## COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ

Citations: | 4 - 3 self |

### BibTeX

@MISC{Ford_commonvalues,

author = {Kevin Ford and Florian Luca and Carl Pomerance},

title = {COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ},

year = {}

}

### OpenURL

### Abstract

ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sum-of-divisors function. This proves a 50-year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c> 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of Heath-Brown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.

### Citations

155 |
Sieve methods
- Halberstam, Richert
- 1974
(Show Context)
Citation Context ... a, P (p + a) > x 1/2−δ }. where For p ∈ U(q, r), we have p + a = qrsb, where s > x1/2−δ is prime and b � x2δ . For fixed b, q, r, a, we estimate the number of possible choices for s using the sieve (=-=[19]-=-, Theorem 3.12). We get #U(q, r) ≪ ∑ x bqr log 2 b (x/bqr) φ(b) ≪ x qr log 2 ∑ 1 x φ(b) ≪ δx qr log x . b�x 2δ b�x 2δ6 KEVIN FORD, FLORIAN LUCA, AND CARL POMERANCE For small enough δ, we then have #U... |

57 | Old and new problems and results in combinatorial number theory, Monographies de L’Enseignement Mathématique (Monographs of L’Enseignement Mathématique) 28, Université de Genève L’Enseignement Mathématique
- Erdős, Graham
(Show Context)
Citation Context ...ful. A short calculation reveals that there are 95145 common values of φ and σ between 1 and 10 6 . This is to be compared with a total of 180184 φ-values and 189511 σ-values in the same interval. In =-=[9]-=-, the authors write that “it is very annoying that we cannot show that φ(a) = σ(b) has infinitely many solutions. . . .” Annoying of course, since it is so obviously correct! Erdős knew (see [18, sec.... |

55 | Sur certaines hypothèses concernant les nombres premiers - Schinzel, Sierpiński - 1958 |

41 | Unsolved Problems in Number Theory, third edition - Guy - 2004 |

41 |
Integers without large prime factors
- Hildebrand, Tenenbaum
- 1993
(Show Context)
Citation Context ...eorem 1 with a method of Erdős [5]. A key estimate is [5, Lemma 2]: (4.1) #{n � x : P (n) � log x} = x o(1) (x → ∞). More results about the distribution of integers n with P (n) small may be found in =-=[21]-=-. Define λ = λ0, α, x0, x1 and η as in the proof of Theorem 1. Without loss of generality, suppose α � 1 1 . Theorem 2 is proved by considering the two cases, x is not (α, )-good and x 500 10 )-good. ... |

35 |
Shifted primes without large prime factors
- Baker, Harman
- 1998
(Show Context)
Citation Context ...he proofs give analogous results for B(n). Numerical values of c have been given by a number of people ([2], [15], [26] and [29]), the largest so far being c = 0.7039 which is due to Baker and Harman =-=[1]-=-. The key to these results is to show that there are many primes p for which p − 1 has only small prime factors. Erdős [6] conjectured that for any constant c < 1 the inequality A(n) > n c holds infin... |

17 |
On the normal behavior of the iterates of some arithmetic functions, Analytic Number Theory (Allerton Park
- Erdős, Granville, et al.
- 1989
(Show Context)
Citation Context ...eorem 5]). For every ε > 0 there is a constant C(ε) so that if q is prime and y > q, then #T (y, q) � C(ε)(y/q) 1+ε . More estimates for counts of prime chains with various properties may be found in =-=[3, 10, 14, 23]-=-. We now proceed to prove Theorem 1. There is an absolute constant λ0 > 0 so that if λ � λ0, then the error term in the conclusion of Lemma 2.3 is at most 0.1z/log 2 z in absolute value. Let α > 0 and... |

17 |
Shifted primes without large prime factors, in Number Theory and Applications
- FRIEDLANDER
- 1989
(Show Context)
Citation Context ...howed that the inequality A(n) > n c holds infinitely often for some positive constant c. The proofs give analogous results for B(n). Numerical values of c have been given by a number of people ([2], =-=[15]-=-, [26] and [29]), the largest so far being c = 0.7039 which is due to Baker and Harman [1]. The key to these results is to show that there are many primes p for which p − 1 has only small prime factor... |

16 | Multiplicative Number Theory, Third Edition, Graduate Texts - Davenport |

13 |
On the normal number of prime factors of p − 1 and some related problems concerning Euler’s φ-function, Quart
- ERDÖS
- 1935
(Show Context)
Citation Context ... in many ways. Let A(n) be the number of solutions of φ(x) = n, and let B(n) be the number of solutions of σ(x) = n. Pillai [25] showed in 1929 that the function A(n) is unbounded, and in 1935, Erdős =-=[5]-=- showed that the inequality A(n) > n c holds infinitely often for some positive constant c. The proofs give analogous results for B(n). Numerical values of c have been given by a number of people ([2]... |

