On the connection between zero-free regions and the error term in the Prime Number Theorem

2025/07/18 by Broucke, Frederik
#11M26 #11M41 #11N05 #11N80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.13780

Abstract

We provide for a wide class of zero-free regions an upper bound for the error term in the Prime Number Theorem, refining works of Pintz (1980), Johnston (2024), and Révész (2024). Our method does not only apply to the Riemann zeta function, but to general Beurling zeta functions. Next we construct Beurling zeta functions having infinitely many zeros on a prescribed contour, and none to the right, for a wide class of such contours. We also deduce an oscillation result for the corresponding error term in the Prime Number Theorem, showing that our aforementioned refinement is close to being sharp.

Citations

• Zero-density estimates and the optimality of the error term in the prime number theorem
• On zero-density estimates for Beurling zeta functions
• Explicit zero density estimate near unity
• Explicit bounds for the Riemann zeta function and a new zero-free region
• Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function
• The distribution of prime numbers

Related

• Oscillation of the remainder term in the prime number theorem of Beurling, "caused by a given zeta-zero"
• Zero-density estimates and the optimality of the error term in the prime number theorem
• The Error in a Smooth Weighted Prime Number Formula and Zero-free Regions for the Riemann Zeta Function
• Some explicit estimates for the error term in the prime number theorem
• Large values of the error term in the prime number theorem