DOCSLIB.ORG

Notes on Riemann's Zeta Function

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. Therefore, we 1 1 have | z+n |≤ n−R for |z| < R and n ≥ R. It follows that for n0 > R, we have ∞ ∞ ∞ ∞ cn |cn| |cn| 1 ≤ ≤ ≤ |cn|. z + n |z + n| n − R n0 − R n=n0 n=n0 n=n0 n=n0 X X X X cn Hence, the series n>R z+n converges absolutely and uniformly on the disk {z | |z| < R} and defines there a holomorphic function. It follows that ∞ cn P n=0 z+n is a meromorphic function on that disk with simple poles at the points of S in ∞ cn P {z | |z| < R}. Therefore, n=0 z+n is a meromorphic function with simple poles at the points in S. Therefore, for any −n ∈ S we have P c c c f(z)= n + m = n + g(z) z + n z + m z + n −m∈XS−{n} where g is holomorphic at −n. This implies that Res(f, −n)= cn. Going back to Γ, we have ∞ 1 ∞ Γ(z)= tz−1e−t dt = tz−1e−t dt + tz−1e−t dt. Z0 Z0 Z1 Clearly, the second integral converges for any complex z and represents an entire function. On the other hand, since the exponential function is entire, its Taylor series converges uniformly on compact sets in C, and we have 1 1 ∞ (−1)p tz−1e−t dt = tz−1 tp dt p! 0 0 p=0 ! Z Z X ∞ (−1)p 1 ∞ (−1)p 1 = tp+z−1 dt = p! p! z + p p=0 0 p=0 X Z X for any z ∈ C. Therefore, ∞ ∞ (−1)p 1 Γ(z)= tz−1e−t dt + p! z + p 1 p=0 Z X for any z in the right half-plane. By 1.2.1, the right side of this equation defines a meromorphic function on the complex plane with simple poles at 0, −1, −2, −3, − · · ·. Hence, we have the following result. 1.2.2. Theorem. The function Γ extends to a meromorphic function on the complex plane. It has simple poles at 0, −1, −2, −3, ···. The residues of Γ are −p are given by (−1)p Res(Γ, −p)= p! for any p ∈ Z+. This result combined with the above calculation immediately implies the follow- ing functional equation. 1.2.3. Proposition. For any z ∈ C we have Γ(z +1)= zΓ(z). NOTESONRIEMANN’SZETAFUNCTION 3 1.3. Another functional equation. Let Re z > 0. Then ∞ Γ(z)= tz−1e−tdt. Z0 Let Re p> 0 and Re q> 0. Then, by change of variable t = u2, we get ∞ ∞ 2 Γ(p)= tp−1e−tdt =2 e−u u2p−1du. Z0 Z0 Analogously we have ∞ 2 Γ(q)=2 e−v v2q−1dv. Z0 Hence, it follows that ∞ ∞ 2 2 Γ(p)Γ(q)=4 e−(u +v )u2p−1v2q−1du dv, Z0 Z0 and by passing to the polar coordinates by u = r cos ϕ, v = r sin ϕ, we have ∞ π 2 2 Γ(p)Γ(q)=4 e−r r2(p+q)−1 cos2p−1 ϕ sin2q−1 ϕ dr dϕ Z0 Z0 ∞ π 2 2 = 2 e−r r2(p+q)−1 dr · 2 cos2p−1 ϕ sin2q−1 ϕ dϕ Z0 Z0 ! π 2 = 2Γ(p + q) cos2p−1 ϕ sin2q−1 ϕ dϕ. Z0 We put s = sin2 ϕ in the integral. Then we have π 2 1 2 cos2p−1 ϕ sin2q−1 ϕ dϕ = sq−1(1 − s)p−1ds. Z0 Z0 If we define 1 B(p, q)= sp−1(1 − s)q−1ds Z0 for Re p> 0, Re q> 0, we get the identity Γ(p)Γ(q) B(p, q)= B(q,p)= . Γ(p + q) Here B is Euler’s Beta function. Let x ∈ (0, 1). Then we have Γ(x)Γ(1 − x) 1 Γ(x)Γ(1 − x)= = B(x, 1 − x)= sx−1(1 − s)−xds. Γ(1) 0 u Z Moreover, if we change the variable s = u+1 , the integral becomes 1 ∞ ux−1 u −x du sx−1(1 − s)−xds = 1 − (u + 1)x−1 u +1 (u + 1)2 Z0 Z0 ∞ ux−1 = du. 1+ u Z0 1.3.1. Lemma. For y ∈ (0, 1) we have ∞ u−y π du = . 