Topic
Mertens conjecture
About: Mertens conjecture is a research topic. Over the lifetime, 48 publications have been published within this topic receiving 479 citations. The topic is also known as: Mertens hypothesis.
Papers
TL;DR: A study of the cumulative sums of the 1\Tbius function on the Atlas Computer of the Science Research Council has revealed certain statistical prop- erties which lead the authors to make a number of conjectures as discussed by the authors.
Abstract: A study of the cumulative sums of the 1\Tbius function on the Atlas Computer of the Science Research Council has revealed certain 'statistical prop- erties which lead the authors to make a number of conjectures. One of these is that any conjecture of the Mertens type, viz. N
45 citations
TL;DR: The Mertens function was computed for all $x \leq 10^{16}$, recording all extrema, all zeros, and $10^8$ values sampled at a regular interval, giving improved lower and upper bounds of-1.837625 and 1.826054.
Abstract: The Mertens function is defined as $M(x) = \sum_{n \leq x} \mu(n)$, where $\mu(n)$ is the Mobius function The Mertens conjecture states $|M(x)/\sqrt{x}| 1$, which was proven false in 1985 by showing $\liminf M(x)/\sqrt{x} 106$ The same techniques used were revisited here with present day hardware and algorithms, giving improved lower and upper bounds of $-1837625$ and $1826054$ In addition, $M(x)$ was computed for all $x \leq 10^{16}$, recording all extrema, all zeros, and $10^8$ values sampled at a regular interval Lastly, an algorithm to compute $M(x)$ in $O(x^{2/3+\varepsilon})$ time was used on all powers of two up to $2^{73}$
19 citations
1 Jan 2009
17 citations
Posted Content•
TL;DR: In this article, the authors study Mertens' own proof (1874) of his theorem on the sum of the reciprocals of the primes and compare it with the modern treatments.
Abstract: We study Mertens’ own proof (1874) of his theorem on the sum of the reciprocals of the primes and compare it with the modern treatments.
17 citations
1 Feb 1971
15 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2018 | 1 |
| 2017 | 2 |
| 2016 | 4 |
| 2015 | 2 |
| 2014 | 3 |