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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Two differential-difference equations arising in number theory


Author: Ferrell S. Wheeler
Journal: Trans. Amer. Math. Soc. 318 (1990), 491-523
MSC: Primary 11N35; Secondary 11N25, 11Q10, 34K05
MathSciNet review: 963247
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Abstract: We survey many old and new results on solutions of the following pair of adjoint differential-difference equations: (1)

$\displaystyle up'(u) = - ap(u) - bp(u - 1),$

(2) (

$\displaystyle (uq(u))' = aq(u) + bq(u + 1).$

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

$\displaystyle \sum\limits_{\begin{array}{*{20}{c}} {1 < n \leqslant x} \\ {{P_2... ... x)}^\alpha }} \qquad (x \to \infty ,\;u \geqslant 1,\;\alpha \in {\mathbf{R}})$

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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0963247-X
PII: S 0002-9947(1990)0963247-X
Keywords: Differential-difference equations, adjoint relation, sieve theory, Dickman function
Article copyright: © Copyright 1990 American Mathematical Society