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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25 views

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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76 views

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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65 views

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 is an approximate closed form for sum of $n^n$ series?

I read about this https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula which is Euler-Maclaurain and I did my calculation to find the constants. However, the answer differ greatly from ...
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1answer
87 views

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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53 views

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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157 views

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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254 views

Prove that $\int _{-\infty }^{+\infty }{\frac {\mathrm {d} z}{(\phi ^{n}z)^{2}+(F_{2n+1}-\phi F_{2n})(e^{\gamma }z^{2}+\zeta (3)z-\pi )^2}}=1$

Recently while dealing with few interesting integrals, I was quite fascinated by this one: $$\int _{-\infty }^{+\infty }{\dfrac {1}{(\phi ^{n}z)^{2}+(F_{2n+1}-\phi F_{2n})(e^{\gamma }z^{2}+\zeta (3)z-\...
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Do these constants appear in other areas of mathematics?

I decided to consider divergent (to infinity) integrals as some new kind of number. Towards this end, I began by establishing certain rules defining equivalence of the integrals. It seems the usual ...
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Dos the Euler-Mascheroni constant $\gamma$ correspond to infinite hyperbolic angle?

So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
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Euler constant as limit of zeta function

I want to prove that $ \lim \limits_{s \to 0} (\zeta(1+s)+\zeta(1-s))= 2\gamma$ , I divide it to two limits, $\lim \limits_{s \to 1^{+}} (\zeta(s)-\frac{1}{s-1}) = \gamma$ which I proved using the ...
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165 views

Bounds on the difference between the polylogarithm with negative base and the gamma function

Trying to understand intuitively the Gamma function I started to think of it as a way to measure how much each factorial power "helps" $x^n$ in the infinite sum of $e^x$, thus trying to ...
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92 views

Does $c_{\ln(x)}=\lim_{n\to\infty}\left(\sum_{k=1}^n \ln(k)-\int_1^n \ln(x)\ dx\right)$ converge?

Does $c_{\ln(x)}=\displaystyle\lim_{n\to\infty}\left(\sum_{k=1}^n \ln(k)-\int_1^n \ln(x)\ dx\right)$ converge? First of all, I always thought Mascheroni was spelled (and pronounced) Masechroni, but ...
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Construction, similar to Chow's EL-numbers? Is it valid? What are the properties?

The idea of EL-numbers, proposed by Chow, impressed me very much, so I decided to build something similar and look what this will turn out. Instead of $\exp(x)$ and $\ln(x)$ functions as the building ...
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Stieltjes constants found in the limit as $x \to \,1^+$

Mathematica 12 evaluates the following sequence of limits thus (tested up to $s=100$) $$\underset{x\to \,1^+}{\text{lim}}\;\left(\frac{\partial ^n }{\partial x^n}\left(\frac{1}{1-x}+\sum _{k=1}^{\...
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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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Set of divergent integrals

So, we take $\frac{\text{sgn}(x-1)}{x}$ and apply $\mathcal{L}_t[t f(t)](x)$ four times. The transform keeps area. These integrals are minus Euler-Mascheroni constant: $$\int_0^\infty \frac{\text{sgn}(...
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Why is $\ln 0\ne-\ln \infty$?

The title of this post is intentionally sensational, but what I am really going to do is to compare the divergent integrals $\int_0^1\frac1xdx$ and $\int_1^\infty\frac1xdx$. Let's consider the ...
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Infinite sum related to hypergeometric series

I was working on some integrals and I came across the following series: $$\sum_{k=1}^{\infty}\frac{1}{k(k+n)!}$$ Wolfram Alpha evaluates it to be $$\frac{_2F_2\left(1,1;2,n+2;1\right)}{(n+1)!}$$ Which ...
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1answer
80 views

On a certain integral related to Euler-Masheroni constant

When trying to derive Mertens' theorem using Perron's formula, I encountered the following integral: $$ \int_0^1{1-e^{-t}\over t}\mathrm dt-\int_1^\infty{e^{-t}\over t}\mathrm dt $$ which, according ...
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Simple approximation for $\pi$ by means of well known constant like $\pi \sim \dfrac{2e-\frac{3\sqrt{2}\gamma^2}{4}}{\phi}$? [closed]

For community wiki, On the occasion of the World day of $\pi$, I want to look for simple formulas which involves $\pi$ using known constants like $e, \phi,\gamma,\cdots$ , For instance just a ...
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Prove the limit equation: $\lim_{x\rightarrow\infty}e^{-x}\sum_{n=0}^{\infty}f_{n}\left(x\right)=-\gamma.$ FYI$,\quad\gamma$ is Euler's constant

Define a function sequence $\left \{ f_{n} \right \}_{n= 0}^{\infty}$ on $\left ( 0, \infty \right )$ as $$f_{0}\left ( x \right )= \ln x,\quad f_{n+ 1}\left ( x \right )= \int_{0}^{x}f_{n}\left ( t \...
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1answer
93 views

Euler-Mascheroni and von Mangoldt function

I know that by their respective definitions: $$\gamma_0 = \sum_{n=1}^{\infty}{\frac{1}{n}-\ln(\frac{n+1}{n})}$$ $$\ln(n)=\sum_{d|n}{\Lambda(d)}$$ and I want to get there: $$\gamma_0 = \sum_{n=1}^{\...
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42 views

Expressing Gamma Function as an euler constant

Q. Use the definition $$ \Gamma(x) = \lim_{n\to \infty} \frac{n!n^x}{x(x+1)\cdots(x+n)} $$ in this problem. a) Show that $ \Gamma (x+1) = x\Gamma(x) $ for $ x>0 $ b) First, express $\mathbb{ln}\...
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74 views

Negative continued fraction of Euler Mascheroni constant

$\gamma$, the Euler-Mascheroni constant, has the following simple regular continued fraction: $$\gamma=[0; 1, 1, 2, 1, 2, 1,\dots]=0+ \cfrac{1}{ 1+\cfrac{1}{ 1+\cfrac{1}{ 2+\cfrac{1}{ ...
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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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211 views

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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1answer
38 views

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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2answers
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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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153 views

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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67 views

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 ...
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1answer
131 views

Prove the de la Vallée-Poussin's formula

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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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^...
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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 ...
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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 $...
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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\...
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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 ...
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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!}=...
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114 views

The Cauchy principal value of the Riemann Zeta function

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 ...

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