Math Problems Vault

Problem 8.28

Let $a,b,c,\ldots ,k,l$ be any non-negative integers. We have

\max (a,b,c,\ldots ,k,l) = a+b+c+\cdots +k+l

{}-\min (a,b)-\min (a,c)-\cdots -\min (k,l)

{}+\min (a,b,c)+\cdots

\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

\pm \min (a,b,c,\ldots ,k,l) .

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Problem 8.29

Let $p$ be any prime number and $a=p^\alpha a'$ , $b=p^\beta b'$ , $c=p^\gamma c'$ , , $k=p^\kappa k'$ , $l=p^\lambda l'$ , where $a',b',c',\ldots ,k',l'$ are not divisible by $p$ . The exponent of $p$ on the left-hand side is equal to $\max (\alpha ,\beta ,\gamma ,\ldots ,\kappa ,\lambda )$ , and the exponent of $p$ on the right-hand side is equal to

\alpha +\beta +\gamma +\cdots +\kappa +\lambda -\min (\alpha ,\beta )-\min (\alpha ,\gamma )-\cdots -\min (\kappa ,\lambda )

{}+\min (\alpha ,\beta ,\gamma )+\cdots

\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

\pm \min (\alpha ,\beta ,\gamma ,\ldots ,\kappa ,\lambda ) .

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Remark.

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The least common multiple $M$ of the positive integers $a,b,c,\ldots ,k,l$ may be represented as follows:

M = abc\cdots kl(a,b)^{-1}(a,c)^{-1}\cdots (k,l)^{-1}(a,b,c)\cdots (a,b,c,\ldots ,k,l)^{\pm 1} .

Problem 8.30

We have

1 = \begin{vmatrix} 1& 0& 0& 0& \ldots & 0\\ 1& 1& 0& 0& \ldots & 0\\ 1& 1& 1& 0& \ldots & 0\\ 1& 1& 1& 1& \ldots & 0\\ \multicolumn {6}{c}{\dotfill }\\ 1& 1& 1& 1& \ldots & 1 \end{vmatrix}^2 = \begin{vmatrix} 1& 1& 1& 1& \ldots & 1\\ 1& 2& 2& 2& \ldots & 2\\ 1& 2& 3& 3& \ldots & 3\\ 1& 2& 3& 4& \ldots & 4\\ \multicolumn {6}{c}{\dotfill }\\ 1& 2& 3& 4& \ldots & n+1 \end{vmatrix} .

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Remark.

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The general element of the first determinant $\eta_{\lambda \mu }=1$ , if $\mu$ is a (proper or improper) part of $\lambda$ , otherwise it $=0$ ( $\lambda ,\mu =0,1,\ldots ,n$ ); in the second determinant the general element $c_{\lambda \mu }$ is equal to the number of common parts of $\lambda$ and $\mu$ , i.e. the smaller of the two numbers $\lambda +1$ and $\mu +1$ ( $\lambda ,\mu =0,1,\ldots ,n$ ).

Problem 8.31

Prove that $\left|c_{\lambda \mu }\right|$ is equal to 1, where $\left|c_{\lambda \mu }\right|$ denotes the determinant whose general element $c_{\lambda \mu }$ is equal to the number of common divisors of $\lambda$ and $\mu$ , in other words to the number of divisors of the greatest common divisor of $\lambda$ and $\mu$ , where $\lambda ,\mu =1,2,\ldots ,n$ .

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Problem 8.32

Let $a_0,a_1,\ldots ,a_n$ be arbitrary, $A_\nu = \sum_{t\leq \nu } a_t$ , $\nu =0,1,\ldots ,n$ . We have

\begin{vmatrix} A_0& A_0& A_0& A_0& \ldots & A_0\\ A_0& A_1& A_1& A_1& \ldots & A_1\\ A_0& A_1& A_2& A_2& \ldots & A_2\\ A_0& A_1& A_2& A_3& \ldots & A_3\\ \multicolumn {6}{c}{\dotfill }\\ A_0& A_1& A_2& A_3& \ldots & A_n \end{vmatrix} = a_0a_1a_2\cdots a_n .

