On a certain asymptotic relationship involving $\vartheta(t) - \lfloor t \rfloor$ and $t^{1/2}$

TL;DR

This paper investigates an asymptotic relationship between the Chebyshev -function difference (t) - \u230a t a0 and the square root of t, providing new insights into their behavior for large t.

Contribution

It establishes a novel asymptotic relationship involving (t) - \u230a t a0 and t^{1/2}, advancing understanding of the Chebyshev -function's properties.

Findings

Derived a new asymptotic formula involving (t) -  t a0 and t^{1/2}

Provided bounds or estimates for the difference (t) -  t a0 in relation to t^{1/2}

Enhanced theoretical understanding of the Chebyshev -function's growth behavior

Abstract

Let $\lfloor t \rfloor$ denote the greatest positive integer less than or equal to a given positive real number $t$ and $\vartheta(t)$ the Chebyshev $\vartheta$-function. In this paper, we prove a certain asymptotic relationship involving $\vartheta(t) - \lfloor t \rfloor $ and $t^{1/2}$.