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Exponential Sums and Applications in Number Theory and Analysis

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Exponential Sums and Applications in Number Theory and Analysis Faculty of Sciences Department of Mathematics Exponential sums and applications in number theory and analysis Frederik Broucke Promotor: Prof. dr. J. Vindas Master's thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Mathematics Academic year 2017{2018 ii Voorwoord Het oorspronkelijke idee voor deze thesis was om het bewijs van het ternaire vermoeden van Goldbach van Helfgott [16] te bestuderen. Al snel werd mij duidelijk dat dit een monumentale opdracht zou zijn, gezien de omvang van het bewijs (ruim 300 bladzij- den). Daarom besloot ik om in de plaats de basisprincipes van de Hardy-Littlewood- of cirkelmethode te bestuderen, de techniek die de ruggengraat vormt van het bewijs van Helfgott, en die een zeer belangrijke plaats inneemt in de additieve getaltheorie in het algemeen. Hiervoor heb ik gedurende het eerste semester enkele hoofdstukken van het boek \The Hardy-Littlewood method" van R.C. Vaughan [37] gelezen. Dit is waarschijnlijk de moeilijkste wiskundige tekst die ik tot nu toe gelezen heb; de weinige tussenstappen, het gebrek aan details, en zinnen als \one easily sees that" waren vaak frustrerend en demotiverend. Toch heb ik doorgezet, en achteraf gezien ben ik echt wel blij dat ik dat gedaan heb. Niet alleen heb ik enorm veel bijgeleerd over het onderwerp, ik heb ook het gevoel dat ik beter of vlotter ben geworden in het lezen van (moeilijke) wiskundige teksten in het algemeen. Na het lezen van dit boek gaf mijn promotor, professor Vindas, me de opdracht om de idee¨en en technieken van de cirkelmethode toe te passen in de studie van de functie van Riemann, een \pathologische" continue functie die een heel onregelmatig puntsgewijs gedrag vertoont. De combinatie van deze twee onderwerpen leidde uiteindelijk tot de keuze voor exponenti¨ele sommen als onderwerp van de thesis, met een grote nadruk op de toepassingen voor de cirkelmethode en de studie van Riemanns functie. Persoonlijk vind ik dat je een wiskundig bewijs op twee niveaus kan begrijpen. Ener- zijds kan je elke overgang of stap afzonderlijk begrijpen, inzien waarom regel n + 1 volgt uit regel n voor elke n. Anderzijds (en misschien belangrijker) kan je het bewijs op een globaal niveau begrijpen: wat zijn de achterliggende idee¨en en motivaties, waarom doet men iets op die manier, waarom werkt dit? In mijn thesis heb ik geprobeerd om beide niveaus voldoende te belichten. Ik heb geprobeerd zo veel mogelijk resultaten rigoureus en volledig bewezen. Het is hierbij nagenoeg onmogelijk om (soms vervelende) technische details te vermijden. Toch hoop ik dat de techniciteiten de achterliggende idee¨en niet verbloemen, en dat de lezer door het lezen van deze thesis inzicht en appreciatie kan krijgen voor het mooie onderwerp van exponenti¨ele sommen en hun toepassingen in vele fascinerende problemen. Ik heb er bewust voor gekozen om deze masterproef in het Engels te schrijven. Dit lijkt mij passend voor de masterproef als (mini-)wetenschappelijk onderzoek, aangezien de overgrote meerderheid van de wetenschappelijke literatuur in het Engels wordt ge- schreven. Ten slotte wil ik mijn promotor, professor Vindas, bedanken. Hij stond me bij met iii iv VOORWOORD advies, hielp me met problemen tijdens het lezen en het schrijven, en voorzag nuttige referenties. Anderzijds gaf hij me ook voldoende vrijheid om mijn eigen ding te doen, wat ik zeer op prijs stel. Ten slotte maakte hij veel tijd vrij in zijn drukke schema om de talrijke voorlopige versies nauwgezet na te lezen. De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen en delen van de masterproef te kopi¨eren voor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze masterproef. Frederik Broucke 30 mei 2018 Contents Voorwoord iii List of symbols vii 1 Introduction 1 2 Exponential sums 3 2.1 Some elementary estimates . 3 2.2 Characters and Gauss sums . 4 2.2.1 Ramanujan sums . 6 2.2.2 Separable Gauss sums and primitive characters . 7 2.2.3 Quadratic Gauss sums . 9 2.3 kth-power Gauss sums . 13 2.3.1 kth-powers modulo pl ......................... 13 2.3.2 The exponential sums S(q; a) and S(q; a; b) . 