1+ u sin πy Z0 4 D. MILICIˇ C´ Proof. We cut the complex plane along the positive real axis. On this region we − z y −y define the function z 7−→ 1+z with the argument of z equal to 0 on the upper side of the cut. This function has a first order pole at z = −1 with the residue e−πiy. CR -1 Cε If we integrate this function along a path which goes along the upper side of the cut from ǫ > 0 to R, then along the circle CR of radius R centered at the origin, then along the lower side of the cut from R to ǫ and finally around the origin along the circle Cǫ of radius ǫ, by the residue theorem we get R u−y z−y R u−y z−y du + dz − e−2πiy du − dz =2πie−πiy. 1+ u 1+ z 1+ u 1+ z Zǫ ZCR Zǫ ZCǫ First we remark that for z 6= 0 we have |z−y| = |e−y log z| = e−y Re(log z) = e−y log |z| = |z|−y. Hence, the integrand in the second and fourth integral satisfies z−y |z|−y |z|−y ≤ ≤ . 1+ z |1+ z| |1 − |z|| Hence, for small ǫ we have z−y ǫ1−y dz ≤ 2π ; 1+ z (1 − ǫ) ZCǫ and for large R we have z−y R1−y dz ≤ 2π ; 1+ z (R − 1) ZCR Clearly, this implies that → 0 as R → ∞ and → 0 as ǫ → 0. This implies CR Cǫ that R ∞ u−y R 1 − e−2πiy du =2πie−πiy. 1+ u Z0 NOTESONRIEMANN’SZETAFUNCTION 5 It follows that ∞ u−y eπiy − e−πiy du =2πi 1+ u Z0 and finally ∞ u−y π du = . 0 1+ u sin πy Z This lemma implies that π π Γ(x)Γ(1 − x)= = . sin π(1 − x) sin πx for x ∈ (0, 1). Since both sides are meromorphic, it follows that the following result holds. 1.3.2. Proposition. For all z ∈ C, we have π Γ(z)Γ(1 − z)= . sin πz Since z 7−→ sin πz is an entire function, the right side obviously has no zeros. So, Γ(z) = 0 is possible only at points where z 7−→ Γ(1 − z) has poles. Since the poles of Γ are at 0, −1, −2, ··· , it follows that poles of z 7−→ Γ(1 − z) are at 1, 2, 3, ··· . At these points Γ(n +1)= n! 6= 0. Therefore, we proved the following result. 1.3.3. Theorem. The function Γ has no zeros. 2. Zeta function 2.1. Meromorphic continuation. Riemann’s zeta function ζ is defined by ∞ 1 ζ(z)= nz n=1 X for Re z > 1. In that region the series converges uniformly on compact sets and represents a holomorphic function. If we consider the expression for the Gamma function ∞ Γ(z)= tz−1e−tdt Z0 for Re z > 0, and change the variable t into t = ns for n ∈ N, we get ∞ Γ(z)= nz sz−1e−nsds, Z0 i.e., we have Γ(z) ∞ = tz−1e−ntdt nz Z0 for any n ∈ N and Re z > 0. This implies that, for Re z > 1, we have ∞ ∞ Γ(z)ζ(z)= tz−1e−ntdt n=1 0 X Z ∞ ∞ ∞ e−t ∞ tz−1 = tz−1 e−nt dt = tz−1 dt = dt. 1 − e−t et − 1 0 n=1 ! 0 0 Z X Z Z This establishes the following integral representation for the zeta function. 6 D. MILICIˇ C´ 2.1.1. Lemma. For Re z > 1 we have ∞ tz−1 Γ(z)ζ(z)= dt. et − 1 Z0 This integral can be split in to two parts ∞ tz−1 1 tz−1 ∞ tz−1 Γ(z)ζ(z)= dt = dt + dt.
