SPECIAL VALUES of MULTIPLE POLYLOGARITHMS 1. Introduction
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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. -
The Unilateral Z–Transform and Generating Functions
The Unilateral z{Transform and Generating Functions Recall from \Discrete{Time Linear, Time Invariant Systems and z-Transforms" that the behaviour of a discrete{time LTI system is determined by its impulse response function h[n] and that the z{transform of h[n] is 1 k H(z) = z− h[k] k=X −∞ If the LTI system is causal, then h[n] = 0 for all n < 0 and 1 k H(z) = z− h[k] Xk=0 Definition 1 (Unilateral z{Transform) The unilateral z{transform of the discrete{time signal x[n] (whether or not x[n] = 0 for negative n's) is defined to be 1 n (z) = z− x[n] X nX=0 When there is any danger of confusing the regular z{transform with the unilateral z{transform, 1 n X(z) = z− x[n] nX= 1 − is called the bilateral z{transform. Example 2 The signal x[n] = anu[n] is zero for all n < 0. So the unilateral z{transform of x[n] is the same as the ordinary z{transform. So, as we saw in Example 7 of \Discrete{Time Linear, Time Invariant Systems and z-Transforms", 1 n n 1 (z) = X(z) = z− a = 1 X 1 z− a nX=0 − 1 provided that z− a < 1, or equivalently z > a . Since the unilateral z{transform of any x[n] is always j j j j j j equal to the bilateral z{transform of the right{sided signal x[n]u[n], the region of convergence of a unilateral z{transform is always the exterior of a circle. -
Introduction to Analytic Number Theory the Riemann Zeta Function and Its Functional Equation (And a Review of the Gamma Function and Poisson Summation)
Math 229: 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: ∞ ∞ X Y X Y 1 [ζ(s) :=] n−s = p−cps = . (1) 1 − p−s n=1 p prime cp=0 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. We already observed that for all real n > 0 we have |n−s| = n−σ, because n−s = exp(−s log n) = n−σe−it log n and e−it 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 ∞ ∞ 1 X Z n+1 X Z n+1 ζ(s) − = n−s − x−s dx = (n−s − x−s) dx, s − 1 n=1 n n=1 n since for x ∈ [n, n + 1] (n ≥ 1) and σ > 0 we have Z x −s −s −1−s −1−σ |n − x | = s y dy ≤ |s|n 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. -
Ordinary Generating Functions
CHAPTER 10 Ordinary Generating Functions Introduction We’ll begin this chapter by introducing the notion of ordinary generating functions and discussing the basic techniques for manipulating them. These techniques are merely restatements and simple applications of things you learned in algebra and calculus. You must master these basic ideas before reading further. In Section 2, we apply generating functions to the solution of simple recursions. This requires no new concepts, but provides practice manipulating generating functions. In Section 3, we return to the manipulation of generating functions, introducing slightly more advanced methods than those in Section 1. If you found the material in Section 1 easy, you can skim Sections 2 and 3. If you had some difficulty with Section 1, those sections will give you additional practice developing your ability to manipulate generating functions. Section 4 is the heart of this chapter. In it we study the Rules of Sum and Product for ordinary generating functions. Suppose that we are given a combinatorial description of the construction of some structures we wish to count. These two rules often allow us to write down an equation for the generating function directly from this combinatorial description. Without such tools, we may get bogged down in lengthy algebraic manipulations. 10.1 What are Generating Functions? In this section, we introduce the idea of ordinary generating functions and look at some ways to manipulate them. This material is essential for understanding later material on generating functions. Be sure to work the exercises in this section before reading later sections! Definition 10.1 Ordinary generating function (OGF) Suppose we are given a sequence a0,a1,.. -
Limiting Values and Functional and Difference Equations †
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 February 2020 doi:10.20944/preprints202002.0245.v1 Peer-reviewed version available at Mathematics 2020, 8, 407; doi:10.3390/math8030407 Article Limiting values and functional and difference equations † N. -L. Wang 1, P. Agarwal 2,* and S. Kanemitsu 3 1 College of Applied Mathematics and Computer Science, ShangLuo University, Shangluo, 726000 Shaanxi, P.R. China; E-mail:[email protected] 2 Anand International College of Engineering, Near Kanota, Agra Road, Jaipur-303012, Rajasthan, India 3 Faculty of Engrg Kyushu Inst. Tech. 1-1Sensuicho Tobata Kitakyushu 804-8555, Japan; [email protected] * Correspondence: [email protected] † Dedicated to Professor Dr. Yumiko Hironaka with great respect and friendship Version February 17, 2020 submitted to Journal Not Specified Abstract: Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function. Keywords: limit values; modular relation; Lerch zeta-function; Hurwitz zeta-function; Laurent coefficients MSC: 11F03; 01A55; 40A30; 42A16 1. Introduction There have appeared enormous amount of papers on the Laurent coefficients of a large class of zeta-, L- and special functions. -
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. -
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. -
INTEGRALS of POWERS of LOGGAMMA 1. Introduction The
