arXiv 0807.13472008-07math.NT

A multimodular algorithm for computing Bernoulli numbers

David Harvey

备注：: 10 pages, 1 table, requires algorithm2e package; many minor edits, updated timings for correct GMP version, added data for calcbn package

英文摘要：

We describe an algorithm for computing Bernoulli numbers. Using a parallel implementation, we have computed B(k) for k = 10^8, a new record. Our method is to compute B(k) modulo p for many small primes p, and then reconstruct B(k) via the Chinese Remainder Theorem. The asymptotic time complexity is O(k^2 log(k)^(2+epsilon)), matching that of existing algorithms that exploit the relationship between B(k) and the Riemann zeta function. Our implementation is significantly faster than several existing implementations of the zeta-function method.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

A note on degenerate poly-Bernoulli numbers and polynomials

In this paper, we consider the degenerate poly-Bernoulli polynomials and present new and explicit formulas for computing them in terms of the degenerate Bernoulli polynomials and Stirling numbers of the second kind.

2015-03-29math.NT

An explicit formula for computing Bernoulli numbers of the second kind in terms of Stirling numbers of the first kind

In the paper, the author finds an explicit formula for computing Bernoulli numbers of the second kind in terms of Stirling numbers of the first kind.

2014-01-20math.CAmath.COmath.NT

Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind

In the paper, by establishing a new and explicit formula for computing the $n$-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind and Stirling numbers of the first kind. As consequences of these formulas, a recursion for Stirling numbers of the first kind and a new representation of the reciprocal of the factorial $n!$ are derived. Finally, the author finds several identities and integral representations relating to Stirling numbers of the first kind.

2013-01-29math.CAmath.COmath.NT

计算第n个伯努利数的次二次算法

A subquadratic algorithm for computing the n-th Bernoulli number

School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）

提出一种新算法，在n^(4/3+o(1))比特操作内计算第n个伯努利数，将复杂度从n^(2+o(1))降低。

2013-05-01cs.DScs.NAmath.NA

代数参数下伯努利卷积的Garsia熵计算

Computing Garsia Entropy for Bernoulli Convolutions with Algebraic Parameters

该研究引入含特定代数整数的参数空间与递增分段连续函数序列给出伯努利卷积Garsia熵下界，证明部分开区域内对应伯努利卷积的Hausdorff维数为1。

2023-11-06math.CAmath.DSmath.NT

形式幂级数第 $k$ 次幂的显式、递推、行列式表达式及其在广义伯努利数中的应用

Explicit, recurrent, determinantal expressions of the $k$th power of formal power series and applications to the generalized Bernoulli numbers

本文给出形式幂级数第k次幂的闭式与递推行列式表达式，应用于广义伯努利数，推导新公式与递推关系，并推广欧拉恒等式，揭示与第二类斯特林数的联系。

2023-05-10math.NT

更多相关论文收起更多论文

用第二类Stirling数表示Bernoulli数的一个显式公式

An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind

该研究发现了一个以第二类Stirling数表示Bernoulli数的显式计算公式，为数论相关计算提供了新的显式表达工具。

2014-01-17math.CAmath.COmath.NT

An explicit formula for Bernoulli polynomials in terms of $r$-Stirling numbers of the second kind

In the paper, the authors establish an explicit formula for computing Bernoulli polynomials at non-negative integer points in terms of $r$-Stirling numbers of the second kind.

2014-02-11math.CAmath.COmath.NT

Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers

In the paper, the authors provide four alternative proofs of an explicit formula for computing Bernoulli numbers in terms of Stirling numbers of the second kind.

2014-06-05math.CAmath.COmath.NT

Fast computation of Bernoulli, Tangent and Secant numbers

We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or Euler) numbers. In particular, we give asymptotically fast algorithms for computing the first n such numbers in O(n^2.(log n)^(2+o(1))) bit-operations. We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n^2) integer operations. These algorithms are extremely simple, and fast for moderate values of n. They are faster and use less space than the algorithms of Atkinson (for Tangent and Secant numbers) and Akiyama and Tanigawa (for Bernoulli numbers).

2011-09-05cs.DMcs.DSmath.CO

Extended Zeilberger's Algorithm for Identities on Bernoulli and Euler Polynomials

We present a computer algebra approach to proving identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Chen, Hou and Mu. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to establish recurrence relations on the integrands. Such recurrence relations have certain parameter free properties which lead to the required identities without computing the integrals.

2011-11-08math.CAmath.COmath.NT

New families of special numbers for computing negative order Euler numbers

The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to the many well-known numbers, which are the Bernoulli numbers, the Fibonacci numbers, the Lucas numbers, the Stirling numbers of the second kind and the central factorial numbers. Our other inspiration of this paper is related to the Golombek's problem \cite{golombek} \textquotedblleft Aufgabe 1088, El. Math. 49 (1994) 126-127\textquotedblright . Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Eu…

2016-04-19math.NT

Generalized Stirling transform

In this paper, algorithms are developed for computing the Stirling transform and the inverse Stirling transform; specifically, we investigate a class of sequences satisfying a two-term recurrence. We derive a general identity which generalizes the usual Stirling transform and investigate the corresponding generating functions also. In addition, some interesting consequences of these results related to classical sequences like Fibonacci, Bernoulli and the numbers of derangements have been derived.

2012-12-05math.COmath.NT

On $p$-adic Euler $L$-functions

In this paper, we define the p-adic Euler L-functions using the fermionic p-adic integral on Zp. By computing the values of the p-adic Euler L-functions at negative integers, we show that for Dirichlet characters with odd conductor, this definition is quivalent to the previous definition following Kubata-Leopoldt and Washington's approach. We also study the behavior of p-adic Euler L-functions at positive integers. An interesting thing is that most of the results in Section 11.3.3 of Cohen's book [H. Cohen, Number Theory Vol. II: Analytic and Modern Tools, Graduate Texts in Mathematics, 240. S…

2020-08-16math.NT

Rapid computation of special values of Dirichlet $L$-functions

We consider computing the Riemann zeta function $ζ(s)$ and Dirichlet $L$-functions $L(s,χ)$ to $p$-bit accuracy for large $p$. Using the approximate functional equation together with asymptotically fast computation of the incomplete gamma function, we observe that $p^{3/2+o(1)}$ bit complexity can be achieved if $s$ is an algebraic number of fixed degree and with algebraic height bounded by $O(p)$. This is an improvement over the $p^{2+o(1)}$ complexity of previously published algorithms and yields, among other things, $p^{3/2+o(1)}$ complexity algorithms for Stieltjes constants and $n^{3/2+o(…

2021-10-20cs.NAmath.CAmath.NA

计算二次不规则性的首次命中分析算法

First-hit analysis of algorithms for computing quadratic irregularity

本文利用均匀分布启发式预测计算二次域中不规则素数所需时间，并探讨收集数据验证该启发式的替代方法。

2000-11-10cs.NAmath.NAmath.NT

实二次zeta函数值的分布

Distribution of Values of Real Quadratic Zeta Functions

本文针对实二次域中zeta函数在负奇数点的值模p分布的计算难题，提出了一种固定D而遍历大量m的高效替代计算方法。

2000-10-29cs.NAmath.NAmath.NT