13 | A large sieve density estimate near σ - Gallagher - 1970 |

12 |
Probability and Random Processes, second edition
- Grimmett, Stirzaker
- 1992
(Show Context)
Citation Context ...n at random, the expected value of ∑ p∈M ′ vq(p − 1) = vq(φ(w(M ′ ∑ ′ ′ ))) is k j�1 jN qj/N ′ (we are now viewing our random variable as vq(p − 1)). By the ( N ′) ′ choices for M with same result in =-=[17]-=-, there are at least 1 2 k ′ vq(φ(w(M ′ ))) � 3 ∑ jN k′ 2 j�1 ′ qj N ′ for all q ∈ Q. For such choices of M ′ , we have vq(φ(w(M ′ ))) ≪ k ′ /q, so if we choose ξ small enough, we have by (4.4) and (4... |

10 |
Remarks on the paper “Sur certaines hypothèses concernant les nombres premiers
- SCHINZEL
(Show Context)
Citation Context ...ee [12], [13]. The famous Carmichael conjecture states that A(n) is never 1, but this is still open. Conjecture 1. For every k � 1 and l � 2, there are integers n with A(n) = l and B(n) = k. Schinzel =-=[27]-=- asserts that this conjecture follows from his Hypothesis H. (2) If, as conjectured by Hardy and Littlewood, the number of pairs of twin primes � x is ∼ Cx/ log 2 x, then the number of common values n... |

9 | The number of solutions of ϕ(x) = m
- Ford
- 1999
(Show Context)
Citation Context ...E ARITHMETIC FUNCTIONS φ AND σ 11 5. FURTHER PROBLEMS (1) It is known that for any integer k � 1, there are integers n with B(n) = k and for any integer l � 2, there are integers n with A(n) = l, see =-=[12]-=-, [13]. The famous Carmichael conjecture states that A(n) is never 1, but this is still open. Conjecture 1. For every k � 1 and l � 2, there are integers n with A(n) = l and B(n) = k. Schinzel [27] as... |

9 |
Primes twins and Siegel zeros
- Heath-Brown
- 1983
(Show Context)
Citation Context ...thank the Mathematics Department for its hospitality. 12 KEVIN FORD, FLORIAN LUCA, AND CARL POMERANCE a definition) creates a major obstacle for the success of our argument. Fortunately, Heath-Brown =-=[20]-=- showed that if Siegel zeros exist, then there are infinitely many pairs of twin primes. However, despite the influence of possible Siegel zeros, our methods are completely effective. Theorem 1. The e... |

9 |
Explicit estimates for the error term in the prime number theorem for arithmetic progressions
- McCurley
- 1984
(Show Context)
Citation Context ...(s, χ2) with β > 1 − c2/log(m1m2). We immediately obtain Lemma 2.2 (Page). For any M � 3, ∏ ∏ m�M χ∈C(m) L(s, χ) has at most one zero in the interval [1 − (c2/2)/log M, 1]. It is known after McCurley =-=[24]-=- that c0 = 1/9.645908801 holds in (2.1), while Kadiri [22] has shown we may take c0 = 1/6.397, and in Lemmas 2.1, 2.2 we may take c2 = 1/2.0452. The Riemann hypothesis for Dirichlet L-functions implie... |

8 |
Irreducible radical extensions and Euler-function chains, Integers 7 (2007), Paper A25. 2010 Mathematics Subject Classification: Primary 11N37. Keywords: largest prime factor, middle prime factor. 9 March 2 2013; revised version received
- Luca, Pomerance
(Show Context)
Citation Context ...eorem 5]). For every ε > 0 there is a constant C(ε) so that if q is prime and y > q, then #T (y, q) � C(ε)(y/q) 1+ε . More estimates for counts of prime chains with various properties may be found in =-=[3, 10, 14, 23]-=-. We now proceed to prove Theorem 1. There is an absolute constant λ0 > 0 so that if λ � λ0, then the error term in the conclusion of Lemma 2.3 is at most 0.1z/log 2 z in absolute value. Let α > 0 and... |

6 |
On some functions connected with φ(n
- Pillai
- 1929
(Show Context)
Citation Context ...that there are infinitely many integers n which are common values of φ and σ in many ways. Let A(n) be the number of solutions of φ(x) = n, and let B(n) be the number of solutions of σ(x) = n. Pillai =-=[25]-=- showed in 1929 that the function A(n) is unbounded, and in 1935, Erdős [5] showed that the inequality A(n) > n c holds infinitely often for some positive constant c. The proofs give analogous results... |

