Number Theory


An Upper Bound for Error Term of Mertens' Formula

Authors: Choe Ryong Gil

In this paper, it is obtained a new estimate for the error term E(t) of the Mertens' formula sum_{p≤t}{p^{-1}}=loglogt+b+E(t), where t>1 is a real number, p is the prime number and b is the well-known Mertens' constant. We , first, provide an upper bound, not a lower bound, of E(p) for any prime number p≥3 and, next, give one in the form as E(t)<logt/√t for any real number t≥3. This is an essential improvement of already known results. Such estimate is very effective in the study of the distribution of the prime numbers.

Comments: 27 pages, 6 tables

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Submission history

[v1] 2017-06-21 02:30:07

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