Sign changes of $\pi(x, q, 1) - \pi(x, q, a)$

TL;DR

Under the assumption of the generalized Riemann hypothesis, this paper provides bounds on the least sign change of certain prime counting functions, using a method that does not rely on zero computations of the zeta function.

Contribution

It establishes bounds on the least sign change of prime counting functions assuming the generalized Riemann hypothesis, without using zero computations.

Findings

Provides an upper bound for the least sign change of $\pi(x, q, 1) - \pi(x, q, a)$.

Gives a lower bound for the number of sign changes of $\pi(x)- ext{li} x$ under the Riemann hypothesis.

Results are weaker than numerical methods but do not require zero computations of the zeta function.

Abstract

It is known, that under the assumption of the generalized Riemannian hypothesis, the function $\pi(x, q, 1) - \pi(x, q, a)$ has infinitely many sign changes. In this article we give an upper bound for the least such sign change. Similarly, assuming the Riemannian hypothesis we give a lower bound for the number of sign changes of $\pi(x)-\li x$. The implied results for the least sign change are weaker then those obtained by numerical methods, however, our method makes no use of computations of zeros of the $\zeta$-function.