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Remark.

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The general element $c_{\lambda \mu }$ of the determinant is $=A_r$ where $r=\min (\lambda ,\mu )$ , $\lambda ,\mu =0,1,\ldots ,n$ . (Generalization of 30.)

Problem 8.33

Let $a_1,a_2,\ldots ,a_n$ be arbitrary, $A_\nu = \sum_{t\mid \nu } a_t$ , $\nu =1,2,\ldots ,n$ . We have

\begin{vmatrix} A_1& A_1& A_1& A_1& \ldots & A_1\\ A_1& A_2& A_1& A_2& \ldots & A_{(2,n)}\\ A_1& A_1& A_3& A_1& \ldots & A_{(3,n)}\\ A_1& A_2& A_1& A_4& \ldots & A_{(4,n)}\\ \multicolumn {6}{c}{\dotfill }\\ A_1& A_{(n,2)}& A_{(n,3)}& A_{(n,4)}& \ldots & A_n \end{vmatrix} = a_1a_2a_3\cdots a_n .

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Remark.

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The general element $c_{\lambda \mu }$ of the determinant is $=A_r$ where $r=(\lambda ,\mu )$ , $\lambda ,\mu =1,2,\ldots ,n$ . (Generalization of 31.)

Problem 8.34

If $a_0,a_1,a_2,\ldots$ are arbitrary and $A_n = \sum_{t\leq n} a_t$ , $n=0,1,2,\ldots$ , then we clearly have

a_0=A_0, \quad a_1=A_1-A_0, \quad a_2=A_2-A_1, \quad \ldots , \quad a_n=A_n-A_{n-1}, \quad \ldots .

If $a_1,a_2,a_3,\ldots$ are arbitrary and $A_n = \sum_{t\mid n} a_t$ , $n=1,2,3,\ldots$ , then we have

a_1=A_1, \quad a_2=A_2-A_1, \quad a_3=A_3-A_1, \quad a_4=A_4-A_2,

a_5=A_5-A_1, \quad a_6=A_6-A_3-A_2+A_1, \quad \ldots

and in general

a_n = \sum _{t\mid n} \mu (t) A_{n/t} , \qquad n=1,2,3,\ldots ,

where $\mu (n)$ is the Möbius function (see definition in § 5 preceding 38).

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Problem 8.35

Denote by $\psi (y)$ an arbitrary function defined for $0 \leq y \leq 1$ . Let

g(n) = \sum _{\nu =1}^n \psi \left(\frac\nu n\right) , \qquad f(n) = \sum _{(r,n)=1} \psi \left(\frac rn\right) ,

where the last sum is extended over the numbers $r$ that are $\leq n$ and are relatively prime to $n$ . We then have

f(n) = \sum _{t\mid n} \mu (t) g\left(\frac nt\right) = \sum _{t\mid n} \mu \left(\frac nt\right) g(t) .

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Problem 8.36

As is well known, we have

\prod _{\nu =1}^n \left(x - e^{\frac{2\pi i\nu }{n}}\right) = x^n - 1 .

Set

\prod _{(r,n)=1} \left(x - e^{\frac{2\pi ir}{n}}\right) = K_n(x) ,

where the product is extended over those numbers $r$ that are $\leq n$ and relatively prime to $n$ . ( $K_n(x)$ is termed the $n$ th cyclotomic polynomial.) The zeros of $x^n-1$ are the $n$ th roots of unity, the zeros of $K_n(x)$ are the primitive $n$ th roots of unity. Derive the formula

K_n(x) = \prod _{t\mid n} \left(x^{n/t}-1\right)^{\mu (t)} .

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Problem 8.37

If $\mu (n)$ is the Möbius function, then we have

\sum _{(r,n)=1} e^{\frac{2\pi ir}{n}} = \mu (n) .

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