Estimating the number of zeros of Dedekind zeta-functions
Victor Amberger
TL;DR
The paper develops a new, explicit method to bound the number of non-trivial zeros of Dedekind zeta-functions up to height $T$, refining previous results and, in particular, improving the error term for the Riemann zeta function. The author adapts a Turing-inspired contour approach to the completed zeta $\xi_K(s)$, leveraging a Weierstrass factorization to express the logarithmic derivative as a zeros-sum plus a constant and then bounds the resulting kernel via a carefully constructed auxiliary operator. The main result provides an explicit bound: $\left|N_K(T)-\frac{T}{\pi}\log\left(d_K\left(\frac{T}{2\pi e}\right)^{n_K}\right)-1.919\right| \leq 0.194\left(\log d_K+n_K\log T\right)+5.543n_K+0.462$ for $T\ge1$, with a corresponding corollary giving the classical $N(T)$ estimate for $\zeta(s)$: $\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)\right|\leq0.097\log T+3.962$. The approach and bounds rely on precise gamma-factor estimates, Euler products, and optimized kernel bounds, and are applicable to general $L$-functions in the Selberg class. Computational aspects are implemented in Mathematica.
Abstract
In this article, I derive a new approach to estimate the number of non-trivial zeros of a given Dedekind zeta function with absolute height at most $T>=1$ counted with multiplicity. The error term in corresponding asymptotic formula improves all previous results, even in the case of the Riemann zeta function.