This discussion focuses on the properties of asymptotic functions, specifically the implications of the notation h(x)~g(x). The participants explore whether h(x+1)~g(x) holds true under certain conditions and examine the behavior of the prime counting function, pi(x), in relation to asymptotic equivalence. The conversation highlights that while some implications may seem intuitive, they do not universally apply, particularly when functions exhibit rapid growth or oscillatory behavior. The discussion concludes that further analysis is required to establish the relationships between these functions definitively.
PREREQUISITESMathematicians, computer scientists, and students studying asymptotic analysis, particularly those interested in the behavior of functions and their growth rates.
If h(x)~g(x), is h(x+1)~g(x)?
I'm not sure that implication holds in general... I can imagine failure can occur if the functions grow sufficiently fast, or if they can do other odd things, like zig-zag back and forth.
I think it might be easier to first decide if pi(x) ~ pi(x + 1), or if (x - 1) / ln (x - 1) ~ x / ln x