These are sufficient to evaluate it numerically, with reasonable efficiency, in all cases. 1. Definition The polylogarithm may be defined as the function z t p 1 Li (z) _____ ______dt, for p > 0 . (1.1) p = t (p) 0 e z In the following, p will represent the real parameter, and z the complex argument. In the important case where the parameter is an integer, it will be represented by n (or n when negative). It is often convenient to write the argument as e w (or sometimes e w); for example 1 t p 1 Li (e w) _____ ________dt . (1.2) p = t w (p) 0 e 1 In this case, w is normally restricted to the range | Im(w) | . 2. Introduction Special cases of this function, particularly the dilogarithm (§ 7), have been studied at least since the time of Euler, usually with small integer parameters (n = 2, 3, . ). Lewin ([1]) gives a detailed account of this and related functions for this case. Truesdell ([2]) covers most of the properties of the function for real parameters. Some work has been done on this function with a complex parameters ([3]). This is not considered here, although most of the formulae hold with real p relaced by p + iq. It arises in physics, often in the form of the Fermi–Dirac or Bose–Einstein functions, usually with 1 parameters of the form n + ⁄2 (§ 18).
    [Show full text]