Haseo Ki

This paper is concerned with a general theorem on the number of nonreal zeros of transcendental functions. J. Fourier formulated the theorem in his work Analyse des equations determineesin 1831, but he did not give a proof. Roughly speaking, the theorem states that if a real entire function f( x)can be expressed as a product of linear factors, then we can count the nonreal zeros of f( x)by observing the behavior of the derivatives of f( x)on the real axis alone. As we shall see in the sequel, th

For X ⊆ [0,1], let $D_X$ denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of $D_X$. For α ≥ 3, X is properly $D_ξ(Π^0_α)$ iff $D_X$ is properl

The Chowla–Selberg formula is applied in approximating a given Epstein zeta function. Partial sums of the series derive from the Chowla–Selberg formula, and although these partial sums satisfy a functional equation, as does an Epstein zeta function, they do not possess an Euler product. What we call partial sums throughout this paper may be considered as special cases concerning a more general function satisfying a functional equation only. In this article we study the distribution of zeros of t

We study the horizontal distribution of zeros of ζ′(s) which are denoted as ρ′ =β′ +iγ′. We assume the Riemann hypothesis which implies β′ ≥ 1/2 for any nonreal zero ρ′, equality being possible only at a multiple zero of ζ (s). In this paper, we prove that lim inf (β′ −1/2)log γ′ ≠ 0 if, and only if, for any c > 0 and s = σ + <it>it</it> with 0 ≤|σ -1/2| ≤c/log t (t>t<inf>0</inf>(c)), we have<fd id="M1"><inline-fig> <link locator="rnn064ueq1"></

Levinson and Montgomery in 1974 proved many interesting formulae on the zeros of derivatives of the Riemann zeta function $\zeta(s)$. When Conrey proved that at least 2/5 of the zeros of the Riemann zeta function are on the critical line, he proved the asymptotic formula for the mean square of $\zeta(s)$ multiplied by a mollifier of length $ T^{4/7}$ near the $1/2$-line. As a consequence of their papers, we study some aspects of zeros of the derivatives of the Riemann zeta function with no assum