计算截断θ函数及其导数和积分的近似最优方法
A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals
摘要：
提出了一种计算截断θ函数及其导数和积分的多对数时间方法。该方法是初等的，严格的，明确的，适合于计算机实现。我们重复地将泊松求和公式应用于截断的θ函数，同时在每次重复之后适当地归一化线性和二次参数。该方法依赖于复指数的周期性，这使得参数的适当归一化成为可能，并且依赖于高斯的自相似性，这确保我们在每次应用泊松求和之后仍然获得截断的θ函数。换句话说，我们的方法依赖于θ函数的模性质。讨论了它在黎曼ζ函数的数值计算和求Waring型丢番图方程的解的个数中的应用。
英文摘要：
A poly-log time method to compute the truncated theta function, its derivatives, and integrals is presented. The method is elementary, rigorous, explicit, and suited for computer implementation. We repeatedly apply the Poisson summation formula to the truncated theta function while suitably normalizing the linear and quadratic arguments after each repetition. The method relies on the periodicity of the complex exponential, which enables the suitable normalization of the arguments, and on the self-similarity of the Gaussian, which ensures that we still obtain a truncated theta function after each application of the Poisson summation. In other words, our method relies on modular properties of the theta function. Applications to the numerical computation of the Riemann zeta function and to finding the number of solutions of Waring type Diophantine equations are discussed.
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