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COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
"... 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 sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."
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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 sumofdivisors function. This proves a 50year 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 HeathBrown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
DIOPHANTINE APPROXIMATION WITH ARITHMETIC FUNCTIONS, I
"... Abstract. We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes. 1. ..."
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Abstract. We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes. 1.
Article 13.8.8 On the Ratio of the Sum of Divisors and Euler’s Totient Function I
"... We prove that the only solutions to the equation σ(n) = 2 · φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n) = 2·φ(n) with Ω(n) ≤ k, and there are at most 22k +k −k squarefree solutions to φ(n) ∣ ∣σ(n) if ω(n) ..."
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We prove that the only solutions to the equation σ(n) = 2 · φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n) = 2·φ(n) with Ω(n) ≤ k, and there are at most 22k +k −k squarefree solutions to φ(n) ∣ ∣σ(n) if ω(n) = k. Lastly the number of solutions to φ(n) ∣ ∣σ(n) as x → ∞ is O ( xexp ( −1 √)) 2 logx. 1