通过莫尔德尔积分计算截断θ函数
Computing the truncated theta function via Mordell integral
AI总结：
本文提出简化Hiary算法，用莫尔德尔积分替代泊松求和公式，高效计算截断θ函数，提升实现便捷性。
摘要：
Hiary[3]提出一种算法，可在O(ln(n/ε)^κ)次算术运算内将截断θ函数∑_{k=0}^n exp(2πi(zk+τk²))计算至±ε精度，适用于数论，尤其用于计算ζ(1/2+it)至±t^{-λ}精度的算法。本文提出简化Hiary算法，避免使用泊松求和公式，改用显式涉及莫尔德尔积分的恒等式，得到高效、概念简单且易实现的算法。
英文摘要：
Hiary [3] has presented an algorithm which allows to evaluate the truncated theta function $\sum_{k=0}^n \exp(2πı(zk+τk^2))$ to within $\pm ε$ in $O(\ln(\tfrac{n}ε)^κ)$ arithmetic operations for any real $z$ and $τ$. This remarkable result has many applications in Number Theory, in particular it is the crucial element in Hiary's algorithm for computing $ζ(\tfrac{1}{2}+ıt)$ to within $\pm t^{-λ}$ in $O_λ(t^{\frac{1}{3}}\ln(t)^κ)$ arithmetic operations, see [2]. We present a significant simplification of Hiary's algorithm for evaluating the truncated theta function. Our method avoids the use of the Poisson summation formula, and substitutes it with an explicit identity involving the Mordell integral. This results in an algorithm which is efficient, conceptually simple and easy to implement.
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