Two differential-difference equations arising in number theory

Wheeler, Ferrell S.

doi:10.1090/S0002-9947-1990-0963247-X

Two differential-difference equations arising in number theory
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Trans. Amer. Math. Soc. 318 (1990), 491-523 Request permission

Abstract:

We survey many old and new results on solutions of the following pair of adjoint differential-difference equations: (1) \[ up’(u) = - ap(u) - bp(u - 1),\] (2) (\[ (uq(u))’ = aq(u) + bq(u + 1).\] We bring together scattered results usually proved only for specific $(a,b)$ pairs, while emphasizing the connections between the two equations. We also point out some of the ways these two equations are used in number theory. We giv s several new integral relationships between (1) and (2) and use them to prove a new application of (2) in number theory, namy el \[ \sum \limits _{\begin {array}{*{20}{c}} {1 < n \leqslant x} \\ {{P_2}(n) \leqslant {P_1}{{(n)}^{1/u}}} \\ \end {array} } {{{(\log {P_1}(n))}^\alpha } \sim {u^\alpha }f(u)x{{(\log x)}^\alpha }} \qquad (x \to \infty ,\;u \geqslant 1,\;\alpha \in {\mathbf {R}})\] where ${P_1}(n)$ and ${P_2}(n)$ are the first and second largest prime divisors of $n$ and $f(u)$ satisfies (2) with $(a,b) = (1 - \alpha , - 1)$.

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On two differential-difference equations arising in analytic number theory