We use the Prime Number Theorem to prove the existence of zero-free strips for the Riemann-zeta function. Precisely, we prove that there exists $\delta>0$ for which if $0\leq r<\delta $ then $\zeta(s)\neq 0$ for Re$(s)>1-r$.
The Prime Number Theorem and Pair Correlation of Zeros of the Riemann Zeta-Function
Sign changes in the prime number theorem
Oscillation of the remainder term in the prime number theorem of Beurling, "caused by a given zeta-zero"
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