Editing List of logarithmic identities (section)

==== Deveci's Proof ====
A fundamental feature of the proof is the accumulation of the [[Subtraction#Notation and terminology|subtrahends]] <math display="inline"> \frac{1}{x}</math> into a unit fraction, that is, <math display="inline"> \frac{m}{x} = \frac{1}{n}</math> for <math>m \mid x</math>, thus <math>m = \omega + 1</math> rather than <math>m = |\mathbb{Z}_{m} \cap \mathbb{N}|</math>, where the [[Maximum and minimum|extrema]] of <math>\mathbb{Z}_{m} \cap \mathbb{N}</math> are <math>[0, \omega]</math> if <math>\mathbb{N} = \mathbb{N}_{0}</math> and <math>[1, \omega]</math> [[Natural number#Notation|otherwise]], with the minimum of <math>0</math> being implicit in the latter case due to the structural requirements of the proof. Since the [[Cardinality#Comparing_sets|cardinality]] of <math>\mathbb{Z}_{m} \cap \mathbb{N}</math> depends on the selection of one of two possible minima, the integral <math>\textstyle \int \frac{1}{t} dt</math>, as a set-theoretic procedure, is a function of the maximum <math>\omega</math> (which remains consistent across both interpretations) plus <math>1</math>, not the cardinality (which is ambiguous<ref>{{Cite arXiv |eprint=1102.0418 |first=Peter |last=Harremoës |title=Is Zero a Natural Number? |year=2011|class=math.HO }} A synopsis on the nature of 0 which frames the choice of minimum as the dichotomy between ordinals and cardinals.</ref><ref>{{Cite journal |last=Barton |first=N. |year=2020 |title=Absence perception and the philosophy of zero |journal=Synthese |volume=197 |issue=9 |pages=3823–3850 |doi=10.1007/s11229-019-02220-x |pmc=7437648 |pmid=32848285}} See section 3.1</ref> due to varying definitions of the minimum). Whereas the harmonic number difference computes the integral in a global sliding window, the double series, in parallel, computes the sum in a local sliding window—a shifting <math>m</math>-tuple—over the harmonic series, advancing the window by <math>m</math> positions to select the next <math>m</math>-tuple, and offsetting each element of each tuple by <math display="inline"> \frac{1}{m}</math> relative to the window's absolute position. The sum <math display="inline">\sum_{n=1}^{k} \sum \frac{1}{x - r}</math> corresponds to <math>H_{km}</math> which scales <math>H_{m}</math> without bound. The sum <math display="inline"> \sum_{n=1}^{k} -\frac{1}{n}</math> corresponds to the prefix <math>H_{k}</math> trimmed from the series to establish the window's moving lower bound <math>k+1</math>, and <math>\ln(m)</math> is the limit of the sliding window (the scaled, truncated<ref>The <math>k+1</math> shift is characteristic of the [[#Riemann Sum|right Riemann sum]] employed to prevent the integral from degenerating into the harmonic series, thereby averting divergence. Here, <math display="inline"> -\frac{1}{n}</math> functions analogously, serving to regulate the series. The successor operation <math>m = \omega + 1</math> signals the implicit inclusion of the modulus <math>m</math> (the region omitted from <math>\mathbb{N}_{1}</math>). The importance of this, from an axiomatic perspective, becomes evident when the residues of <math>m</math> are formulated as <math>e^{\ln(\omega + 1)}</math>, where <math>\omega + 1</math> is bootstrapped by <math>\omega = 0</math> to produce the residues of modulus <math>m = \omega = \omega_{0} + 1 = 1</math>. Consequently, <math>\omega</math> represents a limiting value in this context.</ref> series):

<math display="block">\begin{align}
\sum_{n=1}^k \sum_{r=1}^{\omega} \left( \frac{1}{mn - r} - \frac{1}{mn} \right)
&= \sum_{n=1}^k \sum_{r=0}^{\omega} \left( \frac{1}{mn - r} - \frac{1}{mn} \right) \\
&= \sum_{n=1}^k \left( -\frac{1}{n} + \sum_{r=0}^{\omega} \frac{1}{mn - r} \right) \\
&= -H_k + \sum_{n=1}^k \sum_{r=0}^{\omega} \frac{1}{mn - r} \\
&= -H_k + \sum_{n=1}^k \sum_{r=0}^{\omega} \frac{1}{(n-1)m + m - r} \\
&= -H_k + \sum_{n=1}^k \sum_{j=1}^m \frac{1}{(n-1)m + j} \\
&= -H_k + \sum_{n=1}^k \left( H_{nm} - H_{m(n-1)} \right) \\
&= -H_k + H_{mk}
\end{align}</math>
<math display="block">\lim_{k \to \infty} H_{km} - H_k = \sum_{x \in \langle m \rangle \cap \mathbb{N}} \sum_{r \in \mathbb{Z}_m \cap \mathbb{N}} \left( \frac{1}{x-r} - \frac{1}{x} \right) = \ln(\omega + 1) = \ln(m)</math>