A round of Pintz to celebrate oscillations in sums

We explore a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for $L$-functions. In particular, we make explicit a general result of Pintz of this form; showing how one can use arithmetical information to deduce information about zeroes of $L$-functions, rather than the other way around. As a prototype, we work through an example with the Riemann zeta-function and sums of the Möbius function, but we also outline the utility of this method in general.

💡 Research Summary

The paper, dedicated to János Pintz on the occasion of his diamond jubilee, revisits a classical method originally due to Landau and later refined by Pintz, which links the oscillatory behaviour of partial sums of arithmetic functions to zero‑free regions of L‑functions. The authors make this connection completely explicit, providing a quantitative version of Pintz’s general theorem that allows one to deduce information about the location of zeros from bounds on arithmetic sums, rather than the traditional reverse direction.

The introductory section recalls the prime number theorem and the error term Δψ(x)=ψ(x)−x. It explains how a zero‑free line σ=A<1 for ζ(s) yields the bound Δψ(x)≪x^{A+ε}, and how the Riemann hypothesis would give Δψ(x)≪x^{1/2+ε}, which is essentially optimal because Δψ(x)=Ω(√x). Landau’s observation that the size of the oscillations of Δψ(x) is governed by the real part β₀ of a non‑trivial zero ρ₀=β₀+iγ₀ is reviewed, and the authors point out that Pintz’s framework extends this idea to a wide class of arithmetic functions.

The core of the paper is Theorem 1, an explicit “Landau–Pintz” result for the case of simple zeros (multiplicity ν=1). The theorem assumes a complex‑valued function A(x) satisfying a growth condition |A(x)|≤c_A x^{C} and whose Mellin transform can be written as F(s)·G(s). Both factors are required to obey explicit exponential‑type bounds of the form |F(s)|≤c_F max(1,|t|)^{B_F} e^{σ} and |G(s)|≤c_G max(1,|t|)^{B_G} e^{|σ|} for σ≥β₀−c₀ and σ≥−1 respectively. If G(s) has a simple zero ρ₀=β₀+iγ₀ and F(ρ₀)≠0, then for every Y>e^{C+2} the mean value \