In this area, there is no shortage of open problems. We will simply exhibit a few of them.
Until recently, it was not known if there exist infinitely many pairs of positive integers (m, n) such that σ(m) = φ(n). This was resolved in the affirmative in 2010 by Ford, Luca, and Pomerance [58].
It is still not known if there are any odd perfect numbers. But one can easily show that the sum of the reciprocals of all the perfect numbers
Problems on Chapter 7 109
(including the potential odd ones) is convergent. It is not known whether there exist infinitely many amicable numbers (that is, members of an ami- cable pair). Pomerance [115] showed that the set of n ≤x which are part of an amicable pair is of cardinality O(xexp(−(logx)1/3)) and inferred that the sum of the reciprocals of the members of amicable pairs is convergent.
Let s(n) =σ(n)−n and for k≥ 1, let sk(n) be the k-th fold composition of the function s(n) with itself evaluated at n. If sk(n) = n, then nis said to be part of an aliquot cycle of length k. For example, perfect numbers are in aliquot cycles of length 1, and amicable numbers are in aliquot cycles of length 2. It is conjectured that every positive integer n belongs to an aliquot cycle of finite lengthkfor some k, but we are nowhere near proving anything of this sort. Recent results about aliquot cycles can be found in the paper by Kobayashi, Pollack, and Pomerance [91].
In 1907, Carmichael [21] announced that whenevern is such that n = φ(m) for some m, then n=φ(m) for at least two distinct values of m. In other words, the set φ−1(n) = {m : φ(m) = n} never has cardinality 1.
However, his proof was flawed and since then this is known asCarmichael’s conjecture. This is still an open problem, although Kevin Ford [55] showed that the smallest counterexample, if any, is larger than 101010. (Moral: Don’t attempt to find one with your personal computer !)
Máakowski and Schinzel [103] conjectured that the inequalityσ(φ(n))≥ n/2 holds for all positive integers n. At present, it is only known that σ(φ(n))> cn holds for all n with c = 1/39.4 (see Ford [56]) and that for each ε >0 the inequality σ(φ(n))>(c1−ε)nlog log logn holds for almost all positive integers n, where c1 = e−γ (see Luca and Pomerance [101]).
Lehmer conjectured that if φ(n)|n−1, thenn is prime. It is known that the set of counterexamples n ≤ x to Lehmer’s conjecture has cardinality O(x1/2(logx)−1/2+o(1)) (see Luca and Pomerance [100]). Kˇr´ıˇzek and Luca [92] modified Lehmer’s question and showed that if φ2(n) | n2 −1, then n= 1,2 or 3 (see Problem 13.8).
Problems on Chapter 7
Problem 7.1. Show that the average value of Ω(n)−ω(n) on [1, x] tends to a constant as x tends to infinity.
Problem 7.2. Given a positive integer k, let σk(n) =
d|n
dk
be the sum of the k-th powers of the divisors of n. Find the average value of σk(n) on the interval [1, x]. (Hint: Find the average value of σk(n)/nk by first proving that σk(n)/nk =
d|n1/dk. Then use Abel’s summation formula.)
Problem 7.3. Show that 2ω(n)≤d(n)≤2Ω(n) for all positive integersn.
Problem 7.4. Show thatd(mn)≤d(m)d(n)for all positive integersmand n.
Problem 7.5. Show that the conclusion of Proposition 7.16 remains valid if we replace log logn by f(n), wheref :R+ →R+ is any function which is increasing for x > x0 and tends to infinity with x.
Problem 7.6. Prove the following strengthening of Lemma 7.18: The set A={n:|ω(n)−log logn|>(log logn)2/3}
is of asymptotic density zero. (Hint: Show that if x is large and n ∈ A∩ [x/logx, x], then |ω(n)−log logx|2 > (log logx)4/3. Now use the Tur´an- Kubilius inequality in order to get that the number of such positive integers n≤x is x/(log logx)1/3=o(x) as x→ ∞.)
Problem 7.7. Show that almost all positive integersn have a prime factor p >logn.
Problem 7.8. Consider the set A of those positive integers which do not contain the digit 7 in their decimal expansion.
(i) Show that A is of zero density.
(ii) Use (i) to show that almost all integers contain each of the digits 0,1,. . . ,9.
(iii) Show that the sum of the reciprocals of the elements ofAconverges.
Problem 7.9. Let 0< a1< a2 <ã ã ã be a sequence of positive integers. Let A be the set of elements of this sequence and assume that it is of density δ >0. Show that the “average distance” between consecutive elements of A is equal to 1/δ, in the sense that
xlim→∞
1 A(x)
an≤xn≥2
an−an−1 1/δ = 1.
Problem 7.10. Let A be a set of positive integers for which the counting function A(x) = #{n≤x:n∈A} satisfies the asymptotic relation
A(x) = x
L(x)(1 +o(1)) (x→ ∞),
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where L(x) is a function satisfying L(x) (logx)1+η for some positive constant η >0. Prove that
n∈A
1
n <+∞.
Problem 7.11. Let A be a set of positive integers for which the counting function A(x) = #{n≤x:n∈A} satisfies
A(x) = x
L(x)(1 +o(1)) (x→ ∞), where L(x) is a function satisfying L(x)logx. Prove that
n∈A
1
n = +∞.
