Principal forms X 2+nY 2 representing many integers

Brink, D.M.; Moree, Pieter; Osburn, Robert

doi:10.1007/s12188-011-0059-y

Cited by 7 publications

(4 citation statements)

References 15 publications

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“…. = 0, thus establishing the falsity of Ramanujan's claim (3). Since γ S B < 1/2, it follows by Corollary 1 that actually the Ramanujan approximation is better.…”

Section: Some Euler-kronecker Constants Related To Binary Quadratic F...mentioning

confidence: 78%

“…In the special case where f = X 2 + nY 2 , a remark in a paper of Shanks seemed to suggest that he thought C f would be maximal in case n = 2. However, the maximum does not occur for n = 2, see Brink et al [3].…”

Section: Some Euler-kronecker Constants Related To Binary Quadratic F...mentioning

confidence: 97%

“…Since his work was not published in a mathematical journal it got forgotten and later partially rediscovered by mathematicians such as James and Pall. For a brief description of the proof approach of Bernays see Brink et al [3]. We like to point out that in general the estimate…”

Section: Some Euler-kronecker Constants Related To Binary Quadratic F...mentioning

confidence: 99%

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Counting numbers in multiplicative sets: Landau versus Ramanujan

Moree

2011

Preprint

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A set S of integers is said to be multiplicative if for every pair m and n of coprime integers we have that mn is in S iff both m and n are in S. Both Landau and Ramanujan gave approximations to S(x), the number of n ≤ x that are in S, for specific choices of S. The asymptotical precision of their respective approaches are being compared and related to Euler-Kronecker constants, a generalization of Euler's constant γ = 0.57721566 . . .. This paper claims little originality, its aim is to give a survey on the literature related to this theme with an emphasis on the contributions of the author (and his coauthors).

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“…. = 0, thus establishing the falsity of Ramanujan's claim (3). Since γ S B < 1/2, it follows by Corollary 1 that actually the Ramanujan approximation is better.…”

Section: Some Euler-kronecker Constants Related To Binary Quadratic F...mentioning

confidence: 78%

Section: Some Euler-kronecker Constants Related To Binary Quadratic F...mentioning

confidence: 97%

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Counting numbers in multiplicative sets: Landau versus Ramanujan

Moree

2011

Preprint

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“…Theorem 2.1. (Landau-Ramanujan [5,7,34]) The number of positive integers smaller than n that are the sum of two squares is Θ(n/ √ log n).…”

Section: The Structure Of Point Sets With Few Distinct Distancesmentioning

confidence: 99%

Distinct Distances: Open Problems and Current Bounds

Sheffer

2014

Preprint

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Local Limit Theorems for Hook Lengths in Partitions

Bhowmik,

Tsai

2024

Ann. Comb.

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Partition hook lengths have wide-ranging applications in combinatorics, number theory, physics, and representation theory. We study two infinite families of random variables associated with t-hooks. For fixed $$t\ge 1,$$ t ≥ 1 , if $$Y_{t;\,n}$$ Y t ; n counts the number of hooks of length t in a random integer partition of n, we prove a uniform local limit theorem for $$Y_{t;\,n}$$ Y t ; n on any bounded set of $${\mathbb {R}}.$$ R . To achieve this, we establish an asymptotic formula with a power-saving error term for the number of partitions of n with m many t-hooks. In contrast, we define $${\widehat{Y}}_{t;\,n}$$ Y ^ t ; n as the count of hooks divisible by t in a randomly chosen partition of n. While $${\widehat{Y}}_{t;\,n}$$ Y ^ t ; n converges in distribution, we show that it fails to satisfy the local limit theorem for any $$t \ge 2$$ t ≥ 2 . The proofs employ the multivariable saddle-point method, asymptotic formulas for the number of t-core partitions from Anderson and Lulov–Pittel, and estimates of certain exponential sums. Notably, for $$t=4,$$ t = 4 , the analysis involves the asymptotic behavior of class numbers of imaginary quadratic fields.

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Principal forms X 2+nY 2 representing many integers

Cited by 7 publications

References 15 publications

Counting numbers in multiplicative sets: Landau versus Ramanujan

Counting numbers in multiplicative sets: Landau versus Ramanujan

Distinct Distances: Open Problems and Current Bounds

Local Limit Theorems for Hook Lengths in Partitions

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