论文标题
Riemann Zeta，Lerch和Dirichlet L功能的双指数方法
Double Exponential method for Riemann Zeta, Lerch and Dirichlet L-functions
论文作者
论文摘要
我们提供有效的方法来评估Riemann Zeta，Lerch Zeta和Dirichlet $ L $ functions。该方法使用riemann-siegel（RS）类型公式和适当集成的马鞍点附近的修改后的双指数（MDE）正交方法。我们为Lerch和Dirichlet $ l $ functions提供了RS公式的简化推导，其中包含有限级数和剩余积分。 MDE方法使我们能够删除剩余积分集成轮廓附近的奇异性的贡献。该方法使我们能够评估剩余积分的任何规定准确性。与主系列总和相比，评估它们的数值成本很小。因此，即使是这些功能的高度振荡积分也可以与RS公式相同的复杂性进行评估。尤其是，评估$ζ$和LERCH ZETA功能的数值复杂性，具有$ s =σ+i t $ scales为$ \ sqrt {t} $，对于dirichlet $ l $ - function，它缩放为$ \ max（q，q，\ sqrt {q t}）$。该方法允许自动设置积分截止和有限的离散大小，以实现规定的准确性。此外，它确保了离散时间间隔的每一半几乎都会使结果的准确性增加一倍。因此，它允许进一步检查获得的结果的准确性。
We provide efficient methods to evaluate the Riemann zeta, the Lerch zeta and the Dirichlet $L$-functions. The method uses the Riemann-Siegel (RS) type formulas and a modified double exponential (MDE) quadrature method near the saddle point of appropriate integrands. We provide simplified derivations of the RS formulas, containing finite series sums and residual integrals, for the Lerch and the Dirichlet $L$-functions. The MDE method allows us to remove the contribution of the singularities near the contour of integration for the residual integrals. The method allows us to evaluate the residual integrals to any prescribed accuracy. The numerical cost of evaluating them is minimal compared to the main series sums. Thus even highly oscillatory integrands for these functions can be evaluated with the same complexity as the RS formula. In particular, the numerical complexity to evaluate the $ζ(s)$ and the Lerch zeta function with $s=σ+i t$ scales as $\sqrt{t}$, and for Dirichlet $L$-function it scales as $\max(q, \sqrt{q t})$. The method allows the automatic setting of integral cutoffs and finite discretization size to achieve prescribed accuracy. Furthermore, it ensures that every halving of the discretization interval almost leads to doubling the accuracy of the results. Thus, it allows for a further check on the accuracy of the results obtained.
