Mertens conjecture

Innumber theory, the Mertens function is defined as

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

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

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

Stieltjes claimed in 1885 to have proven a weaker result, namely that ${\displaystyle m($n):=M(n)/{\sqrt {n}}}   was bounded, but did not publish a proof.^[1] (In terms of ${\displaystyle m($n)}  , the Mertens conjecture is that ${\displaystyle -1<m($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][3]

${\displaystyle \liminf m($n)<-1.009}     and   ${\displaystyle \limsup m($n)>1.06.}  

It was later shown that the first counterexample appears below ${\displaystyle e^{3.21\times 10^{64}}\approx 10^{1.39\times 10^{64}}}$ ^[4] but above 10¹⁶.^[5] The upper bound has since been lowered to ${\displaystyle e^{1.59\times 10^{40}}}$ ^[6] or approximately ${\displaystyle 10^{6.91\times 10^{39}},}$  and then again to ${\displaystyle e^{1.017\times 10^{29}}\approx 10^{4.416\times 10^{28}}}$ ^[7], but no explicit counterexample is known.

The law of the iterated logarithm states that if μ is replaced by a random sequence of +1s and −1s then the order of growth of the partial sum of the first n terms is (with probability 1) about √ n log log n, which suggests that the order of growth of m(n) might be somewhere around √log log n. The actual order of growth may be somewhat smaller; in the early 1990s Steve Gonek conjectured^[8] that the order of growth of m(n) was ${\displaystyle (\log \log \log n)^{5/4},}$  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.^[8]

In 1979, Cohen and Dress^[9] found the largest known value of ${\displaystyle m($n)\approx 0.570591}   for M(7766842813) = 50286, and in 2011, Kuznetsov found the largest known negative value (in the sense of absolute value) ${\displaystyle m($n)\approx -0.585768}   for M(11609864264058592345) = −1995900927.^[10] In 2016, Hurst computed M(n) for every n ≤ 10¹⁶ but did not find larger values of m(n).^[5]

In 2006, Kotnik and te Riele improved the upper bound and showed that there are infinitely many values of n for which m(n) > 1.2184, but without giving any specific value for such an n.^[11] In 2016, Hurst made further improvements by showing

${\displaystyle \liminf m($n)<-1.837625}     and   ${\displaystyle \limsup m($n)>1.826054.}  

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

${\displaystyle {\frac {1}{\zeta ($s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}},}  

valid in the region ${\displaystyle {\mathcal {Re}}($s)>1}  . We can rewrite this as a Stieltjes integral

${\displaystyle {\frac {1}{\zeta ($s)}}=\int _{0}^{\infty }x^{-s}dM(x)}  

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

${\displaystyle {\frac {1}{s\zeta ($s)}}=\left\{{\mathcal {M}}M\right\}(-s)=\int _{0}^{\infty }x^{-s}M(x)\,{\frac {dx}{x}}.}  

Using the Mellin inversion theorem we now can express M in terms of 1⁄ζas

${\displaystyle M($x)={\frac {1}{2\pi i}}\int _{\sigma -i\infty }^{\sigma +i\infty }{\frac {x^{s}}{s\zeta (s)}}\,ds}  

which is valid for 1 < σ <2, and valid for 1⁄2 < σ <2 on the Riemann hypothesis. From this, the Mellin transform integral must be convergent, and hence M(x) must be O(x^e) for every exponent e greater than ⁠1/2⁠. From this it follows that

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

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

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

• Kotnik, Tadej; te Riele, Herman (2006). "The Mertens Conjecture Revisited". In Hess, Florian (ed.). Algorithmic number theory. 7th international symposium, ANTS-VII, Berlin, Germany, July 23--28, 2006. Proceedings. Lecture Notes in Computer Science. Vol. 4076. Berlin: Springer-Verlag. pp. 156–167. doi:10.1007/11792086_12. ISBN 3-540-36075-1. Zbl 1143.11345.
• Kotnik, T.; van de Lune, J. (2004). "On the order of the Mertens function" (PDF). Experimental Mathematics. 13 (4): 473–481. doi:10.1080/10586458.2004.10504556. S2CID 2093469. Archived from the original (PDF) on 2007-04-03.
• Mertens, F. (1897). "Über eine zahlentheoretische Funktion". Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, Abteilung 2a. 106: 761–830.
• Odlyzko, A. M.; te Riele, H. J. J. (1985), "Disproof of the Mertens conjecture" (PDF), Journal für die reine und angewandte Mathematik, 1985 (357): 138–160, doi:10.1515/crll.1985.357.138, ISSN 0075-4102, MR 0783538, S2CID 13016831, Zbl 0544.10047
• Pintz, J. (1987). "An effective disproof of the Mertens conjecture" (PDF). Astérisque. 147–148: 325–333. Zbl 0623.10031.
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006), Handbook of number theory I, Dordrecht: Springer-Verlag, pp. 187–189, ISBN 1-4020-4215-9, Zbl 1151.11300
• Stieltjes, T. J. (1905), "Lettre a Hermite de 11 juillet 1885, Lettre #79", in Baillaud, B.; Bourget, H. (eds.), Correspondance d'Hermite et Stieltjes, Paris: Gauthier—Villars, pp. 160–164

• Weisstein, Eric W. "Mertens conjecture". MathWorld.

• "A Prime Surprise (Mertens Conjecture)". Numberphile. January 23, 2020. Archived from the original on 2021-12-21 – via YouTube.