Mertens conjecture explained

M(n) is bounded by \pm\sqrt{n} . Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite (reprinted in), and again in print by, and disproved by .It is a striking example of a mathematical conjecture proven false despite a large amount of computational evidence in its favor.

Definition

In number theory, the Mertens function is defined as

M(n) = \sum _{1} \mu(k),

where μ(k) is the Möbius function; the Mertens conjecture is that for all n > 1,

|M(n)| < \sqrt{n}.

Disproof of the conjecture

Stieltjes claimed in 1885 to have proven a weaker result, namely that

m(n) : = M(n)/\sqrt{n} was bounded, but did not publish a proof.^[1] (In terms of m(n) , the Mertens conjecture is that -1 < m(n) < 1 .)

In 1985, Andrew Odlyzko and Herman te Riele proved the Mertens conjecture false using the Lenstra–Lenstra–Lovász lattice basis reduction algorithm:^[2]

\liminf m(n) < -1 . 009 and \limsup m(n) > 1 . 06 . It was later shown that the first counterexample appears below e ^{3 . 21 x 10 ^{64}} ≈ 10 ^{1 . 39 x 10 ^{64}} ^[3] but above 10¹⁶.^[4] The upper bound has since been lowered to e ^{1 . 59 x 10 ^{40}} ^[5] or approximately 10 ^{6 . 91 x 10 ^{39}} , and then again to e ^{1 . 017 x 10 ^{29}} ≈ 10 ^{4 . 416 x 10 ^{28}} ,^[6] but no explicit counterexample is known.

The law of the iterated logarithm states that if is replaced by a random sequence of +1s and -1s then the order of growth of the partial sum of the first terms is (with probability 1) about which suggests that the order of growth of might be somewhere around . The actual order of growth may be somewhat smaller; in the early 1990s Steve Gonek conjectured that the order of growth of was

( logloglog n) ^{5/4} , which was affirmed by Ng (2004), based on a heuristic argument, that assumed the Riemann hypothesis and certain conjectures about the averaged behavior of zeros of the Riemann zeta function.^[7]

In 1979, Cohen and Dress^[8] found the largest known value of

m(n) ≈ 0 . 570591 for and in 2011, Kuznetsov found the largest known negative value (in the sense of absolute value) m(n) ≈ -0 . 585768 for ^[9] In 2016, Hurst computed for every but did not find larger values of .

In 2006, Kotnik and te Riele improved the upper bound and showed that there are infinitely many values of for which but without giving any specific value for such an .^[10] In 2016, Hurst made further improvements by showing

\liminf m(n) < -1 . 837625 and \limsup m(n) > 1 . 826054 .

Connection to the Riemann hypothesis

The connection to the Riemann hypothesis is based on the Dirichlet seriesfor the reciprocal of the Riemann zeta function,

(1)/(\zeta(s)) = \sum ^{inf ty} _{n =1} (\mu(n))/(n ^{s}) ,

valid in the region

l{Re}(s) > 1 . We can rewrite this as a Stieltjes integral(1)/(\zeta(s)) = \int ^{inf ty} _{0} x ^{-s} dM(x)

and after integrating by parts, obtain the reciprocal of the zeta functionas a Mellin transform

(1)/(s \zeta(s)) = \left\{ l{M} M \right\}( - s) = \int ^{inf ty} _{0} x ^{-s} M(x) (dx)/(x) .

Using the Mellin inversion theorem we now can express in terms of as

M(x) = (1)/(2 \pi i) \int ^{\sigma + i inf ty} _{\sigma - i inf ty} (x ^{s})/(s \zeta(s)) ds

which is valid for, and valid for on the Riemann hypothesis. From this, the Mellin transform integral must be convergent, and hence must be for every exponent e greater than . From this it follows that

M(x) = O ( x ^{\tfrac{1} { 2 } + \epsilon} )

for all positive is equivalent to the Riemann hypothesis, which therefore would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that

M(x) = O ( x ^{\tfrac{1}{2}).}

Further reading

• The Mertens Conjecture Revisited . Tadej . Kotnik . Herman . te Riele . Herman te Riele . 2006 . Lecture Notes in Computer Science . 4076 . . Algorithmic number theory. 7th international symposium, ANTS-VII, Berlin, Germany, July 23--28, 2006. Proceedings . Hess . Florian . 156–167 . Berlin . 10.1007/11792086_12 . 1143.11345 . 3-540-36075-1 .
• Kotnik . T. . van de Lune . J. . 2004 . On the order of the Mertens function . Experimental Mathematics . 13 . 4 . 473–481 . 10.1080/10586458.2004.10504556 . 2093469 . dead . https://web.archive.org/web/20070403143340/http://expmath.org/expmath/volumes/13/13.4/Kotnik.pdf . 2007-04-03 .
• Mertens . F. . 1897 . Über eine zahlentheoretische Funktion . Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, Abteilung 2a . 106 . 761–830 .
  • Pintz . J. . 1987 . An effective disproof of the Mertens conjecture . . 147–148 . 325–333 . 0623.10031 .

External links

• Web site: A Prime Surprise (Mertens Conjecture) . . January 23, 2020 . . https://ghostarchive.org/varchive/youtube/20211221/uvMGZb0Suyc . 2021-12-21 . live.

Notes and References

1. Book: Borwein . Peter . Peter Borwein . Choi . Stephen . Rooney . Brendan . Weirathmueller . Andrea . The Riemann hypothesis. A resource for the aficionado and virtuoso alike . CMS Books in Mathematics . New York, NY . . 2007 . 978-0-387-72125-5 . 1132.11047 . 69.
2. Sandor et al (2006) pp. 188–189.
3. Pintz . J. . 1987 . An effective disproof of the Mertens conjecture . . 147–148 . 325–333 . 0623.10031 .
4. Hurst . Greg . 2016 . Computations of the Mertens function and improved bounds on the Mertens conjecture . 1610.08551 . math.NT.
5. Kotnik and Te Riele (2006).
6. Web site: Rozmarynowycz . John . Kim . Seungki . 2023 . A New Upper Bound On the Smallest Counterexample To The Mertens Conjecture .
7. Web site: The distribution of the summatory function of the Möbius function . Nathan . Ng . 2004.
8. Cohen, H. and Dress, F. 1979. “Calcul numérique de Mx)” 11–13. [Cohen et Dress 1979], Rapport, de I'ATP A12311 ≪ Informatique 1975 ≫
9. Kuznetsov . Eugene . 2011 . Computing the Mertens function on a GPU . 1108.0135 . math.NT.
10. Kotnik & te Riele (2006).