The history of the functional equation of the zeta-function

The zeta-function is one of the most important non-elementary functions in mathematics. It was studied by many great mathematicians and it continues to receive considerable attention from the modern researchers. One of its most important properties is its functional equation, which is traditionally attributed to Bernhard Riemann. The seminar is devoted to the history of this formula and to that of some similar L-functions, as well as to the contributions of various mathematicians in its discovery and proof (Euler, Malmsten, Schlömilch, Kinkelin, Riemann). We will focus our attention more in details on the contributions of Leonhard Euler and on those of a little-known Swedish mathematician Karl Malmsten, who was the first to rigorously prove such a kind of relationships and to obtain a number of other important results (e.g. the Fourier series for the logarithm of the gamma-function), which were later attributed to other mathematicians.