Real Parts Of Zeros Research Articles

On the zeros of linear combinations of derivatives of the Riemann zeta function, II

The relevant number to the Dirichlet series [Formula: see text], is defined to be the unique integer [Formula: see text] with [Formula: see text], which maximizes the quantity [Formula: see text]. In this paper, we classify the set of all relevant numbers to the Dirichlet [Formula: see text]-functions. The zeros of linear combinations of [Formula: see text] and its derivatives are also studied. We give an asymptotic formula for the supremum of the real parts of zeros of such combinations. We also compute the degree of the largest derivative needed for such a combination to vanish at a certain point.

On the zeros of linear combinations of derivatives of the Riemann zeta function

In this paper, the zeros of linear combinations of the Riemann zeta function and its derivatives are studied. We establish an asymptotic formula for the number of zeros in a rectangle of height [Formula: see text]. We also find a sharp asymptotic formula for the supremum of the real parts of zeros of such combinations in a certain family.