# 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.

383 questions
Filter by
Sorted by
Tagged with
51 views

### How to proof these two series of Euler-constant [closed]

Those are called vacca series,but I don’t know how to proof it
1 vote
65 views

### Generalization of $\gamma$ as limit involving primes

The following limit is utilized in Merten’s theorem. $$\gamma = \lim\limits_{n\to\infty}\left(-\log\log n - \sum_{p\text{ prime}}^n\log\left(1-\frac{1}{p}\right)\right)$$ I’m interested in the ...
1 vote
77 views

### Evaluation of $\sum_{n=1}^{\infty}\frac{1}{n(n!)}$

WolframAlpha gives the evaluation as $\text{Ei}(1)-\gamma$. It does not offer a step-by-step solution. Where does this come from?
83 views

### Where does the Euler Mascheroni constant come from in the formula $\sum_{n≤x} 1/n= \log(x)+\gamma +O(1/x)$

From this question, Asymptotic formula for $\sum_{n\le x}\frac 1n$ The author claims without proof that this identity holds. $$\sum_{n≤x} 1/n= \log(x)+\gamma +O(1/x) .$$ It is apparently well known ...
65 views

### How to show that Euler constant lies in $(0,1)$ [duplicate]

Let $$C_n=\sum_{i=1}^n \frac{1}{i}-\log n,\quad n\geq 1$$ It is well-known that the limit of $(C_n)_{n\geq 1}$ exists, where its limit is denoted $\gamma$, Euler's constant. I'll shortly write what ...
116 views

### A sum that's possibly equal to the Euler-Mascheroni Constant $\sum_{n=1}^\infty \frac{\ln n!}{n^3}$

The following interesting sum seems to approach the Euler-Mascheroni constant $\gamma$. $$\sum_{n=1}^\infty \frac{\ln n!}{n^3} \overset{?}{=} \gamma$$ I've looked at the different ways to express the ...
522 views

### Is $\sum_{j=1}^\infty\sum_{n=1}^\infty\left(\frac{e^{-j/n}}{n^2}-\frac{e^{-n/j}}{j^2}\right)=\gamma ?$

A friend proposed the following problem: $$\sum_{j=1}^\infty\sum_{n=1}^\infty\left(\frac{e^{-j/n}}{n^2}-\frac{e^{-n/j}}{j^2}\right)=\gamma,$$ where $\gamma$ is the Euler-Mascheroni constant. The ...
109 views

231 views

99 views

30 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 ...
90 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 ...
113 views

### 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–...
97 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....
79 views

### 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.
84 views

99 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 ... 72 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?
1 vote
167 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 $\gamma$ ...
182 views

### 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 ... 271 views

1 vote
78 views

### 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 ...
1 vote
39 views

### 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, $f(13)\equiv 0.557216896821\mod 1$, ...
52 views

### 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 ...
1 vote
38 views

1 vote
95 views

62 views

### 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 ...
### Closed form of the sum $\sum_{r\ge2}\frac{\zeta(r)}{r^2}$ 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 ...