Riemann Hypothesis: Redheffer Matrix and Semi-Infinite Construction

EasyChair Preprint no. 10582

4 pagesDate: July 17, 2023

Abstract

The Riemann Hypothesis is the conjecture that the Riemann zeta-function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in mathematics (the zeros of the Riemann zeta-function are the key to an analytic expression for the number of primes).

The Riemann Hypothesis is equivalent to the statement about the asymptotics of the Mertens function, the cumulative sum of the Mobius function. The Mertens function, in its turn, can be represented fairly simply as the determinant of a matrix (the Redheffer matrix) defined in terms of divisibility (square matrix, all of whose entries are 0 or 1), where the last can be considered as adjacency matrix, which is associated with a graph. Hence, for each graph it is possible to construct a statistical model.

The paper outlines the above and it presents an algebra (as is customary in the theory of conformal algebras), having manageable and painless relations (unitary representations of the N = 2 superVirasoro algebra appear). The introduced algebra is closely related to the fermion algebra associated with the statistical model coming from the infinite Redheffer matrice (the ith line can be viewed as a part of the thin basis of the statistical system on one-dimensional lattice, where any i consecutive lattice sites carrying at most i − 1 zeroes). It encodes the bound on the growth of the Mertens function.

The Riemann zeta-function is a difficult beast to work with, that’s why a way is to replace the divisibility.

Keyphrases: Conformal algebras, determinant, divisibility, Mertens function, N = 2 superVirasoro algebra, One-dimensional lattice, Redheffer matrix, Riemann hypothesis, Riemann zeta function, semi-infinite, statistical model