Transcendence of Various Infinite Series Applications of Baker's
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Transcendence of Various Infinite Series and Applications of Baker's Theorem by Chester Jay Weatherby A thesis submitted to the Department of Mathematics and Statistics in conformity with the requirements for the degree of Doctor of Philosophy Queen's University Kingston, Ontario, Canada April 2009 Copyright c Chester Jay Weatherby, 2009 Library and Archives Bibliothèque et Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l’édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre référence ISBN: 978-0-494-65340-1 Our file Notre référence ISBN: 978-0-494-65340-1 NOTICE: AVIS: The author has granted a non- L’auteur a accordé une licence non exclusive exclusive license allowing Library and permettant à la Bibliothèque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l’Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans le loan, distribute and sell theses monde, à des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, électronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L’auteur conserve la propriété du droit d’auteur ownership and moral rights in this et des droits moraux qui protège cette thèse. Ni thesis. Neither the thesis nor la thèse ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent être imprimés ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author’s permission. In compliance with the Canadian Conformément à la loi canadienne sur la Privacy Act some supporting forms protection de la vie privée, quelques may have been removed from this formulaires secondaires ont été enlevés de thesis. cette thèse. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n’y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. Abstract We consider various infinite series and examine their arithmetic nature. Se- ries of interest are of the form 1 1 X f(n)A(n) X f(n)A(n) X znA(n) ; ; B(n) B(n) B(n) n=0 n2Z n=0 where f is algebraic valued periodic function, A(x);B(x) 2 Q[x] and z is an algebraic number with jzj ≤ 1. We also examine multivariable extensions 1 X f(n1; : : : ; nk)A1(n1) ··· Ak(nk) B1(n1) ··· Bk(nk) n1;:::;nk=0 and X f(n1; : : : ; nk)A1(n1) ··· Ak(nk) : B1(n1) ··· Bk(nk) n1;:::;nk2Z These series are all very natural things to write down and we would like to understand them better. We calculate closed forms using various techniques. For example, we use relations between Hurwitz zeta functions, digamma func- tions, polygamma functions, Fourier analysis, discrete Fourier transforms, among other objects and techniques. Once closed forms are found, we make use of some of the well-known transcendental number theory including the i theorem of Baker regarding linear forms in logarithms of algebraic numbers to determine their arithmetic nature. In one particular setting, we extend the work of Bundschuh [4] by proving the following series are all transcendental for positive c 2 Q n Z and k a positive integer: X 1 X 1 X 1 X 1 ; ; ; (n2 + c)k (n4 − c4)2k (n6 − c6)2k (n3 ± c3)2k n2Z n2Z n2Z n2Z X 1 X 1 X 1 X 1 ; ; ; : n3 ± c3 n3 − 1 n4 − 1 n6 − 1 n2Z jn|≥2 jn|≥2 jn|≥2 Bundschuh conjectured that the last three series are transcendental, but we offer the first unconditional proofs of transcendence. We also show some conditional results under the assumption of some well- known conjectures. In particular, for Ai(x);Bi(x) 2 Q[x] with each Bi(x) has only simple rational roots, if Schanuel's conjecture is true, the series (avoiding roots of the denominator) 1 X f(n1; : : : ; nk)A1(n1) ··· Ak(nk) B1(n1) ··· Bk(nk) n1;:::;nk=0 is either an effectively computable algebraic number or transcendental. We also show that Schanuel's conjecture implies that the series X A(n) B(n) n2Z is either zero or transcendental, when B(x) has non-integral roots. We develop a general theory, analyzing various infinite series throughout. ii Acknowledgements I would first like to thank Professor M. Ram Murty for his wisdom and helpfulness. His guidance kept this research on track keeping things both interesting and enjoyable. I would also like to thank David Wehlau, Mike Roth, Ernst Kani, and Ole Nielsen from Queen's University and Margaret Beattie from Mount Allison University for their help and guidance over the years