Sums of distinct divisors

Transcription

1 Oberlin College February 3, / 50

2 The anatomy of integers 2 / 50 What does it mean to study the anatomy of integers? Some natural problems/goals: Study the prime factors of integers, their size and their quantity. Obtain good bounds for the number of integers with certain properties (e.g., those with only large prime factors). Understand the distribution of of integers in a given interval.

3 Integers with dense Several papers in this area study the set of integers with z-dense. Let 1 = d 1 (n) < d 2 (n) < < d τ(n) (n) = n denote the sequence of of an integer n. Definition For z 2, an integer n is z-dense if max 1 i<τ(n) d i+1 (n) d i (n) z. Example. 6 is 2-dense, since 2 1, 3 2, / 50

4 Integers with dense F (n) := Theorem (Tenenbaum) For n 2, { 1 n = 1 max{dp (d) : d n, d > 1} n 2. F (n) n = max d i+1 (n) 1 i<τ(n) d i (n). 4 / 50 So n has z-dense if F (n) nz.

5 Integers with dense Let D(x, z) = #{n x : n is z-dense}. Tenenbaum obtained upper and lower bounds for D(x, z), which were later improved by Saias to D(x, z) x log z log x. 5 / 50

6 Proof sketch For any fixed z 2, let D(x) = {n x : F (n) nz}. Let D(x, y) = #{n D(x) : P + (n) y}. Then D(x, y) = 1 + p min(y,h(x)) D(x/p, p) + ET. (E.g., Can take h(x) = x so that ET is negligible.) Smoothed version: D (x, y) = min(y, x) 1 D (x/t, t) dt log t. Can *almost* take D (x, y) = xρ(u 1)/ log x, where ρ is defined by uρ (u) + ρ(u 1) = 0 and ρ(u) = 1 for u 1. 6 / 50

7 Two natural applications Our talk will focus on two natural applications of Tenenbaum s work on integers with z-dense : 1 How often is it the case that every m in [1, n] can be written as a sum of of n? 7 / 50

8 Two natural applications Our talk will focus on two natural applications of Tenenbaum s work on integers with z-dense : 1 How often is it the case that every m in [1, n] can be written as a sum of of n? 2 How often does the polynomial x n 1 have a divisor of every degree between 1 and n in Z[x]? 8 / 50

9 Definition positive integer n is practical if every m with 1 m σ(n) can be written as a sum of of n. Example. n = 6 Divisors: 1, 2, 3, 6 Nonexample. n = 10 Divisors: 1, 2, 5, 10 9 / 50

10 : a short history Theorem (Erdős, 1950) Let P R(X) := #{n X : n is practical}. Then P R(X) = o(x). 10 / 50

11 : a short history Hausman and Shapiro, 1983: P R(X) X (1 1/ log 2)2, β = (log X) β 2 = Margenstern, 1984: P R(X) Tenenbaum, 1986: X exp(α(log log X) 2 ), α = 1 + ε 2 log 2 = X X(log log X)(log log log X) P R(X). (log X)(log log X) 4.21 log X 11 / 50

12 : a short history Theorem (Saias, 1997) For all X 2, P R(X) X log X. 12 / 50

13 vs. ϕ- Definition positive integer n is if every m with 1 m n can be written as d D ϕ(d), where D is a subset of of n. Note: Since x n 1 = d n Φ d(x), this is equivalent to the condition that x n 1 has at least one divisor of every degree between 1 and n. 13 / 50

14 example Example. n = 6 Divisors: 1, 2, 3, 6 ϕ values: 1, 1, 2, 2 Nonexample. n = 66 is practical but not. Divisors: 1, 2, 3, 6, 11, 22, 33, 66 ϕ values: 1, 1, 2, 2, 10, 10, 22, / 50

15 example Example. n = 6 Divisors: 1, 2, 3, 6 ϕ values: 1, 1, 2, 2 Nonexample. n = 66 is practical but not. Divisors: 1, 2, 3, 6, 11, 22, 33, 66 ϕ values: 1, 1, 2, 2, 10, 10, 22, 22 Exercise. Every even is practical. 15 / 50

16 Counting the number of s We can prove the following analogue of Saias result for the : Theorem (T., 2013) Let F (X) = #{n X : n is }. Then F (X) X log X. 16 / 50

17 key obstruction The proofs of Saias et al. relied heavily on the following: Theorem (Stewart, 1954) Let n = p e 1 1 pe j j, n > 1, with p 1 < p 2 < < p j prime and e i 1 for i = 1,..., j. Then n is practical iff for all i = 1,..., j, p i σ(p e 1 1 pe i 1 i 1 ) + 1. Unfortunately, there s no simple method for building up from smaller ones. 17 / 50 Example (2 31 1) is, but none of the 3 2, 3 2 5, , , are.