Load more

Recommended publications
  • 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]
  • 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]
  • Complex Analysis
    Complex Analysis Andrew Kobin Fall 2010 Contents Contents Contents 0 Introduction 1 1 The Complex Plane 2 1.1 A Formal View of Complex Numbers . .2 1.2 Properties of Complex Numbers . .4 1.3 Subsets of the Complex Plane . .5 2 Complex-Valued Functions 7 2.1 Functions and Limits . .7 2.2 Infinite Series . 10 2.3 Exponential and Logarithmic Functions . 11 2.4 Trigonometric Functions . 14 3 Calculus in the Complex Plane 16 3.1 Line Integrals . 16 3.2 Differentiability . 19 3.3 Power Series . 23 3.4 Cauchy's Theorem . 25 3.5 Cauchy's Integral Formula . 27 3.6 Analytic Functions . 30 3.7 Harmonic Functions . 33 3.8 The Maximum Principle . 36 4 Meromorphic Functions and Singularities 37 4.1 Laurent Series . 37 4.2 Isolated Singularities . 40 4.3 The Residue Theorem . 42 4.4 Some Fourier Analysis . 45 4.5 The Argument Principle . 46 5 Complex Mappings 47 5.1 M¨obiusTransformations . 47 5.2 Conformal Mappings . 47 5.3 The Riemann Mapping Theorem . 47 6 Riemann Surfaces 48 6.1 Holomorphic and Meromorphic Maps . 48 6.2 Covering Spaces . 52 7 Elliptic Functions 55 7.1 Elliptic Functions . 55 7.2 Elliptic Curves . 61 7.3 The Classical Jacobian . 67 7.4 Jacobians of Higher Genus Curves . 72 i 0 Introduction 0 Introduction These notes come from a semester course on complex analysis taught by Dr. Richard Carmichael at Wake Forest University during the fall of 2010. The main topics covered include Complex numbers and their properties Complex-valued functions Line integrals Derivatives and power series Cauchy's Integral Formula Singularities and the Residue Theorem The primary reference for the course and throughout these notes is Fisher's Complex Vari- ables, 2nd edition.
    [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]
  • Zeros of Differential Polynomials in Real Meromorphic Functions
    Zeros of differential polynomials in real meromorphic functions Walter Bergweiler∗, Alex Eremenko† and Jim Langley June 30, 2004 Abstract We investigate when differential polynomials in real transcendental meromorphic functions have non-real zeros. For example, we show that if g is a real transcendental meromorphic function, c R 0 n ∈ \{ } and n 3 is an integer, then g0g c has infinitely many non-real zeros.≥ If g has only finitely many poles,− then this holds for n 2. Related results for rational functions g are also considered. ≥ 1 Introduction and results Our starting point is the following result due to Sheil-Small [15] which solved a longstanding conjecture. 2 Theorem A Let f be a real polynomial of degree d. Then f 0 + f has at least d 1 distinct non-real zeros which are not zeros of f . − In the special case that f has only real roots this theorem is due to Pr¨ufer [14, Ch. V, 182]; see [15] for further discussion of the result. The following Theorem B is an analogue of Theorem A for transcendental meromorphic functions. Here “meromorphic” will mean “meromorphic in the complex plane” unless explicitly stated otherwise. A meromorphic function is called real if it maps the real axis R to R . ∪{∞} ∗Supported by the German-Israeli Foundation for Scientific Research and Development (G.I.F.), grant no. G-643-117.6/1999 †Supported by NSF grants DMS-0100512 and DMS-0244421. 1 Theorem B Let f be a real transcendental meromorphic function with 2 finitely many poles. Then f 0 + f has infinitely many non-real zeros which are not zeros of f .
