Editing List of logarithmic identities (section)

== Asymptotic identities ==
=== [[Pronic_number|Pronic numbers]] ===
As a consequence of the [[#Harmonic_number_difference|harmonic number difference]], the natural logarithm is [[Asymptotic analysis|asymptotically]] approximated by a finite [[Series (mathematics)|series]] difference,<ref name="Deveci2022DoubleSeries"/> representing a truncation of the [[#Riemann_Sum|integral]] at <math>k = n</math>:

:<math>H_{{2T[n]}} - H_n \sim \ln(n+1)</math>

where <math>T[n]</math> is the {{mvar|n}}th [[triangular number]], and <math>2T[n]</math> is the sum of the [[Pronic_number#As_figurate_numbers|first {{mvar|n}} even integers]]. Since the {{mvar|n}}th [[pronic number]] is asymptotically equivalent to the {{mvar|n}}th [[Square number|perfect square]], it follows that:

:<math>H_{{n^2}} - H_n \sim \ln(n+1)</math>
=== [[Prime number theorem]] ===
The [[Prime_number_theorem#Statement|prime number theorem]] provides the following asymptotic equivalence:

:<math>\frac{n}{\pi(n)} \sim \ln n</math>

where <math>\pi(n)</math> is the [[Prime-counting function|prime counting function]]. This relationship is equal to:<ref name="Deveci2022DoubleSeries"/>{{rp|2}}

:<math>\frac{n}{H(1, 2, \ldots, x_n)} \sim \ln n</math>

where <math>H(x_1, x_2, \ldots, x_n)</math> is the [[harmonic mean]] of <math>x_1, x_2, \ldots, x_n</math>. This is derived from the fact that the difference between the <math>n</math>th harmonic number and <math>\ln n</math> asymptotically approaches a [[Euler's constant|small constant]], resulting in <math>H_{{n^2}} - H_n \sim H_n</math>. This behavior can also be derived from the [[#Logarithm_of_a_power|properties of logarithms]]: <math>\ln n</math> is half of <math>\ln n^2</math>, and this "first half" is the natural log of the root of <math>n^2</math>, which corresponds roughly to the first <math>\textstyle \frac{1}{n}</math>th of the sum <math>H_{n^2}</math>, or <math>H_n</math>. The asymptotic equivalence of the first <math>\textstyle \frac{1}{n}</math>th of <math>H_{n^2}</math> to the latter <math>\textstyle \frac{n-1}{n}</math>th of the series is expressed as follows:

:<math>\frac{H_n}{H_{n^2}} \sim \frac{\ln \sqrt{n}}{\ln n} = \frac{1}{2}</math>

which generalizes to:

:<math>\frac{H_n}{H_{n^k}} \sim \frac{\ln \sqrt[k]{n}}{\ln n} = \frac{1}{k}</math>

:<math>k H_n \sim H_{n^k}</math>

and:

:<math>k H_n - H_n \sim (k - 1) \ln(n+1)</math>
:<math>H_{{n^k}} - H_n \sim (k - 1) \ln(n+1)</math>
:<math>k H_n - H_{{n^k}} \sim (k - 1) \gamma</math>

for fixed <math>k</math>. The correspondence sets <math>H_n</math> as a [[Unit_of_measurement|unit magnitude]] that partitions <math>H_{n^k}</math> across powers, where each interval <math>\textstyle \frac{1}{n}</math> to <math>\textstyle \frac{1}{n^2}</math>, <math>\textstyle \frac{1}{n^2}</math> to <math>\textstyle \frac{1}{n^3}</math>, etc., corresponds to one <math>H_n</math> unit, illustrating that <math>H_{n^k}</math> forms a [[Harmonic_series_(mathematics)#Definition_and_divergence|divergent series]] as <math>k \to \infty</math>.

=== Real Arguments ===
These approximations extend to the real-valued domain through the [[Harmonic_number#Alternative,_asymptotic_formulation|interpolated harmonic number]]. For example, where <math>x \in \mathbb{R}</math>:
:<math>H_{{x^2}} - H_x \sim \ln x</math>

=== [[Stirling number|Stirling numbers]] ===
The natural logarithm is asymptotically related to the harmonic numbers by the [[Stirling_numbers_of_the_first_kind#Asymptotics|Stirling numbers]]<ref>{{cite arXiv 
| title = New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials
| author = Khristo N. Boyadzhiev
| year = 2022
| eprint = 1911.00186
| pages = 2, 6
| class = math.NT
}}</ref> and the [[Gregory_coefficients#Bounds_and_asymptotic_behavior|Gregory coefficients]].<ref>{{cite book 
| last = Comtet 
| first = Louis 
| title = Advanced Combinatorics 
| publisher = Kluwer 
| year = 1974
}}</ref> By representing <math>H_n</math> in terms of [[Harmonic_number#Identities_involving_harmonic_numbers|Stirling numbers of the first kind]], the harmonic number difference is alternatively expressed as follows, for fixed <math>k</math>:

:<math>\frac{s(n^k+1, 2)}{(n^k)!} - \frac{s(n+1, 2)}{n!} \sim (k-1) \ln(n+1)</math>