关于自然对数的二重级数表示、Hölder平均的渐近性态和素计数函数的初等估计
On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function
摘要：
我们在数论中提出了许多新的结果，包括自然对数的二重级数公式和基于黎曼ζ函数的函数方程的Hölder平均的证明。我们找到了一个调和平均模拟切比雪夫不等式的主要计数函数涉及的欧拉-Mascheroni常数。此外，我们定义了一个函数的所有正整数的Hölder平均到一个给定的数字，并研究其渐近行为，发现两个不同的模式是由调和平均分开。此外，还讨论了该函数在零点处的性质，发现了一个涉及黎曼ζ函数的公式，并利用黎曼函数方程证明了该函数的连续性.受交替调和级数的启发，我们找到了自然对数的二重级数公式，从而得到了涉及黎曼ζ函数、二项式系数和多项式的恒等式。
英文摘要：
We present many novel results in number theory, including a double series formula for the natural logarithm and a proof concerning the Hölder mean based on the functional equation for the Riemann zeta function. We find a harmonic mean analogue of Chebyshev's inequality for the prime counting function involving the Euler-Mascheroni constant. Furthermore, we define a function taking the Hölder mean of all positive integers up to a given number and investigate its asymptotic behavior, finding two different patterns which are separated by the harmonic mean. Additionally, we discuss the behavior of said function at zero and discover a formula involving the Riemann zeta function, whose continuity we prove with Riemann's functional equation. Inspired by the alternating harmonic series, we find a double series formula for the natural logarithm, resulting in identities involving the Riemann zeta function, binomial coefficients, and logarithms.
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