A note on the density of zeros of partial sums of the Dirichlet lambda, beta and eta functions

Abstract The aim of this note is to provide a different and less technical proof of the density of the real parts of zeros of partial sums of the Dirichlet eta function compared to that presented by Farag. To this end, by leveraging Bohr’s equivalence theorem, we establish that the zeros of the partial sums of the Dirichlet lambda and beta functions also exhibit real parts that are dense in the same critical interval as those of the Riemann zeta function. Concurrently, we provide information on the values of the bounds of the critical intervals for the partial sums of the Riemann zeta function and the Dirichlet eta function.