18 Proof of the upper bound Instead, we devise the following workaround: Definition Let n = p e 1 1 pe k k. Let m i = p e 1 1 pe i i. We define an integer n to be weakly if the inequality p i+1 m i + 2 holds for all i. Lemma Every number is weakly. 18 / 50 Note: The converse does not hold. For example, 45 is not but it is weakly.

19 Proof of the upper bound To prove our theorem, we consider two cases: If n is even & then p i+1 m i + 2 σ(m i ) + 1 for all i 1. Hence, each m i satisfies the inequality in Stewart s Condition, so n is practical. 19 / 50

20 Proof of the upper bound To prove our theorem, we consider two cases: If n is even & then p i+1 m i + 2 σ(m i ) + 1 for all i 1. Hence, each m i satisfies the inequality in Stewart s Condition, so n is practical. On the other hand, observe that for every n (0, X], there is a unique k such that 2 k n (X, 2X]. Then, if n is odd &, 2 k n will be practical. 20 / 50

21 Proof of the upper bound To prove our theorem, we consider two cases: If n is even & then p i+1 m i + 2 σ(m i ) + 1 for all i 1. Hence, each m i satisfies the inequality in Stewart s Condition, so n is practical. On the other hand, observe that for every n (0, X], there is a unique k such that 2 k n (X, 2X]. Then, if n is odd &, 2 k n will be practical. Thus, F (X) P R(2X) X log X, by Saias Theorem. 21 / 50

22 Lower Bound Proof Sketch Saias obtains his lower bound by comparing the set of practical with the set of integers with 2-dense : Definition n integer n is 2-dense if max 1 i τ(n) 1 d i+1 (n) d i (n) 2. Note: ll integers with 2-dense are practical, but the same cannot be said about the. For example, n = 66 is 2-dense but it is not. 22 / 50

23 Lower Bound Proof Sketch We obtain our lower bound by comparing the set of with the set of integers with strictly 2-dense : Definition d n integer n is strictly 2-dense if max i+1 (n) 1<i<τ(n) 1 <2 and d 2 (n) d 1 (n) = 2 = d τ(n)(n) d τ(n) 1 (n). d i (n) It turns out that all strictly 2-dense integers are. 23 / 50

24 Lower Bound Proof Sketch Goal: Show that a positive proportion of 2-dense integers are strictly 2-dense, except for some possible obstructions at small primes. 24 / 50

25 Lower Bound Proof Sketch Goal: Show that a positive proportion of 2-dense integers are strictly 2-dense, except for some possible obstructions at small primes. 1 First find an upper bound for the number of integers up to X that are 2-dense but not strictly 2-dense: k>c m (2 k 1,2 k ) m 2-dense p (2 k 1,2 k+1 ) p prime j X/mp mpj 2-dense P (j)>p / 50

26 Lower Bound Proof Sketch 26 / 50 Goal: Show that a positive proportion of 2-dense integers are strictly 2-dense, except for some possible obstructions at small primes. 1 First find an upper bound for the number of integers up to X that are 2-dense but not strictly 2-dense: k>c m (2 k 1,2 k ) m 2-dense p (2 k 1,2 k+1 ) p prime j X/mp mpj 2-dense P (j)>p 2 Use Brun s sieve and other classical techniques from multiplicative number theory to show that the number counted above is ε X log X. 1.

27 Lower Bound Proof Sketch Show that a subset of the strictly 2-dense integers is in one-to-one correspondence with a positive proportion of the 2-dense integers with obstructions at k < C. Corollary (T., 2013) For X sufficiently large, we have #{n X : n is practical but not } X log X. Moreover, we also have #{n X : n is but not practical} X log X. 27 / 50

28 n asymptotic for the practicals Theorem (Weingartner, 2015) There exists a positive constant c such that for X 3, P R(X) = cx ( ( )) log log X 1 + O. log X log X 28 / 50

29 Proof sketch Idea: Count all integers n x according to their practical part: n = (p e 1 1 pe j j )(pe j+1 j+1 pe k k ) = m r where m = p e 1 1 pe j j is practical but p e 1 1 pe j j pe j+1 j+1 is not. 29 / 50

30 Proof sketch Idea: Count all integers n x according to their practical part: n = (p e 1 1 pe j j )(pe j+1 j+1 pe k k ) = m r where m = p e 1 1 pe j j is practical but p e 1 1 pe j j pe j+1 j+1 Then x = Φ(x/m, σ(m) + 1) m x m practical where Φ(x, y) = #{n x : p n p > y}. is not. 30 / 50

