Mertens 関数

Mertens 関数 は任意の正の整数 n において

$M(n)=\sum _{k=1}^{n}\mu (k)$($\mu (k)$は、メビウス関数)

で表される関数のことである。また、次のように定義して正の実数に拡張できます。

$M(x)=M(\lfloor x\rfloor ).$

より形式的には、M（x）は、偶数の素因数 - 奇数を持つものの数を引いたxまでの平方因子をもたない整数です。

Mertens関数は、平均値とピーク値の両方で正負の方向にゆっくりと成長し、$M(n)=0$となる$M(x)$の零点nは、小さいほうから、

2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427, 428, ... オンライン整数列大辞典の数列 A028442.

メビウス関数は、-1, 0, +1のどれかの値しか取らないので、|M(x)| < xとなる。 The Mertens conjecture went further, stating that there would be no x where the absolute value of the Mertens function exceeds the square root of x. The Mertens conjecture was proven false in 1985 by Andrew Odlyzko and Herman te Riele. However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(x), namely M(x) = O(x^{1/2 + ε}). Since high values for M(x) grow at least as fast as the square root of x, this puts a rather tight bound on its rate of growth. Here, O refers to Big O notation.

The true rate of growth of M(x) is not known. An unpublished conjecture of Steve Gonek states that

$0<\limsup _{x\to \infty }{\frac {|M(x)|}{{\sqrt {x}}(\log \log \log x)^{5/4}}}<\infty .$

Probabilistic evidence towards this conjecture is given by Nathan Ng.

Representations

As an integral

オイラー積を使うと、

${\frac {1}{\zeta (s)}}=\prod _{p}(1-p^{-s})=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}$

$\zeta (s)$はリーマンゼータ関数 and the product is taken over primes. 次に,ディリクレ級数とヘロンの公式を使い,次の式が得られる。

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

Conversely, one has the メリン変換

${\frac {1}{\zeta (s)}}=s\int _{1}^{\infty }{\frac {M(x)}{x^{s+1}}}\,dx$($\mathrm {Re} (s)>1$.)

第二チェビシェフ関数を使い以下のようにあらわせる

$\psi (x)=M\left({\frac {x}{2}}\right)\log(2)+M\left({\frac {x}{3}}\right)\log(3)+M\left({\frac {x}{4}}\right)\log(4)+\cdots .$

Assuming that there are not multiple non-trivial roots of $\zeta (\rho )$ we have the "exact formula" by the residue theorem:

$M(x)=\sum _{\rho }{\frac {x^{\rho }}{\rho \zeta '(\rho )}}-2+\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}(2\pi )^{2n}}{(2n)!n\zeta (2n+1)x^{2n}}}.$

Weyl conjectured that the Mertens function satisfied the approximate functional-differential equation

${\frac {y(x)}{2}}-\sum _{r=1}^{N}{\frac {B_{2r}}{(2r)!}}D_{t}^{2r-1}y\left({\frac {x}{t+1}}\right)+x\int _{0}^{x}{\frac {y(u)}{u^{2}}}\,du=x^{-1}H(\log x)$

where H(x) is the Heaviside step function, B are Bernoulli numbers and all derivatives with respect to t are evaluated at t = 0.

There is also a trace formula involving a sum over the Möbius function and zeros of Riemann Zeta in the form

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{\sqrt {n}}}g(\log n)=\sum _{\gamma }{\frac {h(\gamma )}{\zeta '(1/2+i\gamma )}}+2\sum _{n=1}^{\infty }{\frac {(-1)^{n}(2\pi )^{2n}}{(2n)!\zeta (2n+1)}}\int _{-\infty }^{\infty }g(x)e^{-x(2n+1/2)}\,dx,$

where the first sum on the right-hand side is over the nontrivial zeros of the Riemann zeta function, and (g,h) are related by a Fourier transform, such that

$2\pi g(x)=\int _{-\infty }^{\infty }h(u)e^{iux}\,du.$

ファレイ数列

Mertens関数はファレイ数列を使い以下のようにも表せる。

$M(n)=\sum _{a\in {\mathcal {F}}_{n}}e^{2\pi ia}$  ( ${\mathcal {F}}_{n}$ は、オーダーnのファレイ数列)

この公式はFranel–Landau theoremの証明に使われます

Redheffer 行列

Redheffer 行列(a_ijにおいてj = 1もしくは、iがjを割り切るときは1となり、それ以外は、0となる行列)の行列式は M(n) に等しい。

As a sum of the number of points under n-dimensional hyperboloids

$M(x)=1-\sum _{2\leq a\leq x}1+{\underset {ab\leq x}{\sum _{a\geq 2}\sum _{b\geq 2}}}1-{\underset {abc\leq x}{\sum _{a\geq 2}\sum _{c\geq 2}\sum _{b\geq 2}}}1+{\underset {abcd\leq x}{\sum _{a\geq 2}\sum _{b\geq 2}\sum _{c\geq 2}\sum _{d\geq 2}}}1-\cdots$

計算

前述の方法のどちらも、メッテンス関数を計算するための実用的なアルゴリズムを導かない。 