On the Greatest Common Divisor of a Number and Its Sum of Divisors, II

Paul Pollack ()
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Paul Pollack: University of Georgia, Department of Mathematics

A chapter in Number Theory in Memory of Eduard Wirsing, 2023, pp 269-283 from Springer

Abstract: Abstract Let E ( x , y ) = # { n ≤ x : gcd ( n , σ ( n ) ) > y } $$E(x,y) = \#\{n\le x: \gcd (n,\sigma (n)) > y\}$$ . We collect known results about the distribution of E ( x , y ) $$E(x,y)$$ and establish a new, sharp estimate for E ( x , y ) $$E(x,y)$$ when y grows faster than any power of log log x $$\log \log {x}$$ but y = exp ( ( log log x ) o ( 1 ) ) $$y = \exp ((\log \log {x})^{o(1)})$$ . Taken together, these results determine the order of magnitude of log ( E ( x , y ) ∕ x ) $$\log (E(x,y)/x)$$ whenever 1 ≤ y ≤ x 1 − 𝜖 $$1 \le y \le x^{1-\epsilon }$$ .

Keywords: Perfect number; Multiperfect number; Multiply perfect; Sum of divisors (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-31617-3_18

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DOI: 10.1007/978-3-031-31617-3_18

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