Asymptotic Formulaæ in Combinatory Analysis

By G. Hardy, S. Ramanujan

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Differentially Private Password Frequency Lists (2016, positive): “Hardy and Ramanujan [17] showed that asymptotically”
Higher order Turán inequalities for the partition function (2017, unknown): “1 − 1 µ(n) − 1 µ(n)10 , B2(n) = √ 12eµ(n) 24n−1 1 − 1 µ(n) + 1 µ(n)10 , then for n≥ 1206, B1(n) <p(n) <B2(n). (2.6) The proof of Lemma 2.2 relies on the Hardy-Ramanujan-Rademacher formula [14, 26] for p(n) as well as Lehmer’s error bound for the remainder of this formula. The Hardy-Ramanuja”
Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms (2011, methodology): “In celebrated work [24], which gave birth to the “circle method”, Hardy and Ramanujan quantified this rate of growth.”
Log-concavity of the partition function (2013, methodology): “odd values n25. Theorem 1.1. Sequence p(n) is log-concave for all n>25. This intuition behind the theorem comes from explicit calculations for small n, and the Hardy{ Ramanujan asymptotic formula [13] (1) p(n) ˘ 1 4 p 3n eˇ q 2 3 n as n!1: The function on the r.h.s. of (1) is easily shown to be l”
Waring's problem: A survey (2002, unknown): “This has its genesis in a celebrated paper of Hardy and Ramanujan [64] devoted to the partition function.”
Differentially private anonymized histograms (2019, methodology): “Time complexity: By the Hardy-Ramanujan integer partition theorem [41], the number of anonymized histograms with n items is e⇥( p n).”
Partitions: At the Interface of q-Series and Modular Forms (2003, methodology): “In the case k = 1, (4.12) reduces to the modular function transformation used by Hardy and Ramanujan [ 20 ].”
Partition bijections, a survey (2006, methodology): “The first upper bound proof 9.6.4 follows [ 81 ], while the second upper bound proof 9.6.5 follows [65].”
Discrete Analogues in Harmonic Analysis (2009, methodology): “The circle method first arose in 1918 in the study by Hardy and Ramanujan [17] of the generating function of the partition function,”
A Black Hole Farey Tail (2000, methodology): “ormula (1.4) is due to Rademacher. The relatively trivial estimate (1.3) is usually referred to as the Hardy-Ramanujan formula, but in fact, their full formula is much closer to (1.4). See especially [2], eqs. 1.71-1.75. See [4][5] for original papers. For some history see Littlewood’s Miscellany [6”
The Surprising Mathematics of Longest Increasing Subsequences (2015, background): “Ramanujan, who showed in 1918 [56] (see also [91]) that7”
A novel genealogical approach to neutral biodiversity theory (2004, background): “Given a species-abundance dataset, one can in principle estimate the model parameters Q and I (or, equivalently, m) using likelihood maximization on P½~ JS jI ; H; J � . However, when J is large, there is an enormous number of states in the summation (eqn 5) which makes this option unfeasible; for a”