Asymptotics on a class of Legendre formulas

The Legendre Approximation and Arithmetic Bias

An interesting episode in the history of the prime number theorem concerns a formula proposed by Legendre for counting the primes below a given bound. We point out that arithmetic bias likely played an important role in arriving at that formula and in its subsequent widespread, decades-long recognition. We also show that the Legendre constant 1.08366 satisfies a certain simple and natural criterion, and conjecture that this criterion is how Legendre arrived at that erroneous constant in his formula.

Legendre Drinfeld模与泛超奇异多项式

Legendre Drinfeld modules and universal supersingular polynomials

我们引入了一个特定的家庭的Drinfeld模块，我们建议类似的勒让德正规形椭圆曲线。我们展示了明确的公式，在一定时期内的这种Drinfeld模块以及超奇异轨迹在该家庭的公式，建立这两种公式之间的联系。最后，我们还给出了一般Drinfeld模的$\jmath$ -不变量中的超奇异多项式的一个闭公式.

关于某些超分拆和超分拆对的Legendre定理

Legendre theorems for certain overpartitions and overpartition pairs

受超分拆的Legendre型公式启发，推导超分拆对的Legendre定理，得到子类间的等式关系。

A note on Clebsch-Gordan integral, Fourier-Legendre expansions and closed form for hypergeometric series

In this paper we show that a closed form formula for the generalized Clebsch-Gordan integral and the Fourier-Legendre expansion theory allow to evaluate hypergeometric series involving powers of the normalized central binomial coefficient ${\displaystyle \frac{1}{4^{n}}\dbinom{2n}{n}}$.

Legendre椭圆曲线的Brauer-Siegel定理的证明

Analogue of the Brauer-Siegel theorem for Legendre elliptic curves

我们证明了一个类似的Brauer-Siegel定理的勒让德椭圆曲线在$\mathbb{F}_q(t)$。更准确地说，如果$d$是与$q$互质的整数，我们用$E_d$表示$K=\mathbb{F}_q(t)$上的模型为$y^2=x(x+1)(x+t^d)$的椭圆曲线。我们给出了$E_d$的Tate-Shafarevich群（已知是有限的）与它的Néron-Tate调节子的阶的乘积的渐近估计，用$E_d$的指数微分高度表示为$d\to\infty$ .

Legendre's formula and $p$-adic analysis

In number theory, we know Legendre's formula $ v_p(n!) = \sum_{k \ge 1} \lfloor \frac{n}{p^k} \rfloor $, which calculates the $p$-adic valuation of the factorial, i.e. the exponent of the greatest power of a prime $p$ that divides $n!$. There is also the second (or alternative) equality $ v_p (n!) = \frac{n-s_p(n)}{p-1} $ where $s_p(n)$ is the $p$-adic weight of $n$ or the sum of digits of $n$ in base $p$. Both kinds of Legendre's formula allow us to determine valuations of the natural number, the odd factorial, binomial coefficients, Catalan numbers, and other combinatorial objects. The artic…