Local asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron

Ramanujan Summation and the Exponential Generating Function $ \sum_{k=0}^{\infty}\frac{z^{k}}{k!}ζ^{\prime}(-k)$

In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from w…

Applications of an elementary resolution of singularities algorithm to exponential sums and congruences modulo p^n

We use the resolution of singularities algorithm of [G4] to provide new estimates for exponential sums as well as new bounds on how often a function f(x) such as a polynomial with integer coefficients is divisible by various powers of a prime p when x is an integer. They are proved using p-adic analogues of the theorems of [G3] on R^n sublevel set volumes and oscillatory integrals with real phase function. The proofs of these analogues use aspects of the resolution of singularities algorithms of [G4] (but for the most part not the actual resolution of singularities theorems themselves.) Unlike…

三阶Berndt型积分及与Jacobi椭圆函数相关的级数

Berndt-Type Integrals of Order Three and Series Associated with Jacobi Elliptic Functions

本文利用Jacobi椭圆函数的Fourier级数和Maclaurin级数展开及围道积分法，通过Gamma函数特殊值建立了六类双曲和与两族三阶Berndt型积分的显式求值，并给出了相关推论与示例。

A Generalization of Classical Formulas in Numerical Integration and Series Convergence Acceleration

Summation formulas, such as the Euler-Maclaurin expansion or Gregory's quadrature, have found many applications in mathematics, ranging from accelerating series, to evaluating fractional sums and analyzing asymptotics, among others. We show that these summation formulas actually arise as particular instances of a single series expansion, including Euler's method for alternating series. This new summation formula gives rise to a family of polynomials, which contain both the Bernoulli and Gregory numbers in their coefficients. We prove some properties of those polynomials, such as recurrence ide…

从给定通项求任意级数的和

Finding the sum of any series from a given general term

欧拉通过泰勒展开推导欧拉-麦克劳林求和公式，并应用于整数幂和、调和级数部分和及zeta函数近似。

Summability Calculus

In this paper, we present the foundations of Summability Calculus, which places various established results in number theory, infinitesimal calculus, summability theory, asymptotic analysis, information theory, and the calculus of finite differences under a single simple umbrella. Using Summability Calculus, any given finite sum of the form $f(n) = \sum_{k=a}^n s_k\, g(k,n)$, where $s_k$ is an arbitrary periodic sequence, becomes immediately \emph{in analytic form}. Not only can we differentiate and integrate with respect to the bound $n$ without having to rely on an explicit analytic formula …