Articles of fractional part

Fractional part summation

Let us consider the sum $$\displaystyle S_K=\sum_{n \geq \sqrt{K}}^{2 \sqrt{K}} \left\{ \sqrt {n^2-K} \right\} $$ where $K$ is a positive integer and where $\{ \}$ indicates the fractional part. If we calculate the average value of the difference between $S_K$ and $1/2 \, \sqrt{K}$ over all positive integers $K \leq N$, we have $$\frac{1}{N} \sum_{K=1}^{N} […]

The graph of the function $f(x)= \left\{ \frac{1}{2 x} \right\}- \frac{1}{2}\left\{ \frac{1}{x} \right\} $ for $0<x<1$

Let for reals $$\{x\}=\text{Frac}(x)$$ the fractional part function, take for example the more common definition, the first (there is a different definition as you see in this MathWorld’s Page, from Graham et alumni). Thus I know that $$\text{Frac}(x)=x$$ for $0<x<1$. I am interesting in basics about the graphic representation of some of the following functions […]

Fractional part of rational power arbitrary small

I think that $\{a^n\}$ (where $\{x\}$ is $x \pmod 1$), where $a$ is fixed rational greater than 1 and $n$ is positive integer, is dense in $[0,1]$ is unsolved. However what about $\{a^n\}$ is arbitrary small for some $n$ ($a$ is fixed rational as well).

A sum of fractional parts.

I am looking to evaluate the sum $$\sum_{1\leq k\leq mn}\left\{ \frac{k}{m}\right\} \left\{ \frac{k}{n}\right\} .$$ Using matlab, and experimenting around, it seems to be $\frac{(m-1)(n-1)}{4}$ when $m,n$ are relatively prime. How can we prove this, and what about the case where they are not relatively prime? Conjecture: Numerically, it seems that for any $m,n$ we have […]

Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of $x$. Do you have any proof?

A fractional part integral giving $\frac{F_{n-1}}{F_n}-\frac{(-1)^n}{F_n^2}\ln\left(\!\frac{F_{n+2}-F_n\gamma}{F_{n+1}-F_n\gamma}\right)$

I’ve been asked to elaborate on the following evaluation: $$ \begin{align}\\ \displaystyle {\large\int_0^{1}} \!\cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots + \cfrac 1 { 1 + \psi (\left\{1/x\right\}+1)}}}} \:\mathrm{d}x & = \dfrac{F_{n-1}}{F_{n}} – \dfrac{(-1)^{n}}{F_{n}^2} \ln \!\left(\!\dfrac{F_{n+2}-F_{n}\gamma}{F_{n+1}-F_{n}\gamma} \right)\\\\ \end{align} $$ where $\left\{x\right\}=x-\lfloor x\rfloor$ denotes the fractional part of $x$, $\gamma$ is the […]

The fractional part of $n\log(n)$

When I was thinking about my other question on the sequence $$p(n)=\min_a\left\{a+b,\ \left\lfloor\frac {2^a}{3^b}\right\rfloor=n\right\}$$ I found an interesting link with the sequence $$q(n)=\{n\log(n)\}=n\log(n)-[n\log(n)]$$ the fractional part of $n\log(n)$. If we draw the sequence $q$, we get this (for $n$ up to $520$, $5\,000$ and $30\,000$ respectively): We can see some gaps looking like parabolas. What […]

For $x\in\mathbb R\setminus\mathbb Q$, the set $\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$ is dense on $[0,1)$

Let $x\in \mathbb{R}$ an irrational number. Define $X=\{nx-\lfloor nx\rfloor: n\in \mathbb{N}\}$. Prove that $X$ is dense on $[0,1)$. Can anyone give some hint to solve this problem? I tried contradiction but could not reach a proof. I spend part of the day studying this question Positive integer multiples of an irrational mod 1 are dense […]