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PostPosted: Wed May 25, 2016 1:44 pm 

Joined: Sat Nov 14, 2015 6:32 am
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Location: Melbourne, Australia
Let $f:\mathbb{Z} \rightarrow \mathbb{R}$ be a function with period $T>0$, that is $f\left(x+T\right)=f(x)$ forall $x \in \mathbb{Z}$. The least period of a function divides every other period of it.

Let ${\rm a, b}$ be two natural coprime numbers and let $f, g$ be two functions that are onto $\mathbb{Z}$ and have a least period ${\rm a, b}$ respectively. Prove that $f+g$ is periodic and its least period is ${\rm ab}$.

Does the result hold necessary if ${\rm a, b}$ are not coprime? Explain your answer.

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$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$


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