从窗口化方波到双曲正切、偶数ζ值和狄利克雷L值
From a Windowed Square Wave to the Hyperbolic Tangent, Even Zeta Values, and Dirichlet L-Values
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中文总结 AI 辅助
研究通过对延迟周期方波加窗并在零频率评估乘积，利用基本积分和极限，获得双曲正切、欧拉多项式等展开及和的新推导，还得出狄利克雷β值及其他狄利克雷L函数值，以及ζ(4)等，通过归纳得到ζ(2k)。
中文摘要 AI 辅助
我们用衰减符号函数对延迟周期方波进行加窗，并在时域和频域中评估零频率处的乘积。该窗口使出现的每个级数绝对收敛，因此论证仅需基本积分和一个极限。我们获得了双曲正切的米塔格-莱夫勒部分分式展开、第一类欧拉多项式E1的傅里叶展开以及巴塞尔和ζ(2)的新的统一推导。通过相同计算并逐项积分，得到狄利克雷β值β(3)=π³/32以及另外两个狄利克雷L函数在s = 3处的值。该正弦级数的逐项积分产生ζ(4)，并通过归纳法继续该过程得到每个k≥1时的ζ(2k)。
英文摘要
We window a delayed periodic square wave with a decaying signum function and evaluate the product at zero frequency in both the time and frequency domains. The window renders every series that appears absolutely convergent, so the argument requires only elementary integrals and a limit. We obtain new unified derivations for the Mittag-Leffler partial-fraction expansion for the hyperbolic tangent, the Fourier expansion of the first Euler polynomial E1, and the Basel sum, zeta(2). The same calculation, followed by termwise integration, yields the Dirichlet beta value beta(3) = pi^3/32, and the values at s = 3 of two other Dirichlet L-functions. Termwise integration of this sine series produces zeta(4), and an induction continues the process to zeta(2k) for every k greater than or equal to 1.