as well as Damien Roy and Selim Akl for their helpful comments during the defense. I owe many thanks to my parents, Ken and Shelley Weatherby, for their encouragement over the past 27 years, my fianc´eeJennifer Troy for her pa- tience, support, and encouragement while I have worked on this degree, and of course my housemate for the past 5 years, David Byrne, for being there through the entire academic battle. Finally, all of my friends and family have had some part in helping me along the way. iii Statement of Originality The results of this thesis are original research, except of course where credit has been given. Within each chapter it has been spelled out clearly where methods are new and where methods are being adapted from previous work of others. Due credit has been given to those whose work came before this thesis. iv Contents Abstract i Acknowledgements iii Statement of Originality iv 1 Introduction 1 1.1 Overview and Contributions . 4 2 Infinite sums and the Digamma and Polygamma functions 10 2.1 Preliminaries . 11 2.2 Series with conditional convergence . 13 2.3 Series with absolute convergence . 21 2.4 A family of transcendental series . 27 2.5 Infinite sums and polygamma functions . 31 3 Exponential Polynomial Sums 35 3.1 Convergence of the basic case . 36 3.2 Exponential over a linear factor . 37 3.3 Exponential times a polynomial . 40 3.4 Exponential times a rational function . 42 v 4 Multivariable Zeta-type series 46 4.1 Convergence of the multi-indexed series . 47 4.2 Polynomial forms of digamma values . 49 4.3 Fourier analysis of the series . 52 4.4 Multivariable series of rational functions . 58 5 Summation over Z 61 5.1 Elementary methods of analysis . 62 5.2 A multivariable generalization . 76 5.3 Fourier approach . 79 5.4 Schanuel's conjecture and some examples . 90 6 Future Work 127 Bibliography 131 vi Chapter 1 Introduction A formal Dirichlet series is defined as 1 X an f(s) = ns n=1 where s and an; (n = 1; 2; 3;:::) are complex numbers. The most famous of these series is the Riemann zeta function 1 X 1 ζ(s) = ns n=1 which converges for <(s) > 1. A basic question we can ask is \what do these series converge to?" in the case that they converge. Usually we can only say something specific about special cases. For instance, for n a positive integer, we have B (2π)2n ζ(2n) = (−1)n+1 2n 2(2n)! 1 th where B2n is the 2n Bernoulli number. This equation was known to Euler. The formula shows that the values at even arguments of the Riemann zeta function have the property of being transcendental, which we define now. Definition 1.1. A complex number is said to be algebraic if it is the root of a polynomial with integer coefficients. Definition 1.2. A complex number is said to be transcendental if it is not the root of any polynomial with integer cofficients. All complex numbers are either algebraic or transcendental, and this prop- erty is the central interest of this thesis. Other series that are of of interest are Dirichlet L−series defined by 1 X f(n) L(s; f) := ns n=1 for a periodic function f and a complex number s. Much work on special values of these series has originated from a question of Chowla. In 1969, during a conference on number theory at Stony Brook, S. Chowla [5] asked if there was a rational valued periodic function, that is not identi- cally zero, with prime period p such that the series 1 X f(n) n n=1 converges and vanishes. In 1973, Baker, Birch and Wirsing [3] used Baker's theory of linear forms in logarithms of algebraic numbers to answer this question with the following theorem: Theorem 1.3 If f is a non-vanishing function defined on the integers with algebraic values and period q such that (i) f(r) = 0 if 1 < (r; q) < q, (ii) the 2 qth cyclotomic polynomial is irreducible over Q(f(1); : : : ; f(q)), then 1 X f(n) 6= 0: n n=1 In 2001, Adhikari, Saradha, Shorey, and Tijdeman [1] and then in 2007 Murty and Saradha [18], took things further and showed that for an algebraic valued periodic function, f, the sum 1 X f(n) n n=1 is either zero or transcendental when it converges. In [1] and [18], the authors also examined series of the form 1 1 X A(n) X f(n) and also B(n) B(n) n=0 n=0 where A(x) and B(x) are polynomials with rational coefficients, B(x) having only simple rational roots.