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arXiv · 2211.10751

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

Sinan Deveci
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Abstract

We present many novel results in number theory, including a double series formula for the natural logarithm and a proof concerning the H\"older 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 H\"older 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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Sinan Deveci. 2022-11-19. 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. https://arxiv.org/abs/2211.10751

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