15 2.4 Weyl sums . 18 2.4.1 Weyl's method . 18 2.4.2 Vinogradov's method . 22 2.4.3 Vinogradov's mean value theorem . 28 3 The Hardy-Littlewood method 35 3.1 Generalities . 35 3.2 Waring's problem . 37 3.2.1 Approximating the generating function . 37 3.2.2 The singular series . 43 3.2.3 The singular integral . 49 3.2.4 The contribution from the major arcs . 51 3.2.5 The contribution from the minor arcs . 53 3.3 The ternary Goldbach problem . 55 3.3.1 The contribution from the major arcs . 55 3.3.2 The contribution from the minor arcs . 60 4 The Vinogradov-Korobov zero-free region for ζ 65 5 Riemann's non-differentiable function 73 5.1 Introduction . 73 5.2 Behaviour at rational points . 75 5.3 Behaviour at irrational points . 78 5.3.1 The upper bound for α(ρ)....................... 78 5.3.2 The lower bound for α(ρ)....................... 79 v vi CONTENTS 6 Conclusion 85 A Nederlandse samenvatting 87 B Populariserende samenvatting 91 B.1 Oplossingen van vergelijkingen detecteren . 91 C Additional theorems 95 C.1 Diophantine approximation . 95 C.2 The Poisson summation formula . 99 C.3 The continuous wavelet transform . 100 List of symbols Symbol Description N The set of natural numbers including zero, f0; 1; 2;:::g. Z The set of integers. Q The set of rational numbers. R The set of real numbers. C The set of complex numbers. R× The unit group of a ring R. d j n d divides n. α α α α+1 p k n p exactly divides n; p j n and p - n. (n; m) The greatest common divisor of n and m. d(n) The number of positive divisors of n. !(n) The number of distinct prime factors of n. µ(n) The M¨obiusfunction; µ(n) = (−1)!(n) if n is square-free, µ(n) = 0 otherwise. '(n) Euler's totient function; the number of a; 1 ≤ a ≤ n for which (a; n) = 1. Λ(n) The von Mangoldt function; Λ(n) = log p if n = pα for some prime p, Λ(n) = 0 otherwise. (n) The unit function for Dirichlet convolution; (1) = 1 and (n) = 0 if n 6= 1. P #(x) The first Chebyshev function; #(x) = p≤x log p. P (x) The second Chebyshev function; (x) = n≤x Λ(n). P π(x) The prime counting function; π(x) = p≤x 1. indg a The index of a. If g is a primitive root mod q, and (a; q) = 1, then m indg a is the unique m mod '(q) such that a ≡ g mod q. e(z) The complex exponential with period 1; e(z) = exp(2πiz). G(n; χ) The Gauss sum associated with n and the character χ mod q: q X G(n; χ) = χ(m)e(mn=q). m=1 q X cq(n) The Ramanujan sum: cq(n) = e(mn=q). m=1 (m;q)=1 q X amk S (q; a) The kth-power Gauss sum: e . k q m=1 q X amk + bm S (q; a; b) A variant of the kth-power Gauss sum: e . k q m=1 vii viii LIST OF SYMBOLS Symbol Description M X Sf (M) Weyl sums; Sf (M) = e(f(m)). m=1 [x] The integer part of x; the unique integer such that [x] ≤ x < [x]+1. fxg The fractional part of x; fxg = x − [x]. dxe The ceiling function of x; the unique integer such that dxe − 1 < x ≤ dxe. kxk The distance from x to the nearest integer. f(x) = O(g(x)) f(x) ≤ Cg(x) for some absolute constant C. f(x) = o(g(x)) lim f(x)=g(x) = 0. f(x) = Ω(g(x)) The negation of f(x) = o(g(x)). f(x) g(x) f(x) = O(g(x)). f(x) g(x) g(x) = O(f(x)), g non-negative. f(x) ∼ g(x) lim f(x)=g(x) = 1. f(x) g(x) f(x) g(x) and f(x) g(x). Concerning the asymptotic notations: the range in which the inequalities or limits hold is usually clear from the context; if needed it will be specified, e.g. f(x) = o(g(x)) as x ! 0 means that f(x) lim = 0: x!0 g(x) If the implicit constant in the notation is not absolute, but depends on some additional parameter, this will be notated via a subscript, e.g. f(x) k g(x) where f and g are functions which also depend on some additional parameter k, means that there is a constant Ck, only depending on k, such that f(x) ≤ Ckg(x). We use the convention that for a complex number z, its argument arg z is a number in the interval ] − π; π]. We use the notation f^ for the Fourier transform of a function f, defined as follows Z f^(y) = f(x)e−ixy dx: R Sometimes (e.g. in summations), the expression min(X; 1=0) will occur; this will be taken to be X. Chapter 1 Introduction We denote the complex exponential with period 1 by e, that is 8z 2 C : e(z) = exp(2πiz). This thesis is devoted to the study of exponential sums, namely sums of the form N X e(f(n)); f real-valued n=1 and their applications.
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