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Search: MSC category 11P81 ( Elementary theory of partitions [See also 05A17] )

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1. CJM 2009 (vol 61 pp. 465)

Woodford, Roger
On Partitions into Powers of Primes and Their Difference Functions
In this paper, we extend the approach first outlined by Hardy and Ramanujan for calculating the asymptotic formulae for the number of partitions into $r$-th powers of primes, $p_{\mathbb{P}^{(r)}}(n)$, to include their difference functions. In doing so, we rectify an oversight of said authors, namely that the first difference function is perforce positive for all values of $n$, and include the magnitude of the error term.

Categories:05A17, 11P81

2. CJM 1997 (vol 49 pp. 641)

Burris, Stanley; Compton, Kevin; Odlyzko, Andrew; Richmond, Bruce
Fine spectra and limit laws II First-order 0--1 laws.
Using Feferman-Vaught techniques a condition on the fine spectrum of an admissible class of structures is found which leads to a first-order 0--1 law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order 0--1 law. If the condition is satisfied (and hence we have a first-order %% 0--1 law)

Categories:03N45, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81

3. CJM 1997 (vol 49 pp. 468)

Burris, Stanley; Sárközy, András
Fine spectra and limit laws I. First-order laws
Using Feferman-Vaught techniques we show a certain property of the fine spectrum of an admissible class of structures leads to a first-order law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order law. We present three conditions for verifying that the above property actually holds. The first condition is that the count function of an admissible class has regular variation with a certain uniformity of convergence. This applies to a wide range of admissible classes, including those satisfying Knopfmacher's Axiom A, and those satisfying Bateman and Diamond's condition. The second condition is similar to the first condition, but designed to handle the discrete case, {\it i.e.}, when the sizes of the structures in an admissible class $K$ are all powers of a single integer. It applies when either the class of indecomposables or the whole class satisfies Knopfmacher's Axiom A$^\#$. The third condition is also for the discrete case, when there is a uniform bound on the number of $K$-indecomposables of any given size.

Keywords:First order limit laws, generalized number theory
Categories:O3C13, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81

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