Garaev 2011 number common values arithmetic functions below

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theorem_1: Garaev's theorem that for every A > 0 and every x > x_0(A) at least exp((log log x)^A) integers n <= x are values of both Euler's totient phi and the sum-of-divisors function sigma, proved effectively.

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Garaev, Moubariz Z., On the number of common values of arithmetic functions {$\phi$} and {$\sigma$} below {$x$}. Mosc. J. Comb. Number Theory 1 (2011), no. 3, 42--49. The copy read for this card, the journal's reprint of the article, prints "© УРСС, 2011" and "All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission of the publisher." on the issue's front matter (PDF p. 2), every other right reserved.

Ford, Luca and Pomerance had solved an old Erdős conjecture by proving effectively that phi(a) = sigma(b) has infinitely many solutions, with at least exp((log log x)^a) common values below x for some positive a. Theorem 1 of this paper (p. 43) makes the exponent arbitrary: for any A > 0 and x > x_0(A) there are at least exp((log log x)^A) integers n <= x that are common values of phi and sigma, shown effectively. The proof refines Konyagin's alternative approach, which avoids Heath-Brown's link between Siegel zeros and twin primes. Lemma 1 (p. 43), which the paper says follows from Landau's result, separates real zeros of two real primitive L-functions; Lemma 2 (p. 44) produces a scale x at which, for every modulus 3 <= m <= x^alpha and every primitive character chi modulo m, L(s,chi) has no zero in a suitable region. Common values are built as n = sigma(prod p) = prod (p+1) over primes p <= x whose shifts p+1 have all prime factors below x^{1/2-delta}, combined with the implication phi(rad(m)) | m => m = phi(m rad(m)/phi(rad(m))) to realize n as a totient value. The paper does not name Erdős problem 48, which asks whether phi(n) = sigma(m) has infinitely many solutions; Theorem 1 answers it yes, as Ford, Luca and Pomerance's theorem already did, and sharpens their count.

Source: http://mjcnt.phystech.edu/en/article.php?id=24.

Bears on.

• #48: Theorem 1 (p. 43) gives at least exp((log log x)^A) common values of phi and sigma up to x for every A > 0 and x > x_0(A), so phi(n) = sigma(m) has infinitely many solutions and the problem's answer is yes. Ford, Luca and Pomerance proved this first; the paper's contribution is the sharper count.

Contents

• Theorem 1 (p. 43; proof Section 3, pp. 47--49): for any A > 0 and x > x_0(A), at least exp((log log x)^A) integers n <= x are common values of phi and sigma. Lemmas 1 (p. 43) and 2 (p. 44) are summarized in its proof pointer; they bear on no problem and have no pages of their own.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.