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NOTE ON THE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS

  • ADRIAN W. DUDEK (a1), ŁUKASZ PAŃKOWSKI (a2) and VICTOR SCHARASCHKIN (a3)

Abstract

Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of  $n$ . We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$ .

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The second author was partially supported by the Grant no. 2016/23/D/ST1/01149 from the National Science Centre.

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