# Questions tagged [euler-mascheroni-constant]

For questions related to Euler's constant $\gamma$, which is defined to be the limiting difference between the natural logarithm and the harmonic series.

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### Explaining Algebraic Independence of $\delta$ and $\gamma$

I have been attempting to read this paper by Alexander Aptekarev. In it, he proves that $\delta$ and $\gamma$ cannot both be rational simultaneously. He also notes that this result follows from a ...
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### How to derive this series for $\gamma$ that is only involving odd integer values of $\zeta(s)$?

With $\gamma$ being the Euler Mascheroni constant, this series is well known: $$1- \sum_{n=2}^{\infty} \frac{\zeta(n)-1}{n} = \gamma \tag{1}$$ The following series involving $\zeta(2n+1)$ also seems ...
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### Intermediate step in deriving integral representation of Euler–Mascheroni constant: $\int_0^1\frac{1-e^{-t}-e^{-1/t}}{t}dt$

I'm following a complex analysis course and am making an exercise in which I have to derive an integral representation for the Euler–Mascheroni constant. I have the following definition of the Euler–...
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### How to derive Euler-Maclaurin sum formula from Taylor Series?

Page 152 at https://link.springer.com/content/pdf/10.1007%2F978-0-387-73468-2.pdf Hi readers, I have tried substituting y'(0) , y''(0) , y'''(0) and y''''(0) into equation A that is the first equation....
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### Closed form of sum of n^n series? [closed]

Hi readers, may i know how this formula is derived? I thought of using https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula which is Euler-Maclaurian to find the formula but its not.
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### What are some interesting problems in the intersection of Diophantine Approx and Algebraic Geometry?

I am a first year graduate student and I am eager to work on irrationality/transcendental proofs of specific numbers like Euler's constant gamma. Because backgrounds for Elliptic Curves include very ...
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### Sum of 1.5-powers of natural numbers using Euler-Maclaurin Formula?

Hi, I have read this Sum of 1.5-powers of natural numbers and answers posted by Mark Viola. However, I am not too sure on how he got to this. Can anyone show me as I am quite new to series?
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### Why is $\lim\limits_{x\to\infty}\frac{\sum_{i=1}^x(\sum_{j=1}^i\frac1j-\ln i-\gamma)}{\sum_{i=1}^x\frac1i}=\frac12$?

$$\mbox{Why is}\quad\lim_{x\to\infty} \frac{\sum_{i = 1}^{x}\left[\sum_{j = 1}^{i}1/j -\ln\left(i\right)-\gamma\right]}{\sum_{i = 1}^{x}1/i} = \frac{1}{2}\ ?.$$ I learnt Euler's Constant $\gamma$ ...
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### Ergodic Theory and Euler-Mascheroni Constant

Originally posted on mathoverflow but didn't get an answer. I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or ζ(5). A ...
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### Good review article for Euler's Constant $\gamma$

I am writing a paper on Euler's constant (or Euler-Mascheroni constant) $\gamma$ with 4 other people and we are looking for a good review article on the subject. So far we haven't found anything. A ...
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### Does $\lim_{k\to\infty}2^k \log_2(2^k-1)\equiv\gamma\mod 1$ where $k\in\mathbb{N}$ (Euler-Mascheroni Constant)

I was working with the function $f(k)=2^k \log_2(2^k-1)$ and had noticed that it's factional component seemed to converge when $k\in\mathbb{N}$. For example, $f(13)\equiv 0.557216896821\mod 1$, ...
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### What's the typical role of the constant $e^{-\gamma}$?

I often encounter this constant in my research, but I wonder what typical roles does it play in other areas of mathematics? Wikipedia mentions probability theory but nothing exact. Also, I am ...
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### Prove complex logarithm problem involving Euler Mascheroni constant

I want to prove/disprove the following equation: $$\text{Log}{\frac{1}{\sqrt{z^2+1}}} = -\frac12 \ln\left(\frac{1}{z^2}+1\right)-\ln\lvert z\rvert - \gamma$$ Which $\gamma$ is the Euler Mascheroni ...
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### Closed form of the sum $\sum_{r\ge2}\frac{\zeta(r)}{r^2}$

Note:This is the same question, but it doesn't answer my question, the answer doesn't give a closed form. In fact, the answer is not accepted. Moreover, I don't think that a 9 month old inactive ...
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### Generalized Euler Constants

According to Wikipedia, the generalized Euler constants are $$\gamma_\alpha=\lim_{n\to \infty} \Big(\sum_{k=1}^n \frac{1}{k^\alpha} - \int_1^n \frac{1}{x^\alpha} \,dx \Big)$$ for $0<\alpha<1$. I ...
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### Harmonic series approximation and principal branch natural logarithm

I'm studying a paper in which they use an approximation for a harmonic series. $\displaystyle\sum_{i=1}^{t-1} \frac{1}{(t-i){i}} = \frac{1}{t} ( 2Log(t-{\frac{1}{2}}) + 2gamma)$ where 'gamma' is the ...
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### Inequality of harmonic number $\left| \sum_{i=1}^{n}\frac{1}{i}-\log{n}-\gamma \right| \leq \frac{10}{n}$

In my Number Theory textbook, it was quoted without proof that for all positive integers $n$, $$\left| \sum_{i=1}^{n}\frac{1}{i}-\log{n}-\gamma \right| \leq \frac{10}{n}$$ where $\gamma = 0.577...$ is ...
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### Find the limit $\lim\limits_{n \to \infty} \left ( \left (1 + \frac {1} {2} + \cdots + \frac {1} {n} \right ) - \ln n \right ).$ [duplicate]

