Common values of the arithmetic functions φ and σ Kevin
نویسندگان
چکیده
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 fifty-year-old conjecture of Erdős. Moreover, we show that, for some c > 0, there are infinitely many integers n such that φ(a) = n and σ(b) = n, each having more than n solutions. 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.
منابع مشابه
ON COMMON VALUES OF φ(n) AND σ(m), II
For each positive-integer valued arithmetic function f , let Vf ⊂ N denote the image of f , and put Vf (x) := Vf ∩ [1, x] and Vf (x) := #Vf (x). Recently Ford, Luca, and Pomerance showed that Vφ ∩ Vσ is infinite, where φ denotes Euler’s totient function and σ is the usual sum-of-divisors function. Work of Ford shows that Vφ(x) ≍ Vσ(x) as x → ∞. Here we prove a result complementary to that of Fo...
متن کاملCommon Values of the Arithmetic Functions
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 solutions, for some c > 0. The proofs rely on the recent work of the first two authors and K...
متن کاملON COMMON VALUES OF φ(n) AND σ(m), I
We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x/(log x) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sum-of-divisors function.
متن کاملOn the Composition of Some Arithmetic Functions, Ii
We study certain properties and conjuctures on the composition of the arithmetic functions σ, φ, ψ, where σ is the sum of divisors function, φ is Euler’s totient, and ψ is Dedekind’s function.
متن کاملClosure Properties of Weak Systems of Bounded Arithmetic
In this paper we study the properties of systems of bounded arithmetic capturing small complexity classes and state conditions sufficient for such systems to capture the corresponding complexity class tightly. Our class of systems of bounded arithmetic is the class of secondorder systems with comprehension axiom for a syntactically restricted class of formulas Φ ⊂ Σ 1 based on a logic in the de...
متن کامل