Editing Euler's constant (section)

=== Relation to triangular numbers ===
Numerous formulations have been derived that express <math>\gamma</math> in terms of sums and logarithms of [[triangular numbers]].<ref name="Boya2008AnotherRelation">{{Cite journal
| last = Boya
| first = L.J.
| title = Another relation between π, e, γ and ζ(n)
| journal = Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
| volume = 102
| pages = 199–202
| year = 2008
| issue = 2
| url = https://doi.org/10.1007/BF03191819
| doi = 10.1007/BF03191819
| bibcode = 2008RvMad.102..199B
| quote = "γ/2 in (10) reflects the residual (finite part) of ζ(1)/2, of course."
}} See formulas 1 and 10.</ref><ref name="Jonathan2005DoubleIntegrals">{{Cite journal
| last = Sondow
| first = Jonathan
| title = Double Integrals for Euler's Constant and <math>\textstyle \frac{4}{\pi}</math> and an Analog of Hadjicostas's Formula
| journal = The American Mathematical Monthly
| volume = 112
| issue = 1
| year = 2005
| pages = 61–65
| url = https://doi.org/10.2307/30037385
| doi = 10.2307/30037385
| jstor = 30037385
| access-date = 2024-04-27
}}</ref><ref>{{Cite journal
| last = Chen
| first = Chao-Ping
| title = Ramanujan's formula for the harmonic number
| journal = Applied Mathematics and Computation
| volume = 317
| year = 2018
| pages = 121–128
| issn = 0096-3003
| doi = 10.1016/j.amc.2017.08.053
| url = https://www.sciencedirect.com/science/article/pii/S0096300317306112
| access-date = 2024-04-27
}}</ref><ref>{{cite journal
 | last = Lodge
 | first = A.
 | title = An approximate expression for the value of 1 + 1/2 + 1/3 + ... + 1/r
 | journal = Messenger of Mathematics
 | volume = 30
 | year = 1904
 | pages = 103–107
 | url = https://books.google.com/books?id=K4daAAAAYAAJ&dq=%22An%20approximate%20expression%20for%20the%20value%20of%201%2B%22&pg=PA103
}}</ref> One of the earliest of these is a formula<ref>{{Cite arXiv
| last = Villarino
| first = Mark B.
| title = Ramanujan's Harmonic Number Expansion into Negative Powers of a Triangular Number
| year = 2007
| eprint = 0707.3950
| class = math.CA
| quote = It would also be interesting to develop an expansion for n! into powers of m, a new ''Stirling'' expansion, as it were.
}} See formula 1.8 on page 3.</ref><ref>{{Cite journal
| last = Mortici
| first = Cristinel
| year = 2010
| title = On the Stirling expansion into negative powers of a triangular number
| journal = Math. Commun.
| volume = 15
| pages = 359–364
| url = https://www.researchgate.net/publication/228562533
| doi =
}}</ref> for the {{nowrap|<math>n</math>th}} [[harmonic number]] attributed to [[Srinivasa Ramanujan]] where <math>\gamma</math> is related to <math>\textstyle \ln 2T_{k}</math> in a series that considers the powers of <math>\textstyle \frac{1}{T_{k}}</math> (an earlier, less-generalizable proof<ref>{{Cite journal |last=Cesàro |first=E. |title=Sur la série harmonique |journal=Nouvelles annales de mathématiques: Journal des candidats aux écoles polytechnique et normale |volume=4 |pages=295–296 |year=1885 |url=http://eudml.org/doc/100057 |language=fr |publisher=Carilian-Goeury et Vor Dalmont}}</ref><ref>{{cite book
 | last = Bromwich
 | first = Thomas John I'Anson
 | title = An Introduction to the Theory of Infinite Series
 | publisher = American Mathematical Society
 | year = 2005
 | orig-date = 1908
 | edition = 3rd
 | location = United Kingdom
 | url = https://www.dbraulibrary.org.in/RareBooks/An%20introduction%20to%20the%20theory%20of%20infinite%20series.pdf
 | page = 460
}} See exercise 18.</ref> by [[Ernesto Cesàro]] gives the first two terms of the series, with an error term):

:<math>\begin{align} 
    \gamma  
    &= H_u - \frac{1}{2} \ln 2T_u - \sum_{k=1}^{v}\frac{R(k)}{T_{u}^{k}}-\Theta_{v}\,\frac{R(v+1)}{T_{u}^{v+1}}
\end{align}</math>

From [[Stirling's approximation]]<ref name="Boya2008AnotherRelation"/><ref>{{cite book
 | last1 = Whittaker
 | first1 = E.
 | last2 = Watson
 | first2 = G.
 | title = A Course of Modern Analysis
 | edition = 5th
 | orig-date = 1902
 | year = 2021
 | page = 271, 275
 | isbn = 9781316518939
 | doi = 10.1017/9781009004091
}} See Examples 12.21 and 12.50 for exercises on the derivation of the integral form <math>\textstyle \int_{-1}^{0} \ln\Gamma(z+1)\,dz</math> of the series <math>\textstyle \sum_{k=1}^{n} \frac{\zeta(k)}{110_{k}} = \ln(\sqrt{2\pi})</math>.</ref> follows a similar series:

:<math>\gamma = \ln 2\pi - \sum_{k=2}^{\infty} \frac{\zeta(k)}{T_{k}}</math>

The series of inverse triangular numbers also features in the study of the [[Basel problem]]{{sfn|Lagarias|2013|p=13}}<ref>{{cite journal |last=Nelsen |first=R. B. |title=Proof without Words: Sum of Reciprocals of Triangular Numbers |journal=Mathematics Magazine |volume=64 |issue=3 |year=1991 |pages=167|doi=10.1080/0025570X.1991.11977600 }}</ref> posed by [[Pietro Mengoli]]. Mengoli proved that <math>\textstyle \sum_{k = 1}^\infty \frac{1}{2T_k} = 1</math>, a result [[Jacob Bernoulli]] later used to estimate the [[Basel_problem#The_Riemann_zeta_function|value]] of <math>\zeta(2)</math>, placing it between <math>1</math> and <math>\textstyle \sum_{k = 1}^\infty \frac{2}{2T_k} = \sum_{k = 1}^\infty \frac{1}{T_{k}} = 2</math>. This identity appears in a formula used by [[Bernhard Riemann]] to compute [[Eulers_constant#Relation_to_the_zeta_function|roots of the zeta function]],<ref>{{Cite book | last = Edwards | first = H. M. | title = Riemann's Zeta Function | publisher = Academic Press | year = 1974 | series = Pure and Applied Mathematics, Vol. 58 | pages = 67, 159}}</ref> where <math>\gamma</math> is expressed in terms of the sum of roots <math>\rho</math> plus the difference between Boya's expansion and the series of exact [[Unit fraction|unit fractions]] <math>\textstyle \sum_{k = 1}^{\infty} \frac{1}{T_{k}}</math>:

:<math>\gamma - \ln 2 = \ln 2\pi + \sum_{\rho} \frac{2}{\rho} - \sum_{k = 1}^{\infty} \frac{1}{T_k}</math>