Mertens' theorem
(another Prime Pages' Glossary entries)
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GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)

Mertens used Chebyshev's theorem (a weak version of the prime number theorem) to prove that

This is now called Mertens' Theorem.

Assuming the Riemann hypothesis, Schoenfeld showed that when x > 8, we have the following error bound

where is Euler's constant.

Related pages (outside of this work)


BS96 (p. 210,234)
E. Bach and J. Shallit, Algorithmic number theory, Foundations of Computing Vol, I: Efficient Algorithms, The MIT Press, Cambridge, MA, 1996.  pp. xvi+512, MR 97e:11157 (Annotation available)

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