Daniel Hu

• Princeton University

We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous Mertens function, expanding upon work of Ng. Finally, we explore properties of the generalised Mertens function of...

We prove an explicit Chebotarev variant of the Brun–Titchmarsh theorem. This leads to explicit versions of the best known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that limx→∞#{1≤n≤x∣τ(n)≠0}x>1-1.15×10-12,\documentclass[12pt]{minimal} \usepackage{...

We prove an explicit Chebotarev variant of the Brun--Titchmarsh theorem. This leads to explicit versions of the best-known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that $$\lim_{x \to \infty} \frac{\#\{1 \leq n \leq x \mid \tau(n) \neq 0\}}{x} > 1...

Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field L L has infinitely many nontrivial zeros of multiplicity at least 2 if L L has a subfield K K for which L / K L/K is a nonabelian Galois extension. We also extend this to zeros of order 3 when G a l ( L / K ) Gal(L/K) has an irred...

We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous Mertens function, expanding upon work of Ng. Finally, we explore properties of the generalized Mertens function of...

Answering a question of Browkin, we unconditionally establish that the Dedekind zeta function of a number field $L$ has infinitely many nontrivial zeros of multiplicity greater than 1 if $L$ has a subfield $K$ for which $L/K$ is a nonabelian Galois extension.