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Questions tagged [fractional-part]

For questions related to the fractional part of a number.

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2answers
37 views

Finding the limit of this recurrence relation

Let's say we have the equation $$a_{n+1} = Frac\left(\frac{1}{\ln(1+a_n)}\right)$$ with $a_0\ne 0$ and that it is not negative. I've tried finding the limit of $a_n$ as $n \rightarrow \infty$, but I ...
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2answers
36 views

Find the total number of solution of the equation

Find number of solution of :- $$[x]^2 = x + 2\{x\}$$ where $[.]$ represents greatest integer function and $\{.\}$ represents fractional part.
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1answer
56 views

Counting integers with a least prime factor greater than $x$ in a sequence of $x$ consecutive integers.

It is well known from Sylvester-Schur that in any sequence of $x$ consecutive integers, there is always at least one integer divisible by a prime greater than $x$. I am interested in counting the ...
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1answer
19 views

proving floor, and rounding floats in some programming languages

Here is a simple property which is very useful while programming : In most programming languages, ...
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2answers
75 views

Counting the number of integers with their least prime factor greater than $x$ between $ax$ and $ax+x$

Let: $x \ge 2, a \ge 1$ be integers. $x\#$ be the primorial for $x$ $\mu(i)$ be the möbius function. $\text{lpf}(x)$ be the least prime factor of $x$. $p_k$ be the $k$th prime which is the highest ...
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1answer
126 views

Does $\sum_{k=1}^n|\cot \sqrt2\pi k|$ tends to $An\ln n$ as $n\to\infty$?

Question: How can we prove that $$L(n)=\sum_{k=1}^n\left|\cot \sqrt2\pi k\right|=\Theta(n\log n)$$ as $n\to\infty$? Furthermore, if $\sqrt2$ is replaced with a quadratic irrational number, does it ...
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2answers
36 views

Calculate the difference in wavelength of the Balmer-$\alpha$ line ($n = 3$ to $n = 2$) in hydrogen and deuterium

In order to predict correctly the wavelengths of the hydrogen lines it is necessary to use in the expression for $R_{\infty}$ the reduced mass of the electron:$$\mu=\frac{m_e\,m_N}{m_e+m_N}$$ where $...
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1answer
64 views

Doubt on Integration

$$\int_0^4 \lfloor x/2 \rfloor \ d(x-\lfloor x \rfloor)$$ I don't get how we convert the given differential element into normal dx differential element. I plotted the graphs of $$\lfloor x/2 \...
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1answer
53 views

Reasoning with fractional parts and the Möbius function.

Let $S(p_k,x)$ be the set of all elements $s$ where $s \le x$ and gcd$(s,p_k\#)=1$ where $p_k$ is the $k$th prime and $p_k\#$ is the primorial for $p_k$. Let $|S(p_k,x)|$ be the count of elements in $...
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1answer
107 views

Integral $\int_0^1 x^n\left\{\frac{k}{x}\right\}dx$

I am trying to solve the following integral containing fractional part function (denoted by $\{.\}$) $$\int_0^1 x^n\left\{\frac{k}{x}\right\}dx,\ 0<k\le 1,\ n\in \mathrm N^*$$ For $n=0$, it is ...
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1answer
26 views

Fractional part function of $\frac{y}{n} + k$

I am wondering why is $\{\frac{y}{n} + k\} = \{\frac{y}{n}\}$ where $k$ is an integer. $\{x\}$ is a fractional part function where $\{x\} = x - \lfloor x\rfloor$. I know it makes sense logically, but ...
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2answers
78 views

Finding $\lim\limits_{x\to 0} \frac{\lfloor x \rfloor}{x}$ and the different definitions of fractional part function.

