Problems in Analytic and Algebraic Number Theory - Järviniemi, Olli (fi=Turun yliopisto|en=University of Turku|, 2023)
In this thesis we study questions on the distribution of primes and multiplicative orders modulo prime numbers. The problems are attacked using methods from analytic and algebraic number theory. In the first article, we consider the problem of finding primes in “many” short intervals. We improve on a result of Heath-Brown by combining his methods with Harman’s sieve. We further extend the results for shorter intervals, considerably improving on a result of Peck. We give applications to prime-representing functions and binary digits of primes. In the second article, together with J. Teräväinen, we study the distribution of Gaussian almost primes in narrow sectors, demonstrating that the previous results and methods of Teräväinen over the integers may be adapted to Gaussian integers. Our result for products of three Gaussian primes is almost optimal. The result for products of two Gaussian primes is of comparable strength as the best known results over integers. In the third and fourth articles, we study multiplicative orders of integers modulo primes, motivated by Artin’s primitive root conjecture. The results of these articles are conditional on a generalization of the Riemann hypothesis. In the third article, extending previous methods of Lenstra to a multivariable setting, we in particular determine all tuples of integers attaining equal orders modulo infinitely many primes. In the fourth article, together with A. Perucca, we unify many previous variations of Artin’s conjecture into one framework, and give a finite procedure for solving such problems in general.
Keywords: prime numbers; Dirichlet polynomials; Gaussian primes; Artin’s conjecture