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Questions tagged [fractional-part]

For questions related to the fractional part of a number.

2
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1answer
30 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}\}-\...
6
votes
2answers
121 views

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 $\...
13
votes
3answers
253 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 ...
0
votes
0answers
48 views

Trigonometric Integral Involving the fractional part

Let $\{\}$ denote the fractional part function, does the following Integral admit a closed-form ? $$\int_{0}^{\pi/2}\bigg\{\frac{1}{\cos(x)}\bigg\}dx$$
2
votes
2answers
64 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 ? $$\int_{0}^{1}\bigg\{\frac{1}{x^s}\bigg\}dx\,$$
1
vote
0answers
42 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 (...
1
vote
1answer
61 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 $\{z\}=z-\lfloor z\rfloor$...
0
votes
1answer
99 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 $0 < m < n < p$ ...
0
votes
2answers
27 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 ...
5
votes
1answer
89 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" ...
0
votes
1answer
38 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 ...
0
votes
1answer
40 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
votes
1answer
63 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}...
6
votes
3answers
165 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
votes
1answer
76 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 ...
1
vote
3answers
41 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}\...
3
votes
0answers
55 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 ? $$\...
3
votes
5answers
257 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 ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{\,x}\bigg\}\bigg\{\frac{1}{x\,y}\bigg\}dx\,dy\,$$
0
votes
1answer
53 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 ...
1
vote
2answers
100 views

symmetric double-integral on fractional part

Let $\{\}$ denotes the fractional part function, does the following double-integral have a closed-form ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{x}+\frac{1}{y}\bigg\}dx\,dy$$
2
votes
0answers
89 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 ...
0
votes
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 \...
1
vote
2answers
151 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
votes
3answers
113 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
votes
1answer
173 views

Double fractional part integral

Let $\{\}$ denote the fractional part, does the following integral have a closed form ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{x\,y}\bigg\}^2dx\,dy$$
4
votes
3answers
80 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{\...
0
votes
0answers
92 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
votes
1answer
72 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!\...
-1
votes
1answer
59 views

A question about the set of discontinuity of a function related to a series representation for $\zeta(3)$ [closed]

In this post I was inspired in a series used by Apéry, see the MathWorld's article Apéry's Constant to define the following function for $0\leq x\leq 1$ $$f(x)=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^3\...
0
votes
1answer
53 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 $\...
3
votes
2answers
100 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
38 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
64 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 $$7[x]+23\{x\}=191$$ ...
2
votes
4answers
73 views

$\lim_{n \to \infty}\left \langle \sqrt{n} \right \rangle $ doesn't exist

I want to prove that $lim_{n \to \infty}\left \langle \sqrt{n} \right \rangle $ doesn't exists. $\left \langle x \right \rangle $ is defined to be the fractional part. I've managed to prove that if ...
2
votes
1answer
117 views

How to integrate the “fractional part”/sawtooth function?

Let us define $\{t\} = t - \lfloor t \rfloor$ this is also sometimes referred to as frac$(t)$. With this in mind how would I calculate $$I := \int_a^b f(\{t\}) dt$$ for some function $f$? I ask ...
1
vote
0answers
71 views

How to show $a^{\frac{1}{b}}-c^{\frac{1}{d}}$ is rational if and only if $a^{\frac{1}{b}}$ and $c^{\frac{1}{d}}$ are rational and not equal.

Given $a,d,c,d\in\mathbb{N}$, define $$ X = a^\frac{1}{b} $$ $$ Y = c^\frac{1}{d} $$ $$ Z = X-Y $$ I'm pretty sure the following is true: For any $X\neq Y$ that satisfy the above relationships, $...
5
votes
0answers
74 views

Inequality involving power to fractional part of integer multiples of logarithm of integer to coprime base.

