Effective estimates for some functions defined over primes

TL;DR

This paper provides effective bounds and estimates for prime-related functions, including the Chebyshev $ heta$-function and prime counting function, improving existing bounds and establishing conditions for prime distribution in short intervals.

Contribution

It introduces new effective bounds for prime functions and determines the minimal x for certain inequalities to hold, advancing understanding of prime distribution.

Findings

Established the smallest x_0 for Chebyshev's inequality to hold.

Derived new bounds for the prime counting function $(x)$.

Improved existing estimates for prime distribution in short intervals.

Abstract

In this paper we give effective estimates for some classical arithmetic functions defined over prime numbers. First we find the smallest real number $x_0$ so that some inequality involving Chebyshev's $\vartheta$-function holds for every $x \geq x_0$. Then we give some new results concerning the existence of prime numbers in short intervals. Also we derive new upper and lower bounds for some functions defined over prime numbers, for instance the prime counting function $\pi(x)$, which improve current best estimates of similar shape.