PhD Advisor: John Friedlander
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In this thesis, we study the distribution of prime ideals within the Chebotarev Density Theorem. The theorem states that the Artin symbols attached to prime ideals are equidistributed within the Galois group of a given Galois extension.
We exhibit field-uniform unconditional bounds with explicit constants for the least prime ideal in the Chebotarev Density Theorem, that is, the prime ideal of least norm with a specified Artin symbol. Moreover, we provide a new upper bound for the number of prime ideals with a specified Artin symbol which is valid for a wide range and sharp, short of precluding a putative Siegel zero. To achieve these results, we establish explicit statistical information on the zeros of Hecke L-functions and the Dedekind zeta function. Our methods were inspired by works of Linnik, Heath-Brown, and Maynard in the classical case and the papers of Lagarias--Odlyzko, Lagarias--Montgomery--Odlyzko, and Weiss in the Chebotarev setting.
We include applications for primes represented by certain binary quadratic forms, congruences of coefficients for modular forms, and the group structure of elliptic curves reduced modulo a prime. In particular, we establish the best known unconditional upper bounds for the least prime represented by a positive definite primitive binary quadratic form and for the Lang--Trotter conjectures on elliptic curves.