Solutions to the book Introduction to Analytic Number Theory by Tom M. Apostol
Solutions to the book Complex Variables by M. Ya. Antimirov, A. A. Kolyshkin and Remi Vaillancourt
The Italian mathematician Pietro Mengoli (1625–1686), in his 1650 book Novae quadraturae arithmeticae, seu de additione fractionum, posed the following question: What’s the value of the sum

Here’s what Mengoli said:
Having concluded with satisfaction my consideration of those arrangements of fractions,
I shall move on to those other arrangements that have the unit as numerator, and square
numbers as denominators. The work devoted to this consideration has bore some fruit — the question itself still awaiting solution — but it [the work] requires the support of a richer
mind, in order to lead to the evaluation of the precise sum of the arrangement [of fractions]
that I have set myself as a task.Â
First time, Leonhard Euler (1707-1783) in 1735 proved that above series converges to
.
In page 122 of the book “Topics in Number Theory”(1956), by William J. LeVeque there is an exercise for evaluating the following integral in two ways.

First way is to write the integrand as a geometric series,

and the second way by use of a suitable change of variables (
) which is also published by Tom M. Apostol in 1983 in Mathematical Intelligencer.

Note that

and by the monotone convergence theorem we get


We change variables in this by putting
, so that
. Hence


where
is the square with vertices
and
.
Exploiting the symmetry of the square we get


              
Now
, and if
then
and
.
It follows that
and so
. Hence


                        ![\displaystyle=\left[2(\sin^{-1}(u))^2\right]_0^{\frac{1}{2}}+\left[\pi\sin^{-1}(u)-(\sin^{-1}(u))^2\right]_{\frac{1}{2}}^1](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%3D%5Cleft%5B2%28%5Csin%5E%7B-1%7D%28u%29%29%5E2%5Cright%5D_0%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%2B%5Cleft%5B%5Cpi%5Csin%5E%7B-1%7D%28u%29-%28%5Csin%5E%7B-1%7D%28u%29%29%5E2%5Cright%5D_%7B%5Cfrac%7B1%7D%7B2%7D%7D%5E1&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