31 Proof sketch Since x = m x m practical Φ(x/m, σ(m) + 1) 31 / 50

32 Proof sketch Since x = m x m practical 1 = m practical 1 m Φ(x/m, σ(m) + 1) p σ(m)+1 ( 1 1 ) p 32 / 50

33 Proof sketch Since we have 0 = x = m x m practical 1 = m practical m practical 1 m Φ(x/m, σ(m) + 1) p σ(m)+1 ( 1 1 ) p Φ(x/m, σ(m) + 1) x m p σ(m)+1 ( 1 1 p) 33 / 50

34 Proof sketch 0 = m practical Observe that: Φ(x/m, σ(m) + 1) x m p σ(m)+1 ( 1 1 p) 34 / 50

35 Proof sketch 0 = m practical Φ(x/m, σ(m) + 1) x m p σ(m)+1 Observe that: ( ) Φ(x, y) e γ xω log x ( log y 1 1 ) p p y ( 1 1 ) e γ p log y p y log(σ(m) + 1) log(2m) ( 1 1 p) 35 / 50

36 Proof sketch 36 / 50 0 = m practical Φ(x/m, σ(m) + 1) x m p σ(m)+1 Observe that: ( ) Φ(x, y) e γ xω log x ( log y 1 1 ) p p y ( 1 1 ) e γ p log y p y log(σ(m) + 1) log(2m) Use partial summation, get an integral equation, apply a Laplace transform, and invert the Laplace transform. ( 1 1 p)

37 n asymptotic for the s Theorem (Pomerance, T., Weingartner, 2016) There exists a positive constant C such that for X 2, CX 1 F (X) = 1+O. log X log X 37 / 50

38 Proof Sketch: Starters Definition starter is a number m such that either m/p + (m) is not or P + (m) 2 m. 38 / 50 Note: number n is said to have starter m if m is a starter, m is an initial divisor of n, and n/m is squarefree. Examples: 4 is the only starter with squarefull part 4. There are only 3 starters with squarefull part 49: 294 = , 1470 = , 735 = There are infinitely many starters with squarefull part 9.

39 Proof Sketch Proof Sketch: 1 Partition s according to their starters: n = mb. 2 Use Weingartner s machinery to show that, for any fixed m 1, there exist sequences of real c m and r m such that ( ) x B m (x) = c m log x + O x r m log 2, x where B m := #{ n x with starter m}. 3 Show that c m and r m are finite. 39 / 50

40 Estimating the asymptotic constant 40 / 50 We can use Sage to compute F (X)/ X log X : X F (X) F (X)/(X/ log X) F (X)/(Li(X)) Table: Ratios for s

41 Let S f (n) = d n f(d) = f 1(n). Definition positive integer n is called f-practical if for every positive integer m S f (n) there is a set D of of n for which m = d D f(d) holds. Example f = I: practical. 41 / 50 Example f = ϕ:.

42 Densities of f-practicals Example ll positive integers are τ-practical. 42 / 50

43 Densities of f-practicals Example ll positive integers are τ-practical. Example The set of λ-practical has asymptotic density / 50

44 Densities of f-practicals Example ll positive integers are τ-practical. Example The set of λ-practical has asymptotic density 0. Example Let g : N N, where g(1) = 1, g(2 k ) = 2, and g(p k ) = 3 for all p 3 and all k 1. The set of g-practical has asymptotic density 1/2. 44 / 50

45 Densities of f -practicals Theorem (Schwab, T., 2016) For each n N, there is a function fn such that the asymptotic density of fn -practical in N is 1 45 / 50 ϕ(n) n.

46 Densities of f -practicals Theorem (Schwab, T., 2016) For each n N, there is a function fn such that the asymptotic density of fn -practical in N is 1 Corollary (Schwab, T., 2016) The densities of f -practical sets are dense in [0, 1]. 46 / 50 ϕ(n) n.

47 Other results We also: Classified the multiplicative functions f for which the f-practical can be completely determined via a Stewart-like criterion; 47 / 50

48 Other results We also: Classified the multiplicative functions f for which the f-practical can be completely determined via a Stewart-like criterion; Proved Chebyshev-type bounds for certain f-practical sets; 48 / 50

49 Other results We also: Classified the multiplicative functions f for which the f-practical can be completely determined via a Stewart-like criterion; Proved Chebyshev-type bounds for certain f-practical sets; Classified the additive functions f for which all positive integers are f-practical. 49 / 50

50 Thank you! 50 / 50