Connected Components of Complex Divisor Functions

CONNECTED COMPONENTS OF COMPLEX DIVISOR FUNCTIONS COLIN DEFANT Princeton University Fine Hall, 304 Washington Rd. Princeton, NJ 08544 X c Abstract. For any complex number c, define the divisor function σc : N ! C by σc(n) = d . djn Let σc(N) denote the topological closure of the range of σc. Extending previous work of the current author and Sanna, we prove that σc(N) has nonempty interior and has finitely many connected components if <(c) ≤ 0 and c 6= 0. We end with some open problems. 2010 Mathematics Subject Classification: Primary 11B05; Secondary 11N64. Keywords: Divisor function; connected component; spiral; nonempty interior; perfect number; friendly number. 1. Introduction One of the most famous and perpetually mysterious mathematical contributions of the ancient Greeks is the notion of a perfect number, a number that is equal to the sum of its proper divisors. Many early theorems in number theory spawned from attempts to understand perfect numbers. Although few modern mathematicians continue to attribute the same theological or mystical sig- nificance to perfect numbers that ancient people once did, these numbers remain a substantial inspiration for research in elementary number theory [2, 3, 5, 11, 14{16, 18]. arXiv:1711.04244v3 [math.NT] 4 May 2018 P A positive integer n is perfect if and only if σ1(n)=n = 2, where σ1(n) = djn d. More generally, for any complex number c, we define the divisor function σc by X c σc(n) = d : djn These functions have been studied frequently in the special cases in which c 2 {−1; 0; 1g. The divisor functions σc with c 2 N have also received a fair amount of attention, especially because of their close connections with Eisenstein series [12, 21]. Ramanujan, in particular, gave several pleasing identities involving divisor functions [10, 19]. E-mail address: [email protected]. 1 2 CONNECTED COMPONENTS OF COMPLEX DIVISOR FUNCTIONS (a) (b) 6 Figure 1. Plots of σc(n) for 1 ≤ n ≤ 10 . The left image shows a plot when c = −1:3 + i. The right image shows a plot when c = −2:3 + 5i. Around 100 A.D., Nicomachus stated that perfect numbers represent a type of \harmony" be- tween “deficient" and \abundant" numbers. A positive integer n is deficient if σ1(n)=n < 2 and is abundant if σ1(n)=n > 2. A simple argument shows that σ1(n)=n = σ−1(n), so the function σ−1 is often called the \abundancy index." The abundancy index is also used to define friendly and solitary numbers. Two distinct positive integers m and n are said to be friends with each other if σ−1(m) = σ−1(n). An integer is friendly if it is friends with some other integer, and it is solitary otherwise. For example, any two perfect numbers are friends with each other. There are several extremely difficult problems surrounding the notions of friendly and solitary numbers. For example, it is not known whether 10 is friendly or solitary [1, 15, 17, 22]. Motivated by the difficult problems related to perfect and friendly numbers, Laatsch studied σ−1(N), the range of σ−1. He showed that σ−1(N) is a dense subset of the interval [1; 1) and asked if σ−1(N) is in fact equal to the set Q \ [1; 1) [13]. Weiner answered this question in the negative, showing that (Q \ [1; 1)) n σ−1(N) is also dense in [1; 1) [23]. Recently, the author asked what could be said about the topological properties of the ranges of the divisor functions σc for general complex numbers c. More specifically, he has studied sets of the form σc(N), where the overline denotes the topological closure. The fractal-like appearances of these sets make them difficult to describe mathematically (see Figure 1). Nevertheless, the simple number-theoretic definition of the function σc lends itself to proofs of some nontrivial and often surprising facts concerning σc(N). For example, the following theorems summarize some of the main results in this area. Theorem 1.1 is proven in [7] while the various parts of Theorem 1.2 are all proven in [4]. In what follows, let N (c) denote the (possibly infinite) number of connected components of σc(N). Theorem 1.1 ( [7]). Let r 2 R. There exists a constant η ≈ 1:8877909 such that N (−r) = 1 (that is, σ−r(N) is connected) if and only if r 2 (0; η]. Theorem 1.2 ( [4]). Let c 2 C. (1) The set σc(N) is bounded if and only if <(c) < −1. CONNECTED COMPONENTS OF COMPLEX DIVISOR FUNCTIONS 3 (2) If <(c) > 0 or c = 0, then σc(N) is discrete. If <(c) ≤ 0 and c 6= 0, then σc(N) has no isolated points. (3) The set σc(N) is dense in C if and only if −1 ≤ <(c) ≤ 0 and c 62 R. (4) We have N (c) ! 