The Turn of the Century

Abstract

The end of the XIXth century had seen a rapid progress in the theory of complex functions. In the preceding chapter one could notice the impact of Hadamard’s theory of entire functions on the determination of the asymptotic behaviour of functions associated with prime numbers, like π(x) and θ(x). For further development of this theory three results in function theory, all obtained between 1896 and 1910, were of particular importance: Hadamard’s three circle theorem (1896), Jensen’s formula (1899) and the theorem of Phragmén—Lindelöf (1908). With the use of them the main results of Hadamard, von Mangoldt and de la Vallée-Poussin were given simpler proofs, particularly through the activity of Edmund Landau.