A Remark on Prime Divisors of Partition Functions
نویسندگان
چکیده
Let p(n) denote the classical partition function, defined as the number of ways of writing n as a sum of positive integers, where the order of the summands is not taken into account. Hardy and Ramanujan [10] developed the circle method in order to obtain precise estimates of the asymptotic behavior of p(n) as n!1. Their results were later refined by Rademacher [13], who found an exact expression for p(n) as the sum of a rapidly converging series. Taking the first term in Rademacher’s series results in the stunning asymptotic formula
منابع مشابه
A remark on the means of the number of divisors
We obtain the asymptotic expansion of the sequence with general term $frac{A_n}{G_n}$, where $A_n$ and $G_n$ are the arithmetic and geometric means of the numbers $d(1),d(2),dots,d(n)$, with $d(n)$ denoting the number of positive divisors of $n$. Also, we obtain some explicit bounds concerning $G_n$ and $frac{A_n}{G_n}$.
متن کاملσ-sporadic prime ideals and superficial elements
Let $A$ be a Noetherian ring, $I$ be an ideal of $A$ and $sigma$ be a semi-prime operation, different from the identity map on the set of all ideals of $A$. Results of Essan proved that the sets of associated prime ideals of $sigma(I^n)$, which denoted by $Ass(A/sigma(I^n))$, stabilize to $A_{sigma}(I)$. We give some properties of the sets $S^{sigma}_{n}(I)=Ass(A/sigma(I^n))setminus A_{sigma}(I...
متن کاملA remark on boundedness of composition operators between weighted spaces of holomorphic functions on the upper half-plane
In this paper, we obtain a sucient condition for boundedness of composition operators betweenweighted spaces of holomorphic functions on the upper half-plane whenever our weights are standardanalytic weights, but they don't necessarily satisfy any growth condition.
متن کاملSome Algorithms for Prime Testing Using Generalized Lehmer Functions
Let N be an odd integer thought to be prime. The properties of special functions which are generalizations of the functions of Lehmer (Ann. of Math., v. 31, 1930, pp. 419-448) are used to develop algorithms that produce information concerning the possible prime divisors of N. It is shown how the factors of N ± 1, N + 1, 2 N ± N + 1, together with the factor bounds on these numbers, may all be u...
متن کامل2 9 Ju l 2 00 3 Elementary divisors of Gram matrices of certain
The elementary divisors of the Gram matrices of Specht modules S λ over the symmetric group are determined for two-row partitions and for two-column partitions λ. More precisely, the subquotients of the Jantzen filtration are calculated using Schaper's formula. Moreover, considering a general partition λ of n at a prime p > n − λ 1 , the only possible non trivial composition factor of S λ Fp is...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013