Pietro Mengoli and numerical series. The prehistory of Riemann's 'zeta' function
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The article deals with the work of the 17th-century Italian mathematician, Rev. Pietro Mengoli (1625-1686), who was the forerunner of research on numerical series. The legacy of Mengoli, a scientist well-known and well-respected in Italy, but almost altogether forgotten in the West, has never been thoroughly analyzed in Polish historical writing. Yet it was Mengoli who first posed a number of problems related to finding the sums of an infinite number of fractions. He solved most of those problems, but he failed in one case - in the case of the sum of the inverse of squares of successive natural numbers. For fundamental reasons, which had not been understood until several dozen years later, Mengoli was unable to find a compact expression for the sum of this series. He himself, with a humility rarely found in the history of science, admitted that this problem required a 'richer intellect'. This series turned out to be the first example of a fundamental function investigated later by Euler and Riemann, and called, in honour of the latter mathematician, the Riemann 'zeta' function. This function constitutes the key to solving one of the greatest mathematical puzzles of all times - the distribution of prime numbers. Connected with this riddle is also the most important and most difficult of the hitherto unsolved problems of the famous list presented in 1900 by Hilbert at the 2nd International Mathematical Congress in Paris, the Riemann hypothesis. The generalizations of the series considered by Mengoli continue to be researched by mathematicians today. The aim of the article is to show, on the example of Mengoli's achievements and failures, a general regularity: the solution of a given mathematical problem is the result of the subtle interplay between, on one hand, the scientists's knowledge, talent and effort, and, on the other, the level of general knowledge at a given time, which stems from the collective achievements of many previous generations of mathematicians.
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