DOCSLIB.ORG

THE GAMMA FUNCTION and the ZETA FUNCTION Contents 1. The

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
THE GAMMA FUNCTION and the ZETA FUNCTION Contents 1. The THE GAMMA FUNCTION AND THE ZETA FUNCTION PAUL DUNCAN Abstract. The Gamma Function and the Riemann Zeta Function are two special functions that are critical to the study of many different fields of math- ematics. In this paper we will discuss characterizations and properties of each, and how the two are connected. Contents 1. The Gamma Function 1 1.1. Convex Functions 1 1.2. The Gamma Function 4 2. Values of the Riemann Zeta Function 8 3. Characterization of the Zeta Function 9 Acknowledgments 11 References 11 1. The Gamma Function In this paper we will start by characterizing the Gamma function. Its devel- opment is motivated by the desire for a smooth extension of the factorial func- tion to R. Recall the recursive definition of the factorial function, namely that (n + 1)! = (n + 1)n! and 0! = 1. Since it is only defined on the integers, there are many possible continuous extensions to R. However, since we know that en < n! < nn, we can imagine narrowing our options using some notion of "fast" growth. Fortunately, it turns out that we can find a unique extension using the concept of convexity. 1.1. Convex Functions. Recall from calculus that a twice differentiable function is called convex if f 00(x) ≥ 0 for all x. We would like an extension of this definition to functions that are not differentiable. Definition 1.1. For a function f(x) define f(x1) − f(x2) φf (x1; x2) = x1 − x2 which can be thought of as the slope of the secant line between two points. We say that f(x) is convex on an interval if for every x3 on the interval, φ(x1; x3) is a monotonically increasing function of x1. Date: August 29, 2013. 1 2 PAUL DUNCAN Alternatively, f(x) is convex if, for all x1, x2, x3,Ψf ≥ 0, where Ψf is defined as φf (x1; x3) − φf (x2; x3) (1.2) Ψf (x1; x2; x3) = x1 − x2 (x − x )f(x ) + (x − x )f(x ) + (x − x )f(x ) (1.3) = 3 2 1 1 3 2 2 1 3 : (x1 − x2)(x2 − x3)(x3 − x1) Remark 1.4. The values of φf (x1; x2) and Ψf (x1; x2; x3) are invariant under per- mutation of their arguments. First we will need some tools to determine whether a given function is convex. Theorem 1.5. The sum of convex functions, limit function of a convergent se- quence of convex functions, and sum of a convergent series of convex functions are all convex. Proof. From the second statement of the definition of convexity we can see that for two functions f(x) and g(x) if Ψf (x1; x2; x3) ≥ 0 and Ψg(x1; x2; x3) ≥ 0 then Ψf+g(x1; x2; x3) = Ψf (x1; x2; x3) + Ψg(x1; x2; x3) ≥ 0. If we look at the same inequality for a sequence of convex functions f1(x); f2(x)::: that converges to f(x) we see that Ψf (x1; x2; x3) = lim Ψf (x1; x2; x3) ≥ 0 ; n!1 n so f(x) is also convex. The third part of the theorem then follows from the first two since a series is a sequence of partial sums. Definition 1.6. We say that a function is weakly convex on an interval if for every x1+x2+:::+xn 1 x1; x2; :::; xn on that interval f( n ) ≤ n (f(x1) + f(x2) + ::: + f(xn)). Lemma 1.7. If a function is weakly convex and continuous, then it is convex. Proof. Suppose f(x) is weakly convex. Choose x2 < x1 numbers in the interval and 0 ≤ p ≤ n two integers. Apply the definition of weak convexity in the case where p numbers have the value x1 and the other n − p have the value x2. We get p p p p f x + 1 − x ≤ f(x ) + 1 − f(x ) : n 1 n 2 n 1 n 2 Assume f(x) is continuous and let t be a real number such that t 2 [0; 1]. Choose a p sequence of rational numbers converging to t. We can write each number as n , so we can apply the above equation. But f(x) is continuous, so we can take the limit of the sequence to obtain f(t(x1) + (1 − t)x2) ≤ tf(x1) + (1 − t)f(x2) : We want to show that for any x1; x2; x3, we have Ψf (x1; x2; x3) ≥ 0. We can assume x2 < x3 < x1 since