I understand that $\lim\limits_{x\to 0} \frac{\lfloor x \rfloor}{x}$ does not exist because RHL is $0$ and LHL is $\infty$. However, when I tried to calculate the limit of the equivalent expression $1-...
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1answer
53 views

Trigonometric integral involving the fractional part function

I have tried the $u$ substitution yet I could not move forwards on this problem. Could you calculate in closed-form this integral? $$\int_{0}^{\pi/2}\bigg\{\frac{1}{\sin(x)}\bigg\}\mathrm{dx}$$
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0answers
121 views

$| \sin x| > \frac{\sqrt{2-\sqrt{2}}}{2}$ iff $1/8< \{ \frac{x}{\pi}\} < 7/8$ where $\{x\}$ is the fractional part of $x$

This is a problem that arose while reading the book "Putnam and Beyond": Why is $| \sin x| > \frac{\sqrt{2-\sqrt{2}}}{2}$ iff $\frac18 < \{ \frac{x}{\pi}\} < \frac78$ where $\{x\}$ is the ...
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3answers
52 views

Find $\lim_{x \to -1} 1/(\sqrt{|x|-\{-x\}})$ where $\{\}$ denotes the fractional part.

Find $\lim_{x \to -1} 1/(\sqrt{|x|-\{-x\}})$ where $\{\}$ denotes the fractional part. My attempt - $$\lim_{x \to -1} \frac{1}{\sqrt{|x| -\{x\}+1}}\\ \lim_{x \to -1} \frac{1}{\sqrt{|x|+1 -x +[x]}}$$ ...
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1answer
31 views

If $\lim_{x\to c^-}${$\ln x$} and $\lim_{x\to c^+}${$\ln x$} exists finitely but they are not equal

If $\lim_{x\to c^-}${$\ln x$} and $\lim_{x\to c^+}${$\ln x$} exists finitely but they are not equal (where {} denotes fractional part function),then $(a)c$ can take only rational values $(b)c$ can ...
4
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1answer
92 views

On the logarithm of the fractional part Integral

Let $\{\}$ denote the fractional part function, then does the following integral admit a closed-form ? $$\int_{0}^{1}x\ln\bigg(\bigg\{\frac{1}{x}\bigg\}\bigg)dx$$
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1answer
39 views

Fractional part distribution

It is known that the distribution of $\{\sqrt{n} \}$, evaluated over the integer values of $n$, is uniform in the interval $[0,1)$. Let us consider the sum $$S(K)=\sum_{n=1}^K \left(\{\sqrt{n}\}-\...
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2answers
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Evaluating $\liminf_{n\to\infty}n\{n\sqrt2\}$

How can we evaluate $$\liminf_{n\to\infty}n\{n\sqrt2\},$$where $\{\cdot\}$ denotes the fractional part of $\cdot$? The first thing came to my mind is Pell's equation $x^2-2y^2=1$. Knowing that $\...
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3answers
287 views

Sum of $\{n\sqrt{2}\}$

I would like to prove (rigorously, not intuitively) that $$\sum_{n=1}^N \{n\sqrt{2}\}=\frac{N}{2}+\mathcal{O}(\sqrt{N})$$ where $\{\}$ is the "fractional part" function. I understand intuitively why ...
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0answers
61 views

Trigonometric Integral Involving the fractional part

Let $\{\}$ denote the fractional part function, does the following Integral admit a closed-form ? $$\int_{0}^{\pi/2}\bigg\{\frac{1}{\cos(x)}\bigg\}dx$$
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2answers
76 views

Integral involving the fractional part function

Let $\{\}$ denote the fractional part function and $s>1$ be a real number, then does the following integral admit a closed-form ? $$\int_{0}^{1}\bigg\{\frac{1}{x^s}\bigg\}dx\,$$
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1answer
73 views

The functional equation $f(-x+b)=f(x)$

I can solve the (periodic) functional equation $f(x+b)=f(x)$ completely ($x\in \mathbb{R}$ and $b\neq 0$). Indeed, its general solution is $f=\phi o (\; )_b$, where $(\; )_b$ is the $b$-decimal (...
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1answer
73 views

Double integral involving fractional part

I am interested in solving the following definite double integral $$\int_0^1\int_0^1\Bigl\{\frac{1}{x}\Bigr\}\Bigl\{\frac{1}{y}\Bigr\}\frac{(1-x)(1-y)}{1-xy} dx dy$$ where $\{z\}=z-\lfloor z\rfloor$...
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1answer
105 views

Proving that $\left\{ \frac{sm}{p} \right\} < \left\{ \frac{sn}{p} \right\} < \frac{s}{p}$ if and only if $s\not\mid p-1$

I have the following question from USAMO 2006: "Let $p$ be a prime number and let $s$ be an integer with $0 < s < p$. Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ ...
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2answers
31 views

How to get the fractional part of a product of large numbers with machine precision?

