Various Old and New Results in Classical Arithmetic by Special Functions
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VARIOUS OLD AND NEW RESULTS IN CLASSICAL ARITHMETIC BY SPECIAL FUNCTIONS A thesis submitted To Kent State University in partial Fulfillment of the requirements for the Degree of Master of Science By Michael Henry May 2018 c Copyright All rights reserved Except for previously published materials Thesis written by Michael Andrew Henry B.A., The University of Maryland College Park, 2009 M.S., Kent State University 2018 Approved by Gang Yu _____________________________, Advisor Andrew Tonge_________________________, Chair, Department of Mathematical Sciences James L. Blank_________________________, Dean, College of Arts and Sciences Table of Contents Page List of Tables . iv Acknowledgments . v 1. INTRODUCTION . 1 2. BASIC NUMBER THEORY RESULTS . 2 3. IRRATIONALITY . 5 4. TRANSCENDENCE . 14 5. ELEMENTARY CONTINUED FRACTIONS . 20 6. LAMBERT SERIES, #-FUNCTIONS, AND LANDAU’S RECIPROCAL FIBONACCI SUM............................................ 30 7. FIBONACCI POLYNOMIALS AND MORE LANDAU SUMS . 39 8. LATERAL LANDAU SUMS . 44 9. SPECULATIONS AND FINAL REMARKS . 47 iii List of Tables Table Page 5.1 Continuant polynomials . 22 7.1 Fibonacci polynomials . 42 7.2 Values of qn(k) .................................... 43 8.1 A glimpse of the spectrum . 46 iv Acknowledgments I would like to thank Dr. Gang Yu for his assistance in a reading course in number theory that was the basis of this work; I am very grateful for the degree of independence that he has given me throughout my studies. I would like to thank Dr. Ulrike Vorhauer for her encouragement and enthusiasm for me to continue. Furthermore, she has a clarity of thought that is enviable and that has benefited the final form of this thesis tremendously. I have been influenced also by many stimulating conversations with Dr. Omar De la Cruz Cabrera (who is both a statistician and a logician) and Dr. Morley Davidson, the latter having, what I would call, a deeply holistic perspective of mathematics. Also, thanks is due to Dr. Artem Zvavitch, who introduced me to yet another layer of beauty that lies behind mathematical analysis. This work is dedicated to M.E.T. v 1: INTRODUCTION We will prove Dirichlet’s rational approximation theorem, which frames our discussion and approach to Diophantine approximation. We proceed to prove that Liouville’s number L is tran- scendental. Then we complement these results with some theory of simple continued fractions and related results. We state the order of approximation of e, which by Roth’s theorem gives us e is a transcendental number. At this point, we enter what is probably best described as the in- tersection of classical analysis and classical arithmetic, with special attention given to continued fractions. We generalize a result of Landau, giving rise to what we will call Landau sums. These sums generate a countable spectrum of irrational numbers that are computable in quadratic time due to their representation as theta functions; it is still open as to whether these numbers are tran- scendental. Altering the Landau sums we also get another class of sums that we call lateral Landau sums. These sums also give a spectrum of numbers that we speculate are irrational. These results sit nicely within the intersection of classical analysis and classical number theory. We say a few words about this relationship and end with some conjectures. 1 2: BASIC NUMBER THEORY RESULTS The following facts make our exposition of arithmetic self contained. For more a more com- plete and detailed treatment consult [10] or [13]. Definition 1. Let a and b be integers, b 6= 0: Then we say b divides a if bq = a for some integer q: We abbreviate this by writing bja: Definition 2. Let a and b be integers, say a 6= 0: Then the largest positive integer d such that dja and djb is called the greatest common divisor and is denoted gcd(a; b) = d: Definition 3. Suppose that a and b are integers such that gcd(a; b) = 1: Then we say that a and b are coprime or, synonymously, relatively prime. Theorem 1. Let a; b; and c be integers. Then if bjac and gcd(a; b) = 1; then bjc: Proof. Immediate from definitions. Theorem 2 (Division Lemma). Given integers a and b > 0, then there exist unique integers q and r such that a = bq + r; 0 ≤ r < b; (2.1) where r = 0 if and only if bja: 2 Proof. Consider the sequence of integers :::; a − 2b; a − b; a; a + b; a + 