we can permute the arguments of Ψf , so the denominator for expression (1.3) is positive. Set t = x3−x2 . We know that t 2 (0; 1) and x1−x2 1 − t = x1−x3 . So x1−x2 (x3 − x2)x1 + (x1 − x3)x2 tx1 + (1 − t)x2 = = x3 : x1 − x2 Plugging this into the previous equation we get x3 − x2 x1 − x3 f(x3) ≤ f(x1) + f(x2) : x1 − x2 x1 − x2 THE GAMMA FUNCTION AND THE ZETA FUNCTION 3 It is now clear that the numerator of (1.3) is also positive, so f(x) must be convex. Remark 1.8. The continuous condition is necessary, but in order to construct an example of a function that is weakly convex, but not convex, one must use the Axiom of Choice. The converse of the lemma is also true. Definition 1.9. We say f(x) is log convex on an interval if log(f(x)) is defined and convex on that interval. Remark 1.10. If a function is log convex, then it is also convex. From Theorem 1.5 we know that the product of log convex functions is log convex. More surprisingly, the same turns out to be true for their sum. Theorem 1.11. Let f(x) and g(x) be log convex functions defined on a common interval. Then f(x) + g(x) is also log convex. Proof. It is enough to prove the statement for weakly convex functions, since we can add continuity to then obtain the theorem. Assume f(x) and g(x) are weakly log convex. Then both are positive and for every x1; x2 x + x 1 log f( 1 2 ) ≤ (log f(x ) + log f(x )) ; 2 2 1 2 or x + x (f( 1 2 ))2 ≤ f(x )f(x ) : 2 1 2 Similarly, x + x (g( 1 2 ))2 ≤ g(x )g(x ) : 2 1 2 We need to show x + x x + x (f( 1 2 ) + g( 1 2 ))2 ≤ (f(x ) + g(x ))(f(x ) + g(x )) : 2 2 1 1 2 2 Let S = f(a; b; c) 2 R3jac − b2 ≥ 0g. We have x + x x + x (f(x ); f( 1 2 ); f(x2)); (g(x ); g( 1 2 ); g(x )) 2 S; 1 2 1 2 2 and we want to show their sum is in S. We will show this by showing that S is closed under addition. By the quadratic formula we know that ac − b2 ≥ 0 () ax2 + bx + c ≥ 0 8x 2 R ; so S = f(a; b; c) 2 R3jax2 + bx + c ≥ 0 8x 2 Rg, which is clearly closed under addition. Theorem 1.12. If φ(x) is a positive continuous function defined on the interior of the integration interval, then Z b φ(t)tx−1 dt a is a log convex function of x for every interval on which the proper or improper integral exists. 4 PAUL DUNCAN Proof. Suppose f(t; x) is defined and continuous for t 2 [a; b] and arbitrary x, and also log convex as a function of x. For n 2 Z and h = (b − a)=n, construct Fn(x) = h(f(a; x) + f(a + h; x) + ::: + f(a + (n − 1)h; x)) : Since Fn(x) is a sum of log convex functions, it is log convex. If we take the limit as n goes to infinity, we get Z b f(t; x)dt : a If the integral is improper, then it is the limit of a sequence of log convex functions, x−1 so it is also log convex. The theorem is the case in which f(t; x) = φ(t)t . With log convexity, we have enough to define a unique Γ(x). 1.2. The Gamma Function. R 1 −t x−1 Definition 1.13. We define Γ(x) = 0 e t dt for x > 0. Note that the integral converges for positive real x. Theorem 1.14. Suppose a function f(x) satisfies the following three conditions: (1) f(x + 1) = xf(x) (2) The domain of f(x) contains all x > 0, and f(x) is log convex for these x (3) f(1) = 1. Then nxn! f(x) = lim : n!1 x(x + 1):::(x + n) Moreover, Γ(x) satisfies (1), (2), (3) for x > 0. In particular, (1), (2), (3) charac- terize Γ(x) uniquely. Proof. First, we will show that Γ(x) satisfies the three conditions. Integration by parts shows that Γ(x) satisfies the first condition. Z δ Z δ −t x −t x δ −t x−1 Γ(x + 1) = lim e t dt = lim (−e t j + x e t dt) δ!1,!0 δ!1,!0 Z δ = lim (−e−δδx + e−x + x e−ttx−1dt) = xΓ(x) δ!1,!0 The second condition comes from Theorem 1.8 and the third is easily checked. Since the gamma function satisfies the requirements, we know that such a function exists. Let f(x) be a function that satisfies these conditions. Then it is enough to show that f(x) = Γ(x) for x 2 (0; 1], since the first condition then implies that they agree on the full domain.