1 as <(c) ! −∞. (5) If <(c) ≤ −3:02, then N (c) ≥ 2 (so σc(N) is disconnected). The current author has proven analogues of Theorem 1.1 for other families of arithmetic functions that are relatives of the divisor functions (the so-called \unitary divisor functions" and \alternating divisor functions") [6, 9]. He has also proven analogues of this theorem in which one considers the images of particular subsets of N under σ−r [8]. Suppose r > 1 is real. In [20], Sanna provides an algorithm for computing σ−r(N) when r is known with sufficient precision. As a consequence, he proves that N (−r) < 1. This serves as a nice complement to part (4) of Theorem 1.2, which implies that N (−r) ! 1 as r ! 1. Very recently, Zubrilina [24] has obtained asymptotic estimates for N (−r), showing that " # 1 1 r20=9 log log r 1 π(r) + 1 ≤ N (−r) ≤ exp 1 + + O ; 2 2 (log r)29=9 log r − log log r log r where π(r) is the number of primes less than or equal to r. She has also shown that there is no real number r such that N (−r) = 4 (for more about this, see Section 4 below). Part (3) of Theorem 1.2 provides a new natural number-theoretic way of generating countable dense subsets of the plane. Furthermore, these sets come equipped with their own natural enumeration. For example, σ−0:5+i(1); σ−0:5+i(2); σ−0:5+i(3);::: is an enumeration of a dense subset of C (possibly with repetitions). Part (4) of Theorem 1.2 is also noteworthy for the following reason. Let us write c = a + bi for a; b 2 R. Imagine fixing b and letting a decrease without bound. The crude estimate X c X c X a a jσc(n) − 1j = d ≤ jd j = d = O(2 ); 1<djn 1<djn 1<djn which holds uniformly in n, shows that the diameter of the set σc(N) decays at least exponentially in a as a ! −∞. Part (4) of the theorem tells us that while σc(N) is \shrinking" in diameter, it continues to \bud off” into more and more connected components. The preceding paragraph does not exclude the possibility that N (c) = 1 for some complex numbers c. Indeed, parts (1) and (2) of Theorem 1.2 tell us that N (c) = 1 when <(c) > 0 or c = 0. According to Theorem 1.1 and part (3) of Theorem 1.2, N (c) = 1 if −1 ≤ <(c) ≤ 0 and c 6= 0. This leads us to the central question of this article. Question 1.1. Suppose <(c) < −1. Does the set σc(N) have finitely many connected components? As mentioned above, Sanna answered this question in the affirmative in the case in which c is real [20]. The primary purpose of this article is to prove the following theorem, fully answering Question 1.1. Theorem 1.3. If <(c) ≤ 0 and c 6= 0, then σc(N) has finitely many connected components. 4 CONNECTED COMPONENTS OF COMPLEX DIVISOR FUNCTIONS Along the way, we prove that the topological subspace σc(N) of C has nonempty interior (this is equivalent to the claim that σc(N) contains a disk of positive radius) when <(c) < −1 and c 62 R (see Theorem 3.1 below). This seemingly innocuous claim, whose proof is apparently more elusive than one might expect, is key to proving Theorem 1.3. We already know by Sanna's result and part (3) of Theorem 1.2 that Theorem 1.3 is true when c 2 R or −1 ≤ <(c) ≤ 0. Indeed, if −1 ≤ <(c) ≤ 0 and c 62 R, then σc(N) = C is connected! Therefore, we may assume that c 2 C n R and <(c) < −1 in the next section. Since σc(n) = σc(n) for all positive integers n, we assume without loss of generality that =(c) > 0 (in this one instance, bars denote complex conjugation, not topological closure). The paper is organized as follows. In Section 2, we collect some technical yet useful information c about the geometry of the spiral Sc = flog(1 + t ): t ≥ Tcg (for some fixed Tc ≥ 1) and its close c approximation Sec = ft : t ≥ Tcg. The reason for doing this comes from the simple fact that log(σc(p)) lies on Sc for all sufficiently large primes p. In Section 3, we prove that σc(N) contains a disk of positive radius centered at 1. The proof of Theorem 1.3 then boils down to a relatively painless argument showing that the connected components of σc(N) contain disks whose radii are uniformly bounded away from 0 (this proves the desired result since part (1) of Theorem 1.2 tells us that σc(N) is bounded).

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