6 |
Popular values of Euler’s function, Mathematika 27
- Pomerance
- 1980
(Show Context)
Citation Context ...that the inequality A(n) > n c holds infinitely often for some positive constant c. The proofs give analogous results for B(n). Numerical values of c have been given by a number of people ([2], [15], =-=[26]-=- and [29]), the largest so far being c = 0.7039 which is due to Baker and Harman [1]. The key to these results is to show that there are many primes p for which p − 1 has only small prime factors. Erd... |

5 |
On two conjectures of sierpinski concerning the arithmetic functions σ and φ. http://www.math.uiuc.edu/ford/wwwpapers/sigma.pdf
- Ford, Konyagin
(Show Context)
Citation Context ...HMETIC FUNCTIONS φ AND σ 11 5. FURTHER PROBLEMS (1) It is known that for any integer k � 1, there are integers n with B(n) = k and for any integer l � 2, there are integers n with A(n) = l, see [12], =-=[13]-=-. The famous Carmichael conjecture states that A(n) is never 1, but this is still open. Conjecture 1. For every k � 1 and l � 2, there are integers n with A(n) = l and B(n) = k. Schinzel [27] asserts ... |

4 | The Lucas-Pratt primality tree
- Bayless
(Show Context)
Citation Context ...eorem 5]). For every ε > 0 there is a constant C(ε) so that if q is prime and y > q, then #T (y, q) � C(ε)(y/q) 1+ε . More estimates for counts of prime chains with various properties may be found in =-=[3, 10, 14, 23]-=-. We now proceed to prove Theorem 1. There is an absolute constant λ0 > 0 so that if λ � λ0, then the error term in the conclusion of Lemma 2.3 is at most 0.1z/log 2 z in absolute value. Let α > 0 and... |

4 | Some remarks on Euler’s ϕ function - Erdős - 1958 |

3 |
Some remarks on Euler’s ϕ-function and some related problems
- Erdős
- 1945
(Show Context)
Citation Context ...d [29]), the largest so far being c = 0.7039 which is due to Baker and Harman [1]. The key to these results is to show that there are many primes p for which p − 1 has only small prime factors. Erdős =-=[6]-=- conjectured that for any constant c < 1 the inequality A(n) > n c holds infinitely often. Theorem 2. For some positive constant c there are infinitely many n such that both inequalities A(n) > n c an... |

3 | An explicit zero-free region for the Dirichlet L-functions, ArXiv : math.NT/0510570
- Kadiri
(Show Context)
Citation Context ...emma 2.2 (Page). For any M � 3, ∏ ∏ m�M χ∈C(m) L(s, χ) has at most one zero in the interval [1 − (c2/2)/log M, 1]. It is known after McCurley [24] that c0 = 1/9.645908801 holds in (2.1), while Kadiri =-=[22]-=- has shown we may take c0 = 1/6.397, and in Lemmas 2.1, 2.2 we may take c2 = 1/2.0452. The Riemann hypothesis for Dirichlet L-functions implies that no exceptional zeros can exist. If there is an infi... |

2 |
p + a without large prime factors, Sém. Théorie des Nombres Bordeaux (1983-84), exposé 31
- Balog
(Show Context)
Citation Context ...[5] showed that the inequality A(n) > n c holds infinitely often for some positive constant c. The proofs give analogous results for B(n). Numerical values of c have been given by a number of people (=-=[2]-=-, [15], [26] and [29]), the largest so far being c = 0.7039 which is due to Baker and Harman [1]. The key to these results is to show that there are many primes p for which p − 1 has only small prime ... |

1 | Remarks on number theory, II. Some problems on the σ function - Erdős - 1959 |

1 |
The distribution of totients, Paul Erdős (1913–1996
- Ford
- 1998
(Show Context)
Citation Context ...le, if there are infinitely many pairs of twin primes p, p + 2, then φ(p + 2) = p + 1 = σ(p), and if there are infinitely many Mersenne primes 2 p − 1, then σ(2 p − 1) = 2 p = φ(2 p+1 ). Results from =-=[11]-=- indicate that typical values taken by φ and by σ have a similar multiplicative structure; hence, common values should be plentiful. A short calculation reveals that there are 95145 common values of φ... |

1 |
Values taken many times by Euler’s phi-function
- Wooldridge
- 1979
(Show Context)
Citation Context ...inequality A(n) > n c holds infinitely often for some positive constant c. The proofs give analogous results for B(n). Numerical values of c have been given by a number of people ([2], [15], [26] and =-=[29]-=-), the largest so far being c = 0.7039 which is due to Baker and Harman [1]. The key to these results is to show that there are many primes p for which p − 1 has only small prime factors. Erdős [6] co... |