Zeros of the Riemann Zeta Function on Short Intervals of the Critical Line - Khayrulloev, Shamsullo (Karshi Multidisciplinary International Scientific Journal, 2025)
This paper presents a study of the distribution of zeros of the Riemann zeta function, specifically focusing on those of odd order, within short intervals along the critical line. The Riemann zeta function plays a central role in analytic number theory, and its nontrivial zeros are deeply connected with the distribution of prime numbers. The critical line, where the real part of the argument is one-half, is of particular interest due to the famous Riemann Hypothesis, which suggests that all nontrivial zeros lie on this line. Although the hypothesis remains unproven, considerable progress has been made in understanding the behavior and density of these zeros. The objective of this study is to examine the frequency and location of odd-order zeros in small neighborhoods on the critical line. We build upon earlier foundational work by mathematicians such as Hardy, Littlewood, Selberg, and Karatsuba. Their contributions established the groundwork for understanding the occurrence of zeros in specific ranges, and this paper aims to refine and extend those results. In particular, we aim to verify a hypothesis that proposes the presence of a significant number of such zeros in very short intervals. To achieve this, we utilize analytic methods involving trigonometric sums and exponential pair techniques. These approaches allow us to estimate the relevant quantities without requiring explicit evaluation of the zeros themselves. The method employed in this paper refines previous bounds and enables the detection of zeros in intervals that are shorter than those considered in earlier works. Additionally, this study provides sharper criteria under which the existence of such zeros can be guaranteed. Our findings support the hypothesis that zeros of odd order are not only present but relatively frequent in short segments along the critical line. This contributes valuable insight into the fine-scale structure of the zeta function’s zeros and affirms the robustness of analytic techniques based on exponential sums. Moreover, the results have broader implications in number theory, particularly in areas concerned with prime number theorems and related analytic functions.