Determinant of arithmetic functions

Let f(a,b)=\sum_{d|a, \ d|b}d. Let M=(a_{i,j})_{i,j=1}^{n}. Evaluate \det M.

Solution. The idea is utilization of Lower-upper triangular decomposition. Using a software (like Mathematica) one observes that for small n the matrices L and U coulde be intepreted as - L=(l_{i,j})_{i,j=1}^{n}, where l_{i,j}=\begin{cases} 1, & \text{j divides i}\\                            0, & \text{otherwise} \end{cases} and U=(u_{i,j})_{i,j=1}^{n} where u_{i,j}=\begin{cases} i, & \text{i divides j}\\                            0, & \text{otherwise} \end{cases}
It is directly verified that this holds in general (all n).
Obviously \det L=1, while \det U=n! (the diagonal elements are 1,2,\ldots,n). Thus \det M=n!.

The same idea could be employed for similar looking determinants, for example the one having \gcd(i,j) at place (i,j).

Comments

Post a Comment

Popular posts from this blog

Basel problem type sum, proposed by prof. Skordev

Evaluate \sum_{n=1}^{\infty}\frac{1}{2^n n^2}. Proof. The function S(x):=\sum_{n=1}^{\infty}\frac{x^n}{ n^2} is well defined for x\in[-1,1] and thus we need to find S(1/2). Using differentiation, we observe that S(x)=-\int_0^x\frac{\log(1-t)}{t}\text{d}t. If we make change of variables in the latter integral t\to 1-t we obtain S(x)=-\int_{1-x}^1\frac{\log t}{1-t}\text{d}t. \ \ \ \ \ (1) On the other hand, using integration by parts, we obtain S(x)=-\int_0^x\log(1-t)\text{d}\log t=-\log t\log(1-t)\Big|_{t=0}^x-\int_0^x\frac{\log t}{1-t}\text{d}t. Notice that \displaystyle \lim_{t\to 0}\log t \log(1-t)=0 (\log (1-t)\sim t and then L'Hopital), hence S(x)=-\int_0^x\log(1-t)\text{d}\log t=-\log x\log(1-x)-\int_0^x\frac{\log t}{1-t}\text{d}t. \ \ \ \ \ (2) Plugging x=1/2 and equating (1) to (2) we obtain \int_{1/2}^1\frac{\log(1-t)}{t}\text{d} t-\int_{0}^{1/2}\frac{\log(1-t)}{t}\text{d} t= \log\left(\frac{1}{2}\right)^2. On the ot...
Read more

Modification on a sequence from VJIMC

The first two limits of the following problem were proposed at VJIMC, 2005, Category I. The exact value of the last limit was proposed by a user at https://artofproblemsolving.com , where you can also see his solution along other lines.  Let (x_n)_{n\ge 2} be a sequence of real numbers, such that x_2>0 and for every n\ge 2 holds  x_{n+1}=-1+\sqrt[n]{1+nx_n}. Prove consecutively that  1)\ \lim_{n\to\infty}x_n=0,\  \ \ 2)\ \lim_{n\to\infty}nx_n=0,\  \ \ 3)\ \lim_{n\to\infty}n^2x_n=4. \textbf{Proof.}  \textbf{1)} Clearly all the elements of the sequence are positive. The inequality -1+\sqrt[n]{1+nx_n}<x_n is equivalent to (1+nx_n)<(1+x_n)^n, which is seen to be true after expanding, since all the summands on the right are positive. This shows that the sequence is strictly decreasing. Hence 0<x_{n+1}=-1+\sqrt[n]{1+nx_n}\le -1+\sqrt[n]{1+nx_2}, and since the right hand side clearly tends to 0 we obtain $\displaystyle\lim_...
Read more

A problem proposed by prof. Babev

Consider the sequence a_n=\int_{n}^{n+1}\frac{\sin^2(\pi t)}{t}\text{d}t. It could be proven that \lim_{n\to \infty}n a_n=\frac{1}{2}  Now we are going to prove that \lim_{n\to\infty}n\left(na_n-\frac{1}{2}\right)=-\frac{1}{4} and show how to derive all such limits. \textbf{Proof.} Since \displaystyle\int_{n}^{n+1}\sin^2(\pi t)\text{d}{t}=\frac{1}{2} (for any n\in \mathbb{N}) we can rewrite the limit in question as \lim_{n\to \infty}n\left(n\left(\int_{n}^{n+1}\frac{\sin^2(\pi t)}{t}\text{d}t-\int_{n}^{n+1}\frac{\sin^2(\pi t)}{n}\text{d}t\right)\right)=\lim_{n\to \infty}n^2\left(\int_{n}^{n+1}\sin^2(\pi t)\left(\frac{1}{t}-\frac{1}{n}\right)\text{d}t\right) =\lim_{n\to \infty}n\left(\int_{n}^{n+1}\sin^2(\pi t)\frac{n-t}{t}\text{d}t\right) Changing the variables t\to n+s and using periodicity of sine, the last simplifies to \lim_{n\to \infty}n\int_{0}^{1}\sin^2(\pi s)\frac{-s}{n+s}\text{d}s Since for s\in[0,1], $\displaystyle \...
Read more