Let us have about 100 or so random (exact) floats such as: $$ A_1 = 1234123.428\\ A_2 = 13713.4193\\ A_3 = 0.1332\\ A_4 = 123.13213\\ ...$$ Now I want to find an efficient way to get the fractional ...
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1answer
103 views

Definite integral on fractional part

Prove following definite integral: $$\int_0^1\Bigl\{\frac{1}{x}\Bigr\}\ln(x)\,dx = \gamma_0+\gamma_1-1$$ Found in "Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis" ...
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1answer
42 views

Functional equation $f(a)·f(b)=\Bigl\lbrace\frac{1}{ab}\Bigr\rbrace$

I am trying to get all possible solutions of the following functional equation: $$f(a)·f(b)= \Bigl\lbrace\frac{1}{ab}\Bigr\rbrace$$ Where {} mean fractional part function. Solutions only need to be ...
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1answer
49 views

PDF of random variable containing fractional part $x=\{1/r\}$

What happens if we combine fractional part problems and randomness? For a start here are two examples. Let $r$ be a random variable with a given probability distribution function (PDF) $f(r)$ and ...
2
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1answer
74 views

Double integral over fractional part of Einstein-like velocity sum $\int_0^1 \int_0^1 \{\frac{u+v}{1-u v}\} \,dudv$

Looking for more interesting and complicated examples of this type of problem I propose this one inspired by Einstein's addition theorem for relativistic velocities $$i =\int_0^1 \int_0^1 \{\frac{u+v}...
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3answers
195 views

Closed form of integral over fractional part $\int_0^1 \left\{\frac{1}{2}\left(x+\frac{1}{x}\right)\right\}\,dx$

Recenly, several interesting questions have been posted asking for closed forms of integrals over the fractional part of certain functions. For me the story started with Evaluation of $\int_{0}^{1}\...
4
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1answer
79 views

Value of $\lim_{n\to \infty} \frac{1}{2^n} \sum_{k=1}^{2^n} \{\log_{2}k\}$

Is there a closed form expression of $$\lim_{n\to \infty} \frac{1}{2^n} \sum_{k=1}^{2^n} \{\log_{2}k\}$$ , where $\{x\}=x-\lfloor x \rfloor$ denotes the fractional part of $x$? $\text{I think this ...
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3answers
44 views

Solve for $x$ in the equation containing ${\lfloor{x}\rfloor}$ and $\{x\}$

Calculate all possible values of $x$ satisfying, $$\frac{\lfloor{x}\rfloor}{\lfloor{x-2}\rfloor}-\frac{\lfloor{x-2}\rfloor}{\lfloor{x}\rfloor}=\frac{8\{x\}+12}{\lfloor{x}\rfloor \lfloor{x-2}\...
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0answers
70 views

Alternating series combining harmonic number and zeta values

While evaluating the following fractional part integral, I get stuck on an almost euler sum as highlighted in red colour. Could someone evaluate the red series in terms of well-known constants ? $$\...
4
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5answers
288 views

Evaluation of $\int_{0}^{1}\int_{0}^{1}\{\frac{1}{\,x}\}\{\frac{1}{x\,y}\}dx\,dy\,$

Let $\{\}$ denote the fractional part function, does the following double integral have a closed-form ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{\,x}\bigg\}\bigg\{\frac{1}{x\,y}\bigg\}dx\,dy\,$$
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1answer
61 views

Variant of Stieltjes constants

New Identities For any positive integers $k\geq1$ and $j\geq2$, let $x_j=\frac{j+\sqrt{j^2-4}}{2}$. Let us define $(\text{A}_k)_{k\geq1}$ by the following constants "which are variants of Stieltjes ...
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2answers
107 views

symmetric double-integral on fractional part

Let $\{\}$ denotes the fractional part function, does the following double-integral have a closed-form ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{x}+\frac{1}{y}\bigg\}dx\,dy$$
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0answers
131 views

limit of sequence involving the fractional part

Using the pigeonhole principle, any sequence of the form $(\{\frac{n}{r}\})_{n\geq1}$ where $r$ is an irrational number is dense in the unit interval. Then prove that the following limit does not exit ...
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1answer
46 views

How is it possible to solve a second degree polynomial combined with a modulo?