    [Show full text]
  • 4. Complex Analysis, Rational and Meromorphic Asymptotics
    ANALYTIC COMBINATORICS P A R T T W O 4. Complex Analysis, Rational and Meromorphic Asymptotics http://ac.cs.princeton.edu ANALYTIC COMBINATORICS P A R T T W O 4. Complex Analysis, Rational and Meromorphic functions Analytic Combinatorics •Roadmap Philippe Flajolet and •Complex functions Robert Sedgewick OF •Rational functions •Analytic functions and complex integration CAMBRIDGE •Meromorphic functions http://ac.cs.princeton.edu II.4a.CARM.Roadmap Analytic combinatorics overview specification A. SYMBOLIC METHOD 1. OGFs 2. EGFs GF equation 3. MGFs B. COMPLEX ASYMPTOTICS SYMBOLIC METHOD asymptotic ⬅ 4. Rational & Meromorphic estimate 5. Applications of R&M COMPLEX ASYMPTOTICS 6. Singularity Analysis desired 7. Applications of SA result ! 8. Saddle point 3 Starting point The symbolic method supplies generating functions that vary widely in nature. − + √ + + + ...+ − ()= ()= − ()= ()= ... − − − − − − ()= ()= ln + / ( )( ) ...( ) ()= − − − − − Next step: Derive asymptotic estimates of coefficients. − []() [ ]() [ ]() + [ ]()=β ∼ ∼ ∼ (ln ) / √ / / − − − [ ]() [ ]()=ln [ ]() ∼ ! ∼ √ Classical approach: Develop explicit expressions for coefficients, then approximate Analytic combinatorics approach: Direct approximations. 4 Starting point Catalan trees Derangements Construction G = ○ × SEQ( G ) Construction D = SET (CYC>1( Z )) ln ()= ()= − OGF equation () EGF equation − − + √ − Explicit form of OGF Explicit form of EGF = ()= − − ( ) Expansion ()= ( ) Expansion ()= − − − ! " #
    [Show full text]
  • Lecture 8 - the Extended Complex Plane Cˆ, Rational Functions, M¨Obius Transformations
    Math 207 - Spring '17 - Fran¸coisMonard 1 Lecture 8 - The extended complex plane C^, rational functions, M¨obius transformations Material: [G]. [SS, Ch.3 Sec. 3] 1 The purpose of this lecture is to \compactify" C by adjoining to it a point at infinity , and to extend to concept of analyticity there. Let us first define: a neighborhood of infinity U is the complement of a closed, bounded set. A \basis of neighborhoods" is given by complements of closed disks of the form Uz0,ρ = C − Dρ(z0) = fjz − z0j > ρg; z0 2 C; ρ > 0: Definition 1. For U a nbhd of 1, the function f : U ! C has a limit at infinity iff there exists L 2 C such that for every " > 0, there exists R > 0 such that for any jzj > R, we have jf(z)−Lj < ". 1 We write limz!1 f(z) = L. Equivalently, limz!1 f(z) = L if and only if limz!0 f z = L. With this concept, the algebraic limit rules hold in the same way that they hold at finite points when limits are finite. 1 Example 1. • limz!1 z = 0. z2+1 1 • limz!1 (z−1)(3z+7) = 3 . z 1 • limz!1 e does not exist (this is because e z has an essential singularity at z = 0). A way 1 0 1 to prove this is that both sequences zn = 2nπi and zn = 2πi(n+1=2) converge to zero, while the 1 1 0 sequences e zn and e zn converge to different limits, 1 and 0 respectively.
