On the sign changes of ψ(x)-x

2024/08/19 by Grześkowiak, Maciej, Kaczorowski, Jerzy, Pańkowski, Łukasz +1
#11M26 #11N05 #42A75 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2408.10399

Abstract

We improve the lower bound for V(T), the number of sign changes of the error term ψ(x)-x in the Prime Number Theorem in the interval [1,T] for large T. We show that \liminf_T→∞(V(T))/(log T)≥\fracγ₀π+(1)/(60) where γ₀=14.13… is the imaginary part of the lowest-lying non-trivial zero of the Riemann zeta-function. The result is based on a new density estimate for zeros of the associated k-function, over 4⋅10²¹ times better than previously known estimates of this type.

Related

• Sign changes in the prime number theorem
• Some explicit estimates for the error term in the prime number theorem
• Large values of the error term in the prime number theorem
• Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis
• A Proof of the Riemann Hypothesis and Determination of the Relationship Between Non- Trivial Zeros of Zeta Functions and Prime Numbers