This book has a history: it was born after the encounter of two professors from different generations, on the occasion of a series of mathematics seminars organised by the younger of the two at the Lycée Clemenceau in Nantes, in the early part of the years 2000 onwards. The prime objective of these seminars was to allow the professors of this establishment to keep a certain mathematical awareness that the sustained rhythm of preparing students for competitive entrance exams did not always facilitate. The seminars took place roughly once a month, and lasted an hour and a half. Over the years, the professors were joined by an increasing number of students from their classes; a vocation for mathematics was born for many of these, possibly in part due to this initiative. Both authors gave half a dozen talks at these seminars, on themes of their choosing, with a strong emphasis (but not exclusively) on classical analysis.
After the nomination of one of us to Lyon, we thought it would be interesting to assemble and write up these talks in more detail, and to find a connection between them. It seemed to us that a good starting point would be the 1911 paper of Littlewood (Chapter 1), which is at the same time the founding point of what we today call Tauberian theorems, and the beginning of the famous collaboration between Hardy and Littlewood that spanned 35 years, until Hardy's death in 1947. This collaboration produced a large number of remarkable discoveries, not the least of which was that of Ramanujan. The magnificent work of Hardy and Ramanujan on the asymptotic behaviour of the partition function is in fact the subject of an entire chapter (Chapter 8).