Prove that the following limit exists and find the limit $:$ $$\lim\limits_{n \to \infty} \left ( \left (1 + \dfrac {1} {2} + \cdots + \dfrac {1} {n} \right ) - \ln n \right ).$$ I know that the ...
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### Calculate the sum of $\:\frac{x}{5} + \:\frac{x}{11} + \:\frac{x}{17} … + \:\frac{x}{p}$

Let's say $x = 161$, and I want to calculate the sum of all : $$\:\frac{x}{5} + \:\frac{x}{11} + \:\frac{x}{17} .... + \:\frac{x}{p}$$ $$(\text{pattern} = \:\frac{x}{\left(6n-1\right)}\:)$$ but the ...
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### Rewriting a harmonic series

Anyone to help me rewrite this : $$\sum _{n=1}^{p}\:\frac{x}{\left(6n-1\right)}\:+\:\frac{x}{\left(6n+1\right)}$$ Where, $p=\left\lfloor \frac{\sqrt{x}-1}{6}\right\rfloor$ to this ( approximations ...
The Euler-Mascheroni constant is defined as $\gamma := \lim_{x\to\infty}(H_n - \ln\,n)$ where $H_n = \sum_{k=1}^{n}\frac{1}{k}$, and the de la Vallée-Poussin's formula states that: $$\gamma = \lim_{n\... 2answers 125 views ### Prove \gamma = \int_{0}^{1}\frac{1-e^{-u}}{u}\,du - \int_{1}^{+\infty} \frac{e^{-u}}{u}\,du  How do we prove this integral representation of the Euler-Mascheroni constant ?$$\gamma = \int_{0}^{1}\frac{1-e^{-u}}{u} du - \int_{1}^{+\infty} \frac{e^{-u}}{u} du $$Here are the three intermediate ... 1answer 127 views ### Does this \int_{0}^{\infty}(\frac{\log x}{e^x})^n dx always have a closed form for n being positive integer ? what about its irrationality? It is known that \int_{0}^{\infty}\left(\frac{\log x}{e^x}\right)^n dx=-\gamma for n=1 and for n=2 we have :\frac{1}{12}(\pi^2+6(\gamma+\log 2)^2) and for n=3 we have this form , What I have ... 0answers 128 views ### Why is the irrationality of \gamma so ellusive? (Euler–Mascheroni constant) Perhaps I'm in way over my head asking this; regardless, considering the abundance of known formulas and properties of \gamma, seemingly comparable in quantity to the likes of \pi or e, why is ... 3answers 88 views ### Prove that \int_{0}^{\infty} e^{-x}x\log{x}\ dx =1-\gamma [closed] I tried to this equation:$$\int_{0}^{\infty} e^{-x}x\log{x}\ dx =1-\gamma$$but I was stuck because I couldn't avoid the appearance of Ei(0) and 1/0 for my bad way. I want to know how to avoid ... 2answers 85 views ### What is \cos x-\cos2x+\cos3x-\cos4x…\pm\cos(Nx)? I want to arrive at a closed expression for f_N(x)=\frac{2}{π}(\cos x-\cos2x+\cos3x-\cos4x...\pm \cos Nx) (+\cos(Nx) if N is odd, -\cos(Nx) if N is even) Using the fact that \cos(x)=\frac{e^... 1answer 59 views ### Does the green area converge to a known constant when n\to \infty? Let n denote the number of the rectangles in the figure above. We know that the gray area converges to Euler-Mascheroni constant (\gamma) when n\to \infty. I have three questions about the ... 1answer 53 views ### Number of iterations to find number with precision Given the Euler constant, computed by this formula$$\sum_{n=0}^{\infty}\frac{2n+1}{(2n)!} $$n is positive integer, it represents sum of fractions starting from 0 to infinity.We are given a number ... 1answer 77 views ### Imaginary numbers calculation for DFT I am trying to understand the Fourier transformation and the math behind it, so I was trying to use this formula:$$ x_k = \sum_{n=0}^{N-1} x_n e^{-\frac{j2{\pi}kn}{N}} $$to calculate all  x_{0\... 0answers 91 views ### Proving \int_{0}^{1}\int_{0}^{1}\frac{x-1}{(1-xy)\ln(xy)} \mathrm dy \mathrm dx =\gamma The integral is \int_{0}^{1}\int_{0}^{1}\frac{x-1}{(1-xy)\ln(xy)} dydx  This is the special case for the Hadjicosta's formula for s\to -1 . The proof of which was done by Jonathan Sondow. I ... 2answers 59 views ### How to solve \sum _{n=0}^{\infty }\:\frac{n^a}{n!} for any a \geq 1 in general? While doing a problem I encountered this summation$$\sum _{n=1}^{\infty }\:\frac{n^2}{n!}=2e.$$I used$$\sum _{n=0}^{\infty }\:\frac{n}{n!}=e$$and$$\sum _{n=0}^{\infty }\:\frac{(n-1)^2}{n!}=...
In many online sources, you can find $$\zeta(s) \overset{C.P.}{=} \lim_{\epsilon \to 0} \left(\frac{\zeta(s+\epsilon)+\zeta(s-\epsilon)}{2}\right).$$ This seems quite logical, but I neither know how ...