For $x \in \mathbb{R}^+$, let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. Let $k \in \mathbb{N}$. Show that $2^{\{k \log_2(3)\}} < \dfrac{2}{1 + 2^{-k}}$ for $k > 1$. ...
2
votes
1answer
91 views

The behaviour of $\sum_{\mathcal{R}\leq x}\left\{\frac{x}{\mathcal{R}}\right\}$, where $\mathcal{R}$ denote Ramanujan primes

Here we denote with $\left\{x\right\}$ the fractional part function, and being $R_n$ for $n\geq 1$ the sequence of Ramanujan primes, see this Wikipedia, with the notation $\mathcal{R}\leq x$ we mean ...
0
votes
1answer
37 views

Fractional part of integer multiples

If $x\in\mathbb{R}-\mathbb{Q}$, I can prove that $$x-\lfloor x\rfloor,\ 2x-\lfloor 2x\rfloor,\ \dots,\ nx - \lfloor nx\rfloor $$ are all distinct. If we assume otherwise then $$ (i-j)x = \lfloor ix\...
2
votes
1answer
122 views

Justify an approximation of $\int_{0}^1\binom{ \left\{ 1/z\right\}}{z}dz$, where $ \left\{ z \right\}$ denotes the fractional part function

I'm interested in to know an approximation for this integral $$\int_{0}^1\binom{ \left\{ \frac{1}{z} \right\}}{z}dz,\tag{1}$$ that I've created this morning. Here $ \left\{ x \right\}$ denotes the ...
8
votes
4answers
598 views

Compute a division with integer and fractional part

I have a problem that I don't know how to solve: Compute $[\frac{\sqrt{7}}{frac(\sqrt{7})}]$ Here's what I've tried: $[\sqrt{7}]=2 \rightarrow frac(\sqrt{7}) = \sqrt{7}-2 \rightarrow [\frac{\sqrt{7}...
2
votes
0answers
66 views

How get a good approximation of integrals involving the gamma function, exponentials and the fractional part?

While I was playing with Wolfram Alpha online calculator I wondered how to calculate and justify an approximation of an integral involving the fractional part function, the exponential and the gamma ...
5
votes
1answer
130 views

How to solve equations like $2^n = 9… ?$

Consider natural powers of $2$ and their first $k$ digits. For example find the smallest natural number ( positieve integer ) $n$ such that : $$2^n = 3...$$ Is solved by $2^5 = 32 $ and thus $n=5$. ...
7
votes
1answer
99 views

Recurrence Relation involving Fractional Part

I am considering the following sequence: $$a_0=\sqrt 2$$ $$a_{n+1}=a_n+\{a_n\}$$ where $\{x\}:= x-\lfloor x\rfloor$ denotes the fractional part function. Since I have observed that the sequence ...
1
vote
1answer
38 views

For $x \in [0, 1]$, find explicit form for the $n_k$ such that $\{n_k\sqrt{2}\} \to x$.

This is inspired by the comment in Minimum values of the sequence $\{n\sqrt{2}\}$ that, by Kroneckers Approximation Theorem, the fractional part of $n\sqrt{2}$ is dense in $[0, 1]$. My question is ...
21
votes
3answers
353 views

Minimum values of the sequence $\{n\sqrt{2}\}$

I have been studying the sequence $$\{n\sqrt{2}\}$$ where $\{x\}:= x-\lfloor x\rfloor$ is the "fractional part" function. I am particularly interested in the values of $n$ for which $\{n\sqrt{2}\}$ ...
0
votes
1answer
276 views

Continuity of fractional part function for negative integers

Given a function $f(x)=frac(x^2)-(frac(x))^2$ We have to check continuity at $x=-2$ I know that $frac(x)= x-[x]$ where $[x]$ is the greatest integer function less than or equal to x. Checking ...
0
votes
2answers
30 views

A question involving fractional part of a number

Let $n \in \mathbb{N}$ and consider $2^n = r \cdot 10^l +$ terms involving lower powers of $10$. Then we have the following relations: $$r 10^l \le 2^n < (r+1)10^l$$ $$\Leftrightarrow \log_{10} r +...
6
votes
1answer
101 views

Express in terms of constants and series $-\int_0^1\log\left( \left\{ x^{-1/3} \right\} \right)dx$

This morning I've calculated the case $k=2$ of an integral involving the fractional part function $ \left\{ x \right\} $, this $$-\int_0^1\log\left( \left\{ x^{-1/k} \right\} \right)dx,\tag{1}$$ in ...
1
vote
0answers
24 views

the same convergents among 3 irrotations

Let a, b, c be irrational numbers with a < b < c. If a and c have identical convergents $\frac{p_0}{q_0}$, $\frac{p_1}{q_1}$, . . . , up to $\frac{p_n}{q_n}$, how to prove that b also has these ...