I have two equations: a second degree polynomial and one with a modulo. There are two variables: $ x, y \in \mathbb{R}$; $ x, y \geq 0 $, and four constants, which have a known value: $a, b, c, d \in \...
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2answers
159 views

Fractional part of the floor function Integral [closed]

Let $\lfloor\rfloor\ $ and $\{\}$ denote the floor function and the fractional part funtion, respectively. Then calculate in closed-form the following integral $$\int_{0}^{1}\bigg\{\frac{1}{x}\bigg\...
2
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3answers
131 views

Square root of fractional part integral

Does the following integral have a closed form ? $$\int_{0}^{1}\sqrt{\bigg\{\frac{1}{x}\bigg\}}dx$$ Where $\{x\}$ denotes the fractional part of $x$.
2
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1answer
190 views

Double fractional part integral

Let $\{\}$ denote the fractional part, does the following integral have a closed form ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{x\,y}\bigg\}^2dx\,dy$$
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votes
3answers
94 views

Prove that $ \left\lfloor{\frac xn}\right\rfloor= \left\lfloor{\lfloor{x}\rfloor\over n}\right\rfloor$ where $n \ge 1, n \in \mathbb{N}$ [duplicate]

Prove that $ \left\lfloor{\frac xn}\right\rfloor= \left\lfloor{\lfloor{x}\rfloor\over n}\right\rfloor$ where $n \ge 1, n \in \mathbb{N}$ and $\lfloor{.}\rfloor$ represents Greatest Integer $\mathbf{\...
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0answers
107 views

Integrals involving the fractional part function and the W-Lambert function

I am trying make interesting integrals involving the fractional part function and special functions. I wondered if it is possible to deduce a series representation (in the atempt to get a closed-form ...
2
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1answer
73 views

Can $\{n!\log n\}$ converge to $1$?

Since $H_n-\log n\to\gamma$ is it correct to deduce that, if $\gamma$ were a rational $a/b $, then $$\lim_{n\to \infty }\{n!H_n-n!\log n\}=\lim_{n\to \infty }\{-n!\log n\}=1-\lim_{n\to \infty }\{n!\...
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1answer
60 views

An approximation of a definite integral involving the Gudermannian function and the fractional part function

I wondered how to calculate an approximation of next integral involving the fractional part function $\{x\}=x-\lfloor x\rfloor$ and the so-called Gudermannian function denoted in this post as $\...
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1answer
111 views

Integral $\int_{0}^{1}\int_{0}^{1}\sin\big\{ \tfrac{x}{y} \big\}\sin\big\{ \tfrac{y}{x} \big\}dxdy$

This integral was posted on AopS but so far it didn't receive an answer, is there a closed form for this integral? $$\int_{0}^{1}\int_{0}^{1}\sin\bigg\{ \frac{x}{y} \bigg\}\sin\bigg\{ \frac{y}{x} \...
3
votes
2answers
104 views

Limit involving Series and Greatest Integer Function

If $[$.$]$ denotes the greatest integer function, then find the value of $\lim_{n \to \infty} \frac{[x] + [2x] + [3x] + … + [nx]}{n^2}$ What I did was, I wrote each greatest integer function $[x]$ as ...
1
vote
1answer
40 views

On continuity of floor, modulus and fractional part function.

If $$f(x)=\begin{cases}\dfrac{e^{\lfloor x\rfloor}+|x|-1}{\lfloor x\rfloor+\{2x\}}&,\ x\neq0 \\\ \frac{1}{2}&,\ x=0\end{cases}$$comment on continuity of $f(x)$ at $x=0$. Where $\lfloor .\...
1
vote
2answers
70 views

Find the real values of $x$ that satisfy the equation $7[x]+23\{x\}=191$

For any real number $x$, $[x]$ denotes the largest integer less than or equal to $x$ (i.e. floor function) and $ \{x\}=x-[x]$ .Then, the number of real solutions of the equation $$7[x]+23\{x\}=191$$ ...