Problem 7.12. Show that the set A = {n : ω(n) | n} is of asymptotic density zero. (Hint: Let x be a large positive real number and set y = log logx. Use Problem 7.6 to infer that all n ≤ x have ω(n) ∈ I = [y − y2/3, y+y2/3] with at most o(x) exceptions. So, if such an integer n is in A, then k |n for some k ∈ I. The number of such n ≤x for a fixed k is
≤x/k. Thus, the set in question has at most
x
y−y2/3≤k≤y+y2/3
1 k
elements. Now use Theorem 1.7 to deduce that the above sum is o(1) as y→ ∞.)
Problem 7.13. Show that 2ζ(3) =
1
0
1
0
−log(xy) 1−xy dxdy.
(Hint: Use the method used in the proof of ζ(2) =π2/6 in Chapter 3, that is, set
I(σ) = 1
0
1
0
(xy)σ 1−xydxdy.
Then take derivatives with respect to σ and evaluate the result atσ = 0.) Problem 7.14. Show that if σ(n) is odd, then n = m2 or n = 2m2 for some integer m.
Problem 7.15. Show that ifnis an odd perfect number, thenn=p2α+1m2, where p is prime and coprime to m.
Problem 7.16. Show that the sum of the reciprocals of the perfect numbers is convergent. (Hint: Let A be the set of all odd perfect numbers. Show, using the previous problem, that #(A∩[1, x]) =O(x1/2logx) by noting that if n=p2α+1m2, then p|σ(m2). Then use Abel’s summation formula.)
Problem 7.17. Show that ifn >2then22n−1is not of the formp+2a+2b, where p is prime and b > a≥0 are integers. (Hint: Start by noting that if r is such that 2rb−a, then 22r + 1|22n−1−2a−2b.)
Problem 7.18. It is not known if there exist infinitely many multiperfect numbers, that is, positive integers n such that n|σ(n). (See, for instance, Guy’s book [71].) Prove that if s≥2 is a fixed integer, then there exist in- finitely many positive integers nsuch thatn|σs(n), whereσs(n) =
d|nds. Problem 7.19. Show that there are infinitely many positive integersksuch that 2nk−1 is composite for all n≥0.
Problem 7.20. Prove that there are infinitely many odd positive integers k not of the form 2n+p for any positive integer n and odd prime p. Is it easier to show that there are infinitely many odd positive integers k not of the form 10n+p for any positive integer nand prime p?
Problem 7.21. Let a > bbe coprime positive integers, and letx > a. Show
that
n<x
ω(an+b)xlog logx.
(Hint: Note first that an+b <(x+ 1)2, so that for each n there is at most one prime factor p > x+ 1 of an+b. Thus, we may reduce the problem to estimating the contribution coming from primes p < x. Write the remaining sum on the left as a double sum and change the order of summation. Then use the fact that there are at most x/p+ 1≤2x/p values of m≤x such that p|am+b.)
Problem 7.22. Let a > bbe coprime positive integers, and letx > a. Show
that
n<x
d(an+b)xlogx.
Problem 7.23. Let n > 1 and 1 = d1 < d2 < ã ã ã < dd(n) = n be all the d(n) divisors of n. Prove that
1≤i<j≤d(n)
1
dj −di d(n).
Problem 7.24. Show that the sequence{σ(n)/n}n≥1 is dense in [1,∞).
Problem 7.25. Let k≥2be a fixed positive integer. Show that the inequal- ities
ω(n+ 1)< ω(n+ 2)<ã ã ã< ω(n+k) hold for infinitely many positive integers n.
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Problem 7.26. Show that the sequence {ω(n+ 1)/ω(n)}n≥1 is dense in [0,∞). (Hint: Let α be any positive real number. Choose a very largexand writeπ(x) =a+b, wherea/bis nearα. Split the set of all primesp≤xinto two disjoint sets P and Q, one of cardinality a and the other of cardinality b. Then use the Chinese Remainder Theorem to find n0 such thatn0 ≡ −1 (modnP) and n0 ≡ 0 (mod nQ), where nP =
p∈Pp and nQ =
q∈Qq.
Let n∈[e3x, e4x] be such that n≡n0 (modnPãnQ). Deduce that n+ 1 is divisible by the prime factors in P and (perhaps) some other primes larger than x and n by the prime factors in Q and (perhaps) some other prime larger than x. Now show that for most suchn, both n+ 1 and n have only few prime factors larger than x, no more than o(x)as x→ ∞, by using, for example, Problem 7.21. Conclude that ω(n+ 1)/ω(n) = (a/b)(1 +o(1)) as x→ ∞.)
Problem 7.27. Show that the sequence {Ω(n+ 1)/Ω(n)}n≥1 is dense in [0,∞).
Problem 7.28. Prove that there exist infinitely many positive integers k not of the form n−ω(n) for any n ≥ 1. (Hint: Let be large, let P1 = p1p2, P2 = p3p4p5, P3 = p6p7p8p9, . . . , P = ptã ã ãpt+1−1, where t = (−1)/2 is the -th triangular number. Set x = P1ã ã ãP and let k ∈ [x2, x3] be in the arithmetical progressionk≡ −i (mod Pi) for i= 1, . . . , . Show, using the Prime Number Theorem, that 2loglogx, so that (logx/log logx)1/2. Now show that ifk=n−ω(n) for somen, thenω(n) = t ∈ I = [+ 1,logx]. Thus, n should have a lot of prime factors with respect to its size. Afterwards, use Problem 7.21 to show that for most such k in [x2, x3], there is no corresponding t inI such thatn+t has more than t prime factors.)
Chapter 8
The Fascinating Euler Function