An error term of Ramanujan
نویسندگان
چکیده
منابع مشابه
On an Asymptotic Series of Ramanujan
An asymptotic series in Ramanujan’s second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of φ(X) for large λ where X is a Poisson random variable with mean λ and φ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of φ(X) when the distribution of X belongs to a suitable family indexed by...
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Corollary 2, Entry 9, Chapter 4 of Ramanujan's rst notebook claims that 1 X k=1 (?1) k?1 nk x k k! n ln x + as x ! 1. This is known to be correct for the case n = 1, but incorrect for n 3. We show that the result is correct for n = 2. We also consider the order of the error term, and discuss a diierent, correct generalisation of the case n = 1.
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δ|M ∏∞ n=1(1− q )δ = ∑∞ n=0 a(n)q n and three positive integers m, t, p, and which returns true if a(mn + t) ≡ 0 (mod p), n ≥ 0, or false otherwise. A similar algorithm for generating functions of the form ∏∞ n=1(1− q )1 (i.e. the case M = 1) has already been given in [3]. Our original plan was to implement that algorithm in order to prove some congruences from [1]. The algorithm we present her...
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15 صفحه اولAn algorithmic approach to Ramanujan-Kolberg identities
Let M be a given positive integer and r = (rδ)δ|M a sequence indexed by the positive divisors δ of M . In this paper we present an algorithm that takes as input a generating function of the form ∑∞ n=0 ar(n)q n := ∏ δ|M ∏∞ n=1(1−q )δ and positive integers m, N and t ∈ {0, . . . ,m−1}. Given this data we compute a set Pm,r(t) which contains t and is uniquely defined by m, r and t. Next we decide...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1970
ISSN: 0022-314X
DOI: 10.1016/0022-314x(70)90008-9