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The Circle Method Applied to Goldbach's Weak Conjecture

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The Circle Method Applied to Goldbach's Weak Conjecture CARLETON UNIVERSITY SCHOOL OF MATHEMATICS AND STATISTICS HONOURS PROJECT TITLE: The Circle Method Applied to Goldbach’s Weak Conjecture AUTHOR: Michael Cayanan SUPERVISOR: Brandon Fodden DATE: August 27, 2019 Acknowledgements First and foremost, I would like to thank my supervisor Brandon Fodden not only for all of his advice and guidance throughout the writing of this paper, but also for being such a welcoming and understanding instructor during my first year of undergraduate study. I would also like to thank the second reader Kenneth Williams for graciously providing his time and personal insight into this paper. Of course, I am also very grateful for the countless professors and instructors who have offered me outlets to explore my curiosity and to cultivate my appreciation of mathematical abstraction through the courses that they have organized. I am eternally indebted to the love and continual support provided by my family and my dearest friends throughout my undergraduate journey. They were always able to provide assurance and uplift me during my worst and darkest moments, and words cannot describe how grateful I am to be surrounded with so much compassion and inspiration. This work is as much yours as it is mine, and I love you all 3000. THE CIRCLE METHOD APPLIED TO GOLDBACH'S WEAK CONJECTURE Michael Cayanan August 27, 2019 Abstract. Additive number theory is the branch of number theory that is devoted to the study of repre- sentations of integers subject to various arithmetic constraints. One of, if not, the most popular question that additive number theory asks arose in a letter written by Christian Goldbach to Leonhard Euler, fa- mously dubbed the Goldbach conjecture. Dated June 7th, 1742, the translated conjecture states that \Every number n which is a sum of two primes is a sum of as many primes including unity as one wishes (up to n), and that every number > 2 is a sum of three primes". The primary goal of this paper is to tackle a much easier problem concerning the representation of every odd integer greater than 5 as a sum of three primes (appropriately named the weak Goldbach conjecture, or the ternary Goldbach conjecture), under the frame- work of the famed \circle method", mainly attributed to Hardy, Littlewood, Ramanujan and Vinogradov. The circle method will allow us to translate the weak conjecture into a complex-analytic problem relying on the estimation of integrals performed over a constructed partition of [0; 1]. We will also explore the extent and the limitations of the circle method when applied to Goldbach's binary conjecture, as well as its an application concerning Waring's problem. 1. Notation and Number Theory Preliminaries Before we introduce the circle method, some preliminary definitions and results from elementary number theory will be briefly recalled. As usual, Z denotes the set of all integers and N the set of all positive integers. In what follows, p will almost exclusively denote a prime number and P will denote the set of all prime numbers. We say that a nonzero integer d divides an integer n if n = cd for some c 2 Z, and this will be denoted by d j n. The greatest common divisor of integers a and b (not both 0) will always be presented as (a; b). If (a; b) = 1, then we say that a and b are relatively prime, or co-prime. Definition 1.1. An arithmetic function f is a complex valued function defined on the positive integers. If f is not identically zero and f(mn) = f(m)f(n) whenever (m; n) = 1, then f is a multiplicative function. We now briefly describe two elementary arithmetic functions and some of their useful properties. x1 xk Definition 1.2. For n > 1 with prime factorization n = p1 : : : pk , we define the M¨obiusfunction µ(n) as 8 k <>(−1) if x1 = ··· = xk = 1 µ(n) = :>0 otherwise. If n = 1, then we set µ(1) = 1. 