AI 中文总结
该研究利用赫尔维茨ζ函数与勒让德Φ函数的对称性，将黎曼函数方程推广至含三个变量的形式，扩展其定义域至所有实数b，同时解决了勒让德Φ函数有限求和的解析延拓问题。
AI 中文摘要
研究背景：文献中已有的广义黎曼函数方程可扩展至实数b≥0，这一发现源于利用赫尔维茨ζ函数关于移位参数b的新公式的对称性。由此，研究者寻找勒让德Φ函数Φ(e^z,k,b)的类似关系，以一个新公式为起点，再次利用关于b的对称性。方法与细节：该研究得到了含三个变量的黎曼函数方程的推广，借助一个新的自包含勒让德Φ公式，将其定义域扩展至b>1，使该关系对所有实数b成立；同时，作为次要目标，解决了勒让德Φ函数公式中出现的有限求和的解析延拓问题。结论与意义：本文通过清晰的初等数学方法，展示了如何推导函数方程。
英文摘要
The discovery that the generalized Riemann functional equation from the literature can be extended to real $b \ge 0$, by exploiting symmetries of a new formula for the Hurwitz zeta function with respect to the shift parameter $(b)$, led to the search for a similar relation for the Lerch $Φ$ function, $Φ(e^z,k,b)$, where a new formula is used as a starting point and the symmetries with respect to $b$ are exploited again. This search resulted in a generalization of the Riemann functional equation with three variables, whose domain is then extended to $b>1$, with the aid of a new self-contained Lerch $Φ$ formula, making the relation hold for all real $b$. As a secondary objective, we address the problem of obtaining the analytic continuation of a finite summation that appears in a formula for the Lerch $Φ$ function. Above all, this paper demonstrates, with clear elementary mathematics, how a functional equation can be derived.
Comments26 pages
