Computing isolated coefficients of the $j$-function

计算二项式系数 $\binom{mn}{mk}$ 的最大公约数

Computing the Greatest Common Divisor of Binomial Coefficients $\binom{mn}{mk}$

本文研究了二项式系数 $\binom{mn}{mk}$ 的最大公约数，给出了 $m=2$ 时该 GCD 的完整刻画，并展示了在特定条件下高效计算的方法。

On a nonlinear relation for computing the overpartition function

In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function $\overline{p}(n)$. Computing $\overline{p}(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of $\overline{p}(n)$ that requires only the values of $\overline{p}(k)$ with $k\leqslant n/2$. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function $\overline{p}(n)$ to o…

计算 $N^{9/28+o(1)}$ 时间内的除数求和函数

Computing the divisor summatory function in $N^{9/28+o(1)}$

提出一种解析算法，通过 Voronoï 求和加速指数和计算，在 $N^{9/28+o(1)}$ 期望时间内计算除数求和函数孤立值，优于现有 $O(N^{1/3})$ 方法。

Computing the Charlap-Coley-Robbins modular polynomials

Let $\mathcal{E}$ be an elliptic curve over a field $K$ and $\ell$ a prime. There exists an elliptic curve $\mathcal{E}^*$ related to $\mathcal{E}$ by anisogeny (rational map that is also a group homomorphisms) of degree $\ell$ if and only $Φ_\ell(X, j(\mathcal{E})) = 0$, where $Φ_\ell(X, Y)$ is the traditional modular polynomial. Moreover, the modular polynomial gives the coefficients of $\mathcal{E}^*$, together with parameters needed to build the isogeny explicitly. Since the traditional modular polynomial has large coefficients, many families with smaller coefficients can be used instead, …

计算Li系数的有效方法及其性质

Effective method of computing Li's coefficients and their properties

本文提出一种有效计算Li系数的方法，并据此计算出超过三千个系数，发现其序列由严格增长的趋势和微小振荡组成，为黎曼假设研究提供新视角。

Computing Modular Polynomials

We present a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves and are useful in many aspects of computational number theory and cryptography. Our algorithm has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. We avoid computing the exponentially large integral coefficients by working directly modulo a prime and computing isogenies between elliptic curves via Velu's formulas.

On computing factors of cyclotomic polynomials

For odd square-free n > 1 the n-th cyclotomic polynomial satisfies an identity of Gauss. There are similar identity of Aurifeuille, Le Lasseur and Lucas. These identities all involve certain polynomials with integer coefficients. We show how these coefficients can be computed by simple algorithms which require O(n^2) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for the polynomials, and illustrate the application to integer factorization with some numerical examples.

Computing L-series of hyperelliptic curves

We discuss the computation of coefficients of the L-series associated to a hyperelliptic curve over Q of genus at most 3, using point counting, generic group algorithms, and p-adic methods.

A template method for Fourier coefficients of Langlands Eisenstein series

This paper introduces the template method for computing the first coefficient of Langlands Eisenstein series on $\GL(n,\mathbb R)$ and more generally on Chevalley groups over the adele ring of $\mathbb Q.$ In brief, the first coefficient of Borel Eisenstein series can be used as a template to compute the first coefficient of more general Eisenstein series by elementary linear algebra calculations.