On the sign changes of $ψ(x)-x$

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\to\infty}\frac{V(T)}{\log T}\geq\frac{γ_{0}}π+\frac{1}{60} ] where $γ_{0}=14.13\ldots$ 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 $4\cdot10^{21}$ times better than previously known estimates of this type.

💡 Research Summary

The paper investigates the oscillatory behaviour of the error term ψ(x)−x in the Prime Number Theorem, focusing on the number V(T) of sign changes of ψ(x)−x in the interval