Abstract
In this paper, Leibniz triangle and suitable binomial coefficients were used to get the bounds of ψ (x) . Using the generalized convolution and the differentiation on generalized convolution of arithmetical functions, we get to prove Tatuzawa-Izeki identity. Selberg’s asymptotic formula is included as a special case, which is the beginning of certain elementary proofs of the Prime Number Theorem. Integration is used on some related inequalities to provide a smoother elementary proof of the Prime Number Theorem.
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