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 139, Number 2, February 2011, Pages 535–545 S 0002-9939(2010)10589-0 Article electronically published on August 18, 2010 INTEGRALS OF POWERS OF LOGGAMMA TEWODROS AMDEBERHAN, MARK W. COFFEY, OLIVIER ESPINOSA, CHRISTOPH KOUTSCHAN, DANTE V. MANNA, AND VICTOR H. MOLL (Communicated by Ken Ono) Abstract. Properties of the integral of powers of log Γ(x) from 0 to 1 are con- sidered. Analytic evaluations for the first two powers are presented. Empirical evidence for the cubic case is discussed. 1. Introduction The evaluation of definite integrals is a subject full of interconnections of many parts of mathematics. Since the beginning of Integral Calculus, scientists have developed a large variety of techniques to produce magnificent formulae. A partic- ularly beautiful formula due to J. L. Raabe [12] is 1 Γ(x + t) (1.1) log √ dx = t log t − t, for t ≥ 0, 0 2π which includes the special case 1 √ (1.2) L1 := log Γ(x) dx =log 2π. 0 Here Γ(x)isthegamma function defined by the integral representation ∞ (1.3) Γ(x)= ux−1e−udu, 0 for Re x>0. Raabe’s formula can be obtained from the Hurwitz zeta function ∞ 1 (1.4) ζ(s, q)= (n + q)s n=0 via the integral formula 1 t1−s (1.5) ζ(s, q + t) dq = − 0 s 1 coupled with Lerch’s formula ∂ Γ(q) (1.6) ζ(s, q) =log √ . ∂s s=0 2π An interesting extension of these formulas to the p-adic gamma function has recently appeared in [3]. -
The Bloch-Wigner-Ramakrishnan Polylogarithm Function
Math. Ann. 286, 613424 (1990) Springer-Verlag 1990 The Bloch-Wigner-Ramakrishnan polylogarithm function Don Zagier Max-Planck-Insfitut fiir Mathematik, Gottfried-Claren-Strasse 26, D-5300 Bonn 3, Federal Republic of Germany To Hans Grauert The polylogarithm function co ~n appears in many parts of mathematics and has an extensive literature [2]. It can be analytically extended to the cut plane ~\[1, ~) by defining Lira(x) inductively as x [ Li m_ l(z)z-tdz but then has a discontinuity as x crosses the cut. However, for 0 m = 2 the modified function O(x) = ~(Liz(x)) + arg(1 -- x) loglxl extends (real-) analytically to the entire complex plane except for the points x=0 and x= 1 where it is continuous but not analytic. This modified dilogarithm function, introduced by Wigner and Bloch [1], has many beautiful properties. In particular, its values at algebraic argument suffice to express in closed form the volumes of arbitrary hyperbolic 3-manifolds and the values at s= 2 of the Dedekind zeta functions of arbitrary number fields (cf. [6] and the expository article [7]). It is therefore natural to ask for similar real-analytic and single-valued modification of the higher polylogarithm functions Li,. Such a function Dm was constructed, and shown to satisfy a functional equation relating D=(x-t) and D~(x), by Ramakrishnan E3]. His construction, which involved monodromy arguments for certain nilpotent subgroups of GLm(C), is completely explicit, but he does not actually give a formula for Dm in terms of the polylogarithm. In this note we write down such a formula and give a direct proof of the one-valuedness and functional equation. -
A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function
mathematics Article A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function Daeyeoul Kim 1,* and Yilmaz Simsek 2 1 Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju 54896, Korea 2 Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya TR-07058, Turkey; [email protected] * Correspondence: [email protected] Abstract: In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials. Keywords: Bernoulli numbers and polynomials; Euler numbers and polynomials; Apostol–Bernoulli and Apostol–Euler numbers and polynomials; Hurwitz–Lerch zeta function; Hurwitz zeta function; alternating Hurwitz zeta function; generating function; Mellin transformation MSC: 05A15; 11B68; 26C0; 11M35 Citation: Kim, D.; Simsek, Y. A New Family of Zeta Type Function 1. Introduction Involving the Hurwitz Zeta Function The families of zeta functions and special numbers and polynomials have been studied and the Alternating Hurwitz Zeta widely in many areas. They have also been used to model real-world problems. -
Functions of Random Variables
Names for Eg(X ) Generating Functions Topic 8 The Expected Value Functions of Random Variables 1 / 19 Names for Eg(X ) Generating Functions Outline Names for Eg(X ) Means Moments Factorial Moments Variance and Standard Deviation Generating Functions 2 / 19 Names for Eg(X ) Generating Functions Means If g(x) = x, then µX = EX is called variously the distributional mean, and the first moment. • Sn, the number of successes in n Bernoulli trials with success parameter p, has mean np. • The mean of a geometric random variable with parameter p is (1 − p)=p . • The mean of an exponential random variable with parameter β is1 /β. • A standard normal random variable has mean0. Exercise. Find the mean of a Pareto random variable. Z 1 Z 1 βαβ Z 1 αββ 1 αβ xf (x) dx = x dx = βαβ x−β dx = x1−β = ; β > 1 x β+1 −∞ α x α 1 − β α β − 1 3 / 19 Names for Eg(X ) Generating Functions Moments In analogy to a similar concept in physics, EX m is called the m-th moment. The second moment in physics is associated to the moment of inertia. • If X is a Bernoulli random variable, then X = X m. Thus, EX m = EX = p. • For a uniform random variable on [0; 1], the m-th moment is R 1 m 0 x dx = 1=(m + 1). • The third moment for Z, a standard normal random, is0. The fourth moment, 1 Z 1 z2 1 z2 1 4 4 3 EZ = p z exp − dz = −p z exp − 2π −∞ 2 2π 2 −∞ 1 Z 1 z2 +p 3z2 exp − dz = 3EZ 2 = 3 2π −∞ 2 3 z2 u(z) = z v(t) = − exp − 2 0 2 0 z2 u (z) = 3z v (t) = z exp − 2 4 / 19 Names for Eg(X ) Generating Functions Factorial Moments If g(x) = (x)k , where( x)k = x(x − 1) ··· (x − k + 1), then E(X )k is called the k-th factorial moment. -
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.