Load more

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]
  • 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.
    [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]
  • 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].
    [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]
  • Sums of Powers and the Bernoulli Numbers Laura Elizabeth S
    Eastern Illinois University The Keep Masters Theses Student Theses & Publications 1996 Sums of Powers and the Bernoulli Numbers Laura Elizabeth S. Coen Eastern Illinois University This research is a product of the graduate program in Mathematics and Computer Science at Eastern Illinois University. Find out more about the program. Recommended Citation Coen, Laura Elizabeth S., "Sums of Powers and the Bernoulli Numbers" (1996). Masters Theses. 1896. https://thekeep.eiu.edu/theses/1896 This is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Masters Theses by an authorized administrator of The Keep. For more information, please contact [email protected]. THESIS REPRODUCTION CERTIFICATE TO: Graduate Degree Candidates (who have written formal theses) SUBJECT: Permission to Reproduce Theses The University Library is rece1v1ng a number of requests from other institutions asking permission to reproduce dissertations for inclusion in their library holdings. Although no copyright laws are involved, we feel that professional courtesy demands that permission be obtained from the author before we allow theses to be copied. PLEASE SIGN ONE OF THE FOLLOWING STATEMENTS: Booth Library of Eastern Illinois University has my permission to lend my thesis to a reputable college or university for the purpose of copying it for inclusion in that institution's library or research holdings. u Author uate I respectfully request Booth Library of Eastern Illinois University not allow my thesis
    [Show full text]
  • Notes on Riemann's Zeta Function
    NOTES ON RIEMANN’S ZETA FUNCTION DRAGAN MILICIˇ C´ 1. Gamma function 1.1. Definition of the Gamma function. The integral ∞ Γ(z)= tz−1e−tdt Z0 is well-defined and defines a holomorphic function in the right half-plane {z ∈ C | Re z > 0}. This function is Euler’s Gamma function. First, by integration by parts ∞ ∞ ∞ Γ(z +1)= tze−tdt = −tze−t + z tz−1e−t dt = zΓ(z) Z0 0 Z0 for any z in the right half-plane. In particular, for any positive integer n, we have Γ(n) = (n − 1)Γ(n − 1)=(n − 1)!Γ(1). On the other hand, ∞ ∞ Γ(1) = e−tdt = −e−t = 1; Z0 0 and we have the following result. 1.1.1. Lemma. Γ(n) = (n − 1)! for any n ∈ Z. Therefore, we can view the Gamma function as a extension of the factorial. 1.2. Meromorphic continuation. Now we want to show that Γ extends to a meromorphic function in C. We start with a technical lemma. Z ∞ 1.2.1. Lemma. Let cn, n ∈ +, be complex numbers such such that n=0 |cn| converges. Let P S = {−n | n ∈ Z+ and cn 6=0}. Then ∞ c f(z)= n z + n n=0 X converges absolutely for z ∈ C − S and uniformly on bounded subsets of C − S. The function f is a meromorphic function on C with simple poles at the points in S and Res(f, −n)= cn for any −n ∈ S. 1 2 D. MILICIˇ C´ Proof. Clearly, if |z| < R, we have |z + n| ≥ |n − R| for all n ≥ R.
    [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]
  • Special Values of the Gamma Function at CM Points
    Special values of the Gamma function at CM points M. Ram Murty & Chester Weatherby The Ramanujan Journal An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan ISSN 1382-4090 Ramanujan J DOI 10.1007/s11139-013-9531-x 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media New York. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Ramanujan J DOI 10.1007/s11139-013-9531-x Special values of the Gamma function at CM points M. Ram Murty · Chester Weatherby Received: 10 November 2011 / Accepted: 15 October 2013 © Springer Science+Business Media New York 2014 Abstract Little is known about the transcendence of certain values of the Gamma function, Γ(z). In this article, we study values of Γ(z)when Q(z) is an imaginary quadratic field. We also study special values of the digamma function, ψ(z), and the polygamma functions, ψt (z).
    [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]