    [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]
  • The Gamma Function the Interpolation Problem
    The Interpolation Problem; the Gamma Function The interpolation problem: given a function with values on some discrete set, like the positive integer, then what would the value of the function be defined for all the reals? Euler wanted to do this for the factorial function. He concluded that Z 1 n! = (− ln x)ndx: 0 Observe that (in modern calculation) Z 1 1 − ln x dx = −x ln x + x 0 0 = 1 + lim x ln x x!0+ ln x = 1 + lim 1 x!0+ x 1 x = 1 + lim 1 x!0+ − x2 −x = 1 + lim = 1 − 0 = 1: x!0+ 1 Which is consistent with 1! = 1. Now consider the following integration, Z 1 (− ln x)ndx: 0 We apply the (Leibnitz version) of the integration by parts formula R u dv = uv − R v du. dv = 1 v = x 1 u = (− ln x)n du = n(− ln x)n−1 − dx x Z 1 1 Z 1 n n n−1 (− ln x) dx = x(− ln x) + n (− ln x) dx 0 0 0 Since (in modern notation and using the same limit techniques as above) lim x(ln x)n = 0 x!0+ 1 (Euler would have probably said that 0(ln(0))n = 0.) We have: Z 1 Z 1 (− ln x)ndx = n (− ln x)n−1dx 0 0 which is consistent with our inductive definition of the factorial function: n! = n · (n − 1)!; so for integers the formula clearly works. Then he used the transformation t = − ln x so that x = e−t to obtain the following.
    [Show full text]
  • 20 the Modular Equation
    18.783 Elliptic Curves Spring 2017 Lecture #20 04/26/2017 20 The modular equation In the previous lecture we defined modular curves as quotients of the extended upper half plane under the action of a congruence subgroup (a subgroup of SL2(Z) that contains a b a b 1 0 Γ(N) := f c d 2 SL2(Z): c d ≡ ( 0 1 )g for some integer N ≥ 1). Of particular in- ∗ terest is the modular curve X0(N) := H =Γ0(N), where a b Γ0(N) = c d 2 SL2(Z): c ≡ 0 mod N : This modular curve plays a central role in the theory of elliptic curves. One form of the modularity theorem (a special case of which implies Fermat’s last theorem) is that every elliptic curve E=Q admits a morphism X0(N) ! E for some N 2 Z≥1. It is also a key ingredient for algorithms that use isogenies of elliptic curves over finite fields, including the Schoof-Elkies-Atkin algorithm, an improved version of Schoof’s algorithm that is the method of choice for counting points on elliptic curves over a finite fields of large characteristic. Our immediate interest in the modular curve X0(N) is that we will use it to prove the first main theorem of complex multiplication; among other things, this theorem implies that the j-invariants of elliptic curve E=C with complex multiplication are algebraic integers. There are two properties of X0(N) that make it so useful. The first, which we will prove in this lecture, is that it has a canonical model over Q with integer coefficients; this allows us to interpret X0(N) as a curve over any field, including finite fields.
    [Show full text]
  • 1. the Gamma Function 1 1.1. Existence of Γ(S) 1 1.2
    CONTENTS 1. The Gamma Function 1 1.1. Existence of Γ(s) 1 1.2. The Functional Equation of Γ(s) 3 1.3. The Factorial Function and Γ(s) 5 1.4. Special Values of Γ(s) 6 1.5. The Beta Function and the Gamma Function 14 2. Stirling’s Formula 17 2.1. Stirling’s Formula and Probabilities 18 2.2. Stirling’s Formula and Convergence of Series 20 2.3. From Stirling to the Central Limit Theorem 21 2.4. Integral Test and the Poor Man’s Stirling 24 2.5. Elementary Approaches towards Stirling’s Formula 25 2.6. Stationary Phase and Stirling 29 2.7. The Central Limit Theorem and Stirling 30 1. THE GAMMA FUNCTION In this chapter we’ll explore some of the strange and wonderful properties of the Gamma function Γ(s), defined by For s> 0 (or actually (s) > 0), the Gamma function Γ(s) is ℜ ∞ x s 1 ∞ x dx Γ(s) = e− x − dx = e− . x Z0 Z0 There are countless integrals or functions we can define. Just looking at it, there is nothing to make you think it will appear throughout probability and statistics, but it does. We’ll see where it occurs and why, and discuss many of its most important properties. 1.1. Existence of Γ(s). Looking at the definition of Γ(s), it’s natural to ask: Why do we have re- strictions on s? Whenever you are given an integrand, you must make sure it is well-behaved before you can conclude the integral exists.
    [Show full text]