On the $a$-points of the derivatives of the Riemann zeta function

TL;DR

This paper investigates the distribution and properties of $a$-points of the derivatives of the Riemann zeta function, providing formulas, estimates, and density results in specific regions of the complex plane.

Contribution

It introduces new formulas and estimates for the $a$-points of the derivatives of the Riemann zeta function, including an analogue of the zero density theorem.

Findings

Derived a Riemann-von Mangoldt type formula for $a$-points.

Estimated the number of $a$-points in certain regions.

Established an analogue of the zero density theorem.

Abstract

We prove three results on the $a$-points of the derivatives of the Riemann zeta function. The first result is a formula of the Riemann-von Mangoldt type; we estimate the number of the $a$-points of the derivatives of the Riemann zeta function. The second result is on certain exponential sum involving $a$-points. The third result is an analogue of the zero density theorem. We count the $a$-points of the derivatives of the Riemann zeta function in $1/2-(\log\log T)^2/\log T<\Re s<1/2+(\log\log T)^2/\log T$.