Entropy of Riemann zeta zero sequence

Abstract

One of the key unsolved problems in mathematics is the proof or refutation of Riemann’s remarkable 1858 hypothesis about the location of the roots of the Riemann zeta function. The statistical properties of the zero distribution have been studied intensively to get insight into the phenomenon. In this work we study the entropy of the sequence of zeros, which tells us how constrained is the pattern of zeros. A high value of the entropy would imply that the sequence has relatively low structure, and hence predicting the zeros is a dicult problem. A low value would give us encouragement that techniques in machine learning like neural networks would be helpful in studying the phenomenon.