Sharper bounds for the Chebyshev function 𝜃(𝑥)

Abstract

In this article, we provide explicit bounds for the prime counting functions θ ( x ) \theta (x) for all ranges of x x . The bounds for the error term for θ ( x ) − x \theta (x)- x are of the shape ε x \varepsilon x and c k x ( log ⁡ x ) k \frac {c_k x}{(\log x)^k} , for k = 1 , … , 5 k=1,\ldots ,5 . Tables of values for ε \varepsilon and c k c_k are provided.