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Erdős and Diamond proved in [1] that Chebyshev could have achieved sharper bounds for the asymptotic behavior of the prime counting function. Nevertheless, their proof does not shed any light on the first question that you posed because they took the PNT for granted throughout their note. References: [1] H. G. Diamond; P. Erdős. On sharp elementary prime number estimates, Enseign. Math. (2) 26 (1980) 313-321. |
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Erdős and Diamond proved in 1 that Chebyshev could have achieved sharper bounds for the asymptotic relation that you mentionbehavior of the prime counting function. Nevertheless, their proof does not shed any light on the first question that you posed because they took the PNT for granted throughout their note. References: [1] H. G. Diamond, ; P. Erdős: . On sharp elementary prime number estimates, Enseign. Math. (2) 26 (1980) 313-321. |
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Erdős and Diamond proved in 1 that Chebyshev could have achieved sharper bounds for the asymptotic relation that you mention. Nevertheless, their proof does not shed any light on the first question that you posed because they took the PNT for granted throughout their note. References: [1] H. G. Diamond, P. Erdős: On sharp elementary prime number estimates, Enseign. Math. (2) 26 (1980) 313-321, 1981. |
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