2 THE CIRCLE METHOD APPLIED TO GOLDBACH'S WEAK CONJECTURE 3 Definition 1.3. For n ≥ 1, the Euler totient function φ(n) is equal to the number of positive integers k ≤ n such that (n; k) = 1. That is, n X φ(n) = 1: k=1 (n;k)=1 It is well known that both µ and φ are multiplicative functions. Evaluating the totient function at prime k k k−1 k 1 Q 1 powers yields φ(p ) = p − p = p 1 − p , and more generally, φ(n) = n pjn 1 − p . Summing over all of the divisors d of n for φ(n) and µ(n), this gives 8 X X <>1 if n = 1 φ(d) = n, and µ(d) = djn djn :>0 if n > 1: All of these results may be found in any elementary number theory texts such as Apostol [1] (Chapters 2.2 and 2.3) or Nathanson [15] (Sections A.5 and A.6). Proposition 1.4. Given " > 0, we have n1−" < φ(n) < n for all sufficiently large n. Proof. Clearly, φ(n) < n, 8n > 1. Now, we want to show that n1−" lim = 0: n!1 φ(n) Since φ is multiplicative, it suffices to show that the result holds for prime powers. Observe that for every p prime p, we have that p−1 ≤ 2, hence pm(1−") pm(1−") p pm(1−") 2 = = ≤ : φ(pm) pm − pm−1 p − 1 pm pm" Thus, pm(1−") lim = 0: pm!1 φ(pm) For notational convenience, for x 2 R, we will set e(x) := e2πix. We will now define an exponential sum and prove an important reformulation using the M¨obius function which we will make use of. Definition 1.5. Let q; n 2 Z and q ≥ 1. We define the Ramanujan sum as the exponential sum q X kn c (n) = e : q q k=1 (k;q)=1 Theorem 1.6. The Ramanujan sum can be expressed in the form X q c (n) = µ d: q d dj(q;n) Furthermore, if (q; n) = 1, we have cq(n) = µ(q). 4 Michael Cayanan P Proof. As noted above, we know that dj1 µ(d) = 1, hence q q q X kn X kn X kn X c (n) = e = e · 1 = e µ(d): q q q q k=1 k=1 k=1 dj(k;q) (k;q)=1 (k;q)=1 Therefore, q q=d q=d X X kn X X dln X X ln c (n) = µ(d) e = µ(d) e = µ(d) e : q q q q=d djq k=1 djq l=1 djq l=1 djk k=dl Pd ln Define fd(n) := l=1 e d , and note that 8 <>d if d j n fd(n) = :>0 otherwise. From this, we then have that q=d X X ln c (n) = µ(d) e q q=d djq l=1 X = µ(d)fq=d(n) djq X q = µ f (n) d d djq X q = µ d d djq djn X q = µ d: d dj(q;n) If (q; n) = 1, then the formula simplifies to cq(n) = µ(q), as required. Using this result, we may now show that cq(n) is a multiplicative function in q. For if q1; q2 ≥ 1 with (q1; q2) = 1, we have X q1 X q2 X q1 q2 cq1 (n) · cq2 (n) = µ d1 µ d2 = µ µ d1d2: d1 d2 d1 d2 d1j(q1;n) d2j(q2;n) d1j(q1;n) d2j(q2;n) Writing d = d1d2 and noting that µ is multiplicative, we rewrite the previous summation as X q1q2 X q1q2 µ d = µ d = c (n): d d q1·q2 dj(q1;n)·(q2;n) dj(q1·q2;n) To conclude this introductory section, we introduce the last few number theoretic results and definitions which will greatly aid us in the proof of Vinogradov's Theorem. Excluding Theorem 1.11 and its following remark, the proofs of all of these results may be found in the appendix of Nathanson [15]. THE CIRCLE METHOD APPLIED TO GOLDBACH'S WEAK CONJECTURE 5 P1 Theorem 1.7 (Euler Products). Let f be a multiplicative function. Suppose that the series n=1 f(n) converges absolutely. Then 1 1 X Y Y X f(n) = (1 + f(p) + f(p2) + ··· ) = 1 + f(pk): n=1 p p k=1 If f is completely multiplicative, then 1 X Y f(n) = (1 − f(p))−1: n=1 p Definition 1.8. Let f be any complex valued function, and let g be any positive function. Then f = O(g) if and only if there exists C > 0 such that jf(x)j ≤ Cg(x) for all x large enough. The constant C is called the implied constant. We may occasionally write f g (notation due to Vinogradov) in place of f = O(g) as well. Definition 1.9. We say that f is asymptotic to g, denoted f ∼ g, if f(x) lim = 1: x!1 g(x) Theorem 1.10 (Partial Summation). Let a(n) be an arithmetic function, and suppose that f is a function P with continuous derivative on an interval [y; x] with 0 ≤ y < x. Define the sum function A(x) = n≤x a(n), and A(x) = 0 if x < 1. Then X Z x a(n)f(n) = A(x)f(x) − A(y)f(y) − A(t)f 0(t)dt: y<n≤x y In particular, taking 0 < y < 1 yields X Z x a(n)f(n) = A(x)f(x) − A(t)f 0(t)dt: n≤x 1 P Theorem 1.11 (Prime Number Theorem).
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