  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.
