Abstract
Rewriting PNT as \(\displaystyle \pi (x) = \frac{x}{\log x} + o\left( \frac{x}{\log x}\right) \), it is clear from Theorem 8.1 that \(\int _2^x\frac{dt}{\log t}\) is a better approximation than \(\frac{x}{\log x}\) to \(\pi (x)\). We have given a proof of PNT in the previous chapter by a method different from that of J. Hadamard and de la Vallée Poussin.
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Shorey, T.N. (2020). The Prime Number Theorem with an Error Term. In: Complex Analysis with Applications to Number Theory. Infosys Science Foundation Series(). Springer, Singapore. https://doi.org/10.1007/978-981-15-9097-9_8
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DOI: https://doi.org/10.1007/978-981-15-9097-9_8
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