Mertens conjecture
However, Herman J. J. te Riele and Andrew Odlyzko have proven that there exist counterexamples beyond , but have yet to find one specific counterexample.
The Mertens conjecture is related to the Riemann hypothesis
, since
is another way of stating the Riemann hypothesis.
Given the Dirichlet series of the reciprocal of the Riemann zeta function, we find that
is true for . Rewriting as Stieltjes integral,
suggests this Mellin transform:
Then it follows that
for .
References
- 1 G. H. Hardy and S. Ramanujan, Twelve Lectures on Subjects Suggested by His Life and Work 3rd ed. New York: Chelsea, p. 64 (1999)
- 2 A. M. Odlyzko and H. J. J. te Riele, “Disproof of the Mertens Conjecture.” J. reine angew. Math. 357, pp. 138 - 160 (1985)
Title | Mertens conjecture |
---|---|
Canonical name | MertensConjecture |
Date of creation | 2013-03-22 16:04:25 |
Last modified on | 2013-03-22 16:04:25 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 10 |
Author | PrimeFan (13766) |
Entry type | Conjecture |
Classification | msc 11A25 |
Synonym | Mertens’ conjecture |
Synonym | Mertens’s conjecture |