The article discusses two classic hypotheses of the theory of numbers: those of Riemann (1859) and of Mertens (1897), in the context of recent research using computer techniques.The validity of Riemann's hypothesis remains an extremely difficult research problem of fundamental importance for the question of the distribution of prime numbers. The search for a possible counterexample to the hypothesis has led to the development of effective methods for the numerical calculation of 'high' complex zeroes of the Riemann zeta function, and along with time, to the accumulation of high quantities of zeroes. The numerical material has made it possible to put forward a number of interesting statistical hypotheses, which, in the future, are likely to throw more light on the new and very promising field of physics: quantum chaos. Mertens' hypothesis attracted the attention of mathematicians ever since it was announced, for if the hypothesis were true, it would mean that Riemann's hypothesis was true as well (although not vice versa). However, it was shown already in the 1940s that the hypothesis entails some paradoxical properties of zeroes in the zeta function. Since then it was suspected that the hypothesis might be false. The falsity of the hypothesis was indeed proved in 1985. A key role in proving the hypothesis false was played by modern numerical techniques used in cryptography and by the use of fast computers, without which the proof would not have been possible. This fact has important implications for the traditional understanding of the notion of mathematical proof.
Financed by the National Centre for Research and Development under grant No. SP/I/1/77065/10 by the strategic scientific research and experimental development program:
SYNAT - “Interdisciplinary System for Interactive Scientific and Scientific-Technical Information”.