2b; ::: : (2.2) From this sequence select the smallest non-negative element and denote it r: Hence, r = a − qb and r = 0 or 0 < r < b, otherwise r was not chosen to be least non-negative element. To prove uniqueness of q and r, suppose that there is another pair q0; r0 satisfying the same conditions. Without loss of generality, suppose r0 > r: Therefore, 0 < r0 − r < b and r0 − r = (q0 − q)b; implying bj(r0 − r): But this is impossible since 0 < r0 − r < b. Therefore, r0 = r and q0 = q: Applying Theorem 2 repeatedly, we have Euclid’s Algorithm, a process worthy of study in its own right. We describe explicitly the algorithm and prove an important property in the next theorem. Theorem 3 (Euclid’s Algorithm). Given integers a; b with b > 0, let a = bq1 + r1; 0 < r1 < b; b = r1q2 + r2; 0 < r2 < r1; r1 = r2q3 + r3; 0 < r3 < r2 . rk−2 = rk−1qk + rk; 0 < rk < rk−1; rk−1 = rk + qk+1: Then gcd(a; b) = rk; the least non-zero remainder in the algorithm described above. Proof. Induction. 3 Theorem 4 (Bezout’s Lemma). Let a; b with b > 0 be integers. Then there exists integers x; y such that ax + by = gcd(a; b): Proof. Follows from Theorem 3. Definition 4. We call an integer p > 1 prime if there is no divisor d of p such that 1 < d < p: Theorem 5. Every integer n > 1 can be expressed as a product of primes. Proof. If n is prime, then we are done. If n is not prime, then it can be factored into n = n1n2 where 1 < n1; n2 < n: If n1 and n2 are prime, we are finished. Otherwise, repeat process with n1 and n2: This process must terminate because each factor is greater than 1 and less than n. Theorem 6 (Unique factorization). Let n be positive integer. Suppose that p1; :::; pk are ordered m1 mk u1 uj and distinct primes, and likewise for p1; :::; pj: Then if n = p1 ··· pk = p1 ··· pj ; then k = j mi ul and pi = pl if and only if i = l: Proof. Follows from the fact that any two non-zero integers have a greatest common divisor. As- sume non-uniqueness and derive a contradiction. Theorem 7 (Euclid’s Lemma). If a prime number p divides the integer ab, then pja or pjb: Proof. If p does not divide a; then gcd(a; p) = 1: Hence, b = b gcd(a; p) = gcd(ab; pb): Since pjab by assumption, and we see pjb: Theorem 8 (Euclid). There are infinitely many prime numbers. Proof. Assume there are only finitely many prime numbers p1; :::; pk: Consider the number N = p1p2 ··· pk + 1; a number clearly larger than 1. Hence, there is a prime pi for some 1 ≤ i ≤ k such that pijN: Therefore, pijp1 ··· pk, so pijN. Hence, pij(N − p1 ··· pk) = 1; a contradiction. 4 3: IRRATIONALITY We start with the definition and some concrete observations. Definition 5. A number α is rational if α = p=q for p; q 2 Z; q 6= 0: Otherwise, α is irrational. It is readily observed that rational numbers are dense in R: In other words, for any chosen > 0 and real number α we can find some rational number p=q such that jα − p=qj < . Informally, it is just there exists a rational number arbitrarily close to any real number we choose. If we are interested in properties of the numbers we are approximating with, density alone does not tell us much. But we could ask this same question, now applying restrictions to . For example, suppose we make the value of dependent on p=q. This changes the problem to a study of the relative closeness of p=q to α: Only a minor change in the problem can lead to some surprising results. Indeed, making = 1=q2 opens up some deep mathematics that gives information about the local behavior of numbers in the continuum. An early theorem was stated and proven by Dirichlet in 1842 (see [20], p.34). We replicate it after first introducing some useful functions. We introduce the floor function and its companion, which we will use in the proof of the following theorem. Definition 6. For any real number α, we denote bαc = k to be the unique integer k such that k ≤ α < k + 1: Furthermore, we let fαg = α − bαc: We call these the floor and fractional part of α; respectively. 5 Theorem 9 (Dirichlet). Let α be any real number and N be any positive integer. Then there exists a rational number p=q such that 0 < q ≤ N and p 1 α − ≤ : (3.1) q (N + 1)q j j+1 Proof. First, we write [0; 1) as the union of N + 1 disjoint sub-intervals: writing Ij = [ N+1 ; N+1 ), then N [ [0; 1) = Ij: j=0 Define n o S = fαg; f2αg; :::; fNαg ; then S is a set of N real numbers in [0; 1) and thus each falls into some Ij: Considering extremal intervals first, suppose for some q 2 Z that 1 ≤ q ≤ N where fqαg 2 I0.