For a smooth compactly supported whose Mellin transform has suitable vertical decay, Mellin inversion andgiveMoving the contour through the pole at and into the classical zero-free region gives main term and error when .
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The Prime Number Theorem (PNT) is a fundamental result in number theory that describes the asymptotic distribution of prime numbers. It states that the number of prime numbers less than a given number \( n \), denoted as \( \pi(n) \), is approximately equal to \( \frac{n}{\log(n)} \), where \( \log(n) \) is the natural logarithm of \( n \).
