**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 (1625â€“1686), in his 1650 book Novae quadraturae arithmeticae, seu de additione fractionum, posed the following question: Whatâ€™s the value of the sum**

** Hereâ€™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 â€” the question itself still awaiting solution â€” 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**

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**Now** **, and if** ** then** **and** **. **

**It follows that** **and so** **. Hence**

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