ON A SUM INVOLVING THE PRIME COUNTING FUNCTION π(x)
نویسنده
چکیده
An asymptotic formula for the sum of reciprocals of π(n) is derived, where π(x) is the number of primes not exceeding x. This result improves the previous results of De Koninck-Ivi´c and L. Panaitopol. Let, as usual, π(x) = px 1 denote the number of primes not exceeding x. The prime number theorem (see e.g., [2, Chapter 12]) in its strongest known form states that (1) π(x) = li x + R(x), with (2) li x := x 2 dt log t = x 1 log x + 1! log 2 x + · · · + m! log m+1 x + O 1 log m+2 x for any fixed integer m 0, and (3) R(x) ≪ x exp(−Cδ(x)), δ(x) := (log x) 3/5 (log log x)
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تاریخ انتشار 2003