Prime and Möbius correlations for very short intervals in Fq[x]

Dr. Lior Rosenzweig’s study:

Published in American Journal of Mathematics

In collaboration with Prof. Pär Kurlberg of the Mathematics Department, KTH, Sweden

This paper describes several proofs for the distribution of irreducible polynomials over finite large fields. The motivation for these questions stems from conjectures in number theory concerning the characteristics of prime numbers, such as the number of twin primes in short intervals, the Möbius conjecture on special sums, and others. By replacing the concept of a prime with an irreducible polynomial, one can describe appropriate conjectures for these polynomials, as an analogy to conjectures for integers. In this paper, we’ve proven that for the generic case, the appropriate conjectures are indeed true, which among other things also indicates the truth of the conjectures in the case of prime numbers; however, examples can also be found for which the conjecture is untrue and even counterintuitive. We also describe potential applications in the context of building prime numbers and irreducible polynomials for encryption and decryption.

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