Problems from the discrete to the continuous. Probability, number theory, graph theory, and combinatorics (Q2250288)

As its title shows correctly, the book under review collects a number of problems of discrete nature and with solutions utilizing continuous and analytic tools. While the problems are chosen mainly from number theory, graph theory, and combinatorics, their solutions involve probability theory. The book is sorted in ten chapters, and ends with four appendices including a primer on discrete probability, a brief review of the required results from power series and generating functions, a proof of the so-called Stirling's approximation for \(n!\), and an elementary proof of the Euler's celebrated identity \(\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}\) based on changing variables in a certain double integration. Each chapter starts by introducing and clearing the problem, and then giving ideas for a detailed solution. Usually, similar and related problems also are studied. Each chapter ends with some exercises, and a brief paragraph giving some notes. The number theoretic problems studied in this book include the money changing problem related to partition theory, finding the asymptotic density of coprime pairs of integers and of square-free numbers, which have same solution \(\frac{6}{\pi^2}\). The author also considers Chebyshev's result on the number of primes counting function \(\pi(x)\), and Mertens' theorems on the asymptotic behavior of the summations \(\sum_{p\leq x}\frac{1}{p}\) and \(\sum_{p\leq x}\frac{\log p}{p}\) running over the primes. While the author also studies the Hardy-Ramanujan theorem on the number of distinct prime divisors, it seems that he is not aware of the so-called Erdös' multiplication table theorem, which is highly related to this theorem and fits the style of his book. The probabilistic problems studied in the book include a one-dimensional probabilistic packing problem, which originally is initiated by a Nobel Prize winner chemist. The solution to this problem is finally the quantity expressed by \[ p_k=k\exp\left(-2\sum_{j=1}^{k-1}\frac{1}{j}\right)\int_0^1\exp\left(2\sum_{j=1}^{k-1}\frac{s^j}{j}\right)ds, \] for any integer \(k\geq 2\). As an open problem the author asks about monotonicity of \(p_k\) and also its limit value \(\lim_{k\to\infty}p_k\). He also studies the arcsine laws for the one-dimensional simple symmetric random walk. The problems from graph theory and combinatorics considered in the present book include studying the distribution of cycles in random permutations, the largest clique in a random graph and applications to tampering detection and Ramsey theory, and also the phase transition concerning the giant component in a sparse random graph, focusing on a result due to \textit{P. Erdős} and \textit{A. Rényi} [Publ. Math. Inst. Hung. Acad. Sci., Ser. A 5, 17--61 (1960; Zbl 0103.16301)]. The ``continuous'' solutions to the above problems use only concepts from calculus and elementary analysis, and also from elementary probability. Thus the book is suitable for undergraduate students to have an excursion on some selected problems and interesting theorems, and also it is suitable for instructors to use it to introduce some good and meaningful examples.