**Preface to "Number Theory and Symmetry"**

"Number Theory and Symmetry" deals with topics connecting numbers (integers, algebraic integers, transcendental numbers, p-adic numbers) and symmetries. First of all, symmetry became part of number theory when Riemann investigated the distribution of prime numbers and for that purpose introduced the complex functional equation and the related Riemann hypothesis (RH) that non-trivial zeros of the Riemann zeta function lie on the symmetry axis s = 1/2. Then, in a quest to justify RH on physical grounds, the Hilbert–Polya conjecture claimed that the imaginary part of the Riemann zeros on the symmetry axis should correspond to the eigenvalues of a Hermitian operator. This topic is covered by German Sierra.

Besides these classical areas, number fields offer clues to the connection between numbers and symmetries through arithmetic groups, geometry, and topology. I have in mind the Poincare´ conjecture and the whole work of Thurston about 3-manifolds. This topic is the kernel of the two papers by Michel Planat and co-authors and Torsten Asselmeyer Maluga. The aforementioned three papers highlight a strong connection between number theory and quantum physics.

The range of the three subsequent papers in this series is about more standard topics of number theory. A modified Sieve procedure by Bruno Aiazzi and coauthors, the Miller–Rabin primality test by Shamil Ishmukhametov and co-authors, and the 4-Pascal's triangle by Atsushi Yamagami and Kazuki Taniguchi are investigated.

The paper by Pavel Trojovsky offers clues to the relation between algebraic and transcendental numbers through polynomials. Atsushi Yamagami and Yuki Matsui's paper is in the field of b-adic numbers. The last paper by Ilwoo Cho covers the topic of p-adic numbers thanks to C\*-algebras and Banach\*-probability spaces.

The rich panel of mathematical concepts involved in this Special Issue illustrates the continuous interest of scholars in the relationship between numbers, their symmetries, and physics.

> **Michel Planat** *Editor*
