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We present many novel results in number theory, including a double seriesformula for the natural logarithm and a proof concerning the H\"older meanbased on the functional equation for the Riemann zeta function. We find aharmonic mean analogue of Chebyshev's inequality for the prime countingfunction involving the Euler-Mascheroni constant. Furthermore, we define afunction taking the H\"older mean of all positive integers up to a given numberand investigate its asymptotic behavior, finding two different patterns whichare separated by the harmonic mean. Additionally, we discuss the behavior ofsaid function at zero and discover a formula involving the Riemann zetafunction, whose continuity we prove with Riemann's functional equation.Inspired by the alternating harmonic series, we find a double series formulafor the natural logarithm, resulting in identities involving the Riemann zetafunction, binomial coefficients, and logarithms.

On a Double Series Representation of the Natural Logarithm, the  Asymptotic Behavior of H\"older Means, and an Elementary Estimate for the  Prime Counting Function