论文标题
在自然对数的双重序列表示上，霍尔德均值的渐近行为和素数函数的基本估计值
On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function
论文作者
论文摘要
我们在数字理论中介绍了许多新颖的结果，包括自然对数的双重序列公式，以及基于Riemann Zeta函数的功能方程的有关Hölder平均值的证明。我们发现Chebyshev不平等的谐波平均值类似于涉及Euler-Mascheroni常数的主要计数函数。此外，我们定义了一个功能，将所有正整数中的Hölder平均值达到给定数字并研究其渐近行为，发现了两种不同的模式，这些模式被谐波平均值分开。此外，我们讨论了零时所述函数的行为，并发现涉及Riemann Zeta函数的公式，我们用Riemann的功能方程证明了其连续性。受到交替谐波系列的启发，我们找到了自然对数的双重系列公式，导致涉及Riemann Zeta函数，二项式系数和对数的身份。
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.
