### 数学代写|数论作业代写number theory代考|MATH538101

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• Foundations of Data Science 数据科学基础

## 数学代写|数论作业代写number theory代考|IRRATIONALITY OF THE EXPONENTIAL CONSTANT

Once we get beyond radical expressions and decimals, irrationality proofs, for the most part, become significantly harder. A notable exception is the irrationality of the exponential constant $e$. Apart from the intrinsic interest of the result, its proof provides our first glimpse of an idea which will recur again and again in irrationality arguments, and which we shall employ extensively in Chapters 2 and $5 .$

Theorem 1.9. The exponential constant $e$ is irrational.
Proof. Assume that $e=p / q$ is rational. That is,
$$\frac{p}{q}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots,$$
and for any positive integer $n$, we have
$$\frac{p n !}{q}=n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\cdots+1+R,$$
where $R$ (which depends on $n$ ) is given by
$$R=\frac{n !}{(n+1) !}+\frac{n !}{(n+2) !}+\cdots$$
We can estimate $R$ in terms of a geometric series:
$$R=\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\cdots<\frac{1}{n+1}+\frac{1}{(n+1)^{2}}+\cdots=\frac{1}{n} .$$
In particular, choose $n=q$. Then
$$R=\frac{p n !}{q}-\left(n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\cdots+1\right)$$
is clearly an integer; but using (1.1), we have $0<R<1$. This is impossible, and so $e$ is irrational.

Observe that this proof relies essentially on an infinite series for $e$, and therefore has to involve concepts of calculus. In some sense this may be surprising, as number theory is usually thought of as studying discrete systems while calculus is the science of the continuous; in another sense there should be no surprise, as it is not even possible to define the number $e$ without recourse to calculus techniques. Whether it is in fact a surprise or not, we shall find that many of our future proofs will be expressed in terms of calculus.

## 数学代写|数论作业代写number theory代考|OTHER RESULTS, AND SOME OPEN QUESTIONS

It is known that $\pi$ is irrational: we shall prove this in the next chapter. It is not hard to see that at least one of the numbers $\pi+e$ and $\pi e$ must be irrational (in fact, at least one must be transcendental – see Chapter 3 ); although, most likely, both are irrational, this has not been proved for either one individually. As a consequence of a difficult result due to Gelfond and Schneider (Theorem $5.18$ ) we know that $e^{\pi}$ is irrational; however it is still unknown whether or not $\pi^{e}$ is irrational. It can also be shown that various numbers such as, for example, $e^{\sqrt{2}}$ and $2^{\sqrt{2}}$ are irrational. However, the irrationality of $\pi^{\sqrt{2}}$ and $2^{e}$, and that of the Euler-Mascheroni constant
$$\gamma=\lim {n \rightarrow \infty}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\log n\right)=0.57721 \cdots$$ remain undecided. Another problem which has attracted much attention is to investigate the irrationality of the numbers $\zeta(n)$. Here $n \geq 2$ is an integer and $\zeta$ is the Riemann zeta function defined by $$\zeta(s)=\sum{k=1}^{\infty} \frac{1}{k^{s}}=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\cdots$$
for $s>1$. By methods of complex integration we can show that if $n$ is even then $\zeta(n)$ is a rational number times $\pi^{n}$, and this is known to be irrational. On the other hand, it is much harder to find out anything of interest about $\zeta(n)$ for odd $n$. In 1978 the French mathematician R. Apéry sensationally proved that $\zeta(3)$ is irrational. His complicated argument had the appearance of being completely unmotivated, and all of the techniques he had used would have been available two centuries earlier: for these reasons, few people believed that the proof could possibly be correct. Nevertheless it was found possible eventually to confirm all of Apéry’s assertions and thereby establish what has been called “a proof that Euler missed”. A brief (but not easy!) account of Apéry’s work is given in [66].

## 数学代写|数论作业代写number theory代考|IRRATIONALITY OF THE EXPONENTIAL CONSTANT

$$\frac{p}{q}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\cdots$$

$$\frac{p n !}{q}=n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\cdots+1+R$$

$$R=\frac{n !}{(n+1) !}+\frac{n !}{(n+2) !}+\cdots$$

$$R=\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\cdots<\frac{1}{n+1}+\frac{1}{(n+1)^{2}}+\cdots=\frac{1}{n}$$

$$R=\frac{p n !}{q}-\left(n !+\frac{n !}{1 !}+\frac{n !}{2 !}+\cdots+1\right)$$

## 数学代写|数论作业代写number theory代考|OTHER RESULTS, AND SOME OPEN QUESTIONS

$$\gamma=\lim n \rightarrow \infty\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\log n\right)=0.57721 \cdots$$

$$\zeta(s)=\sum k=1^{\infty} \frac{1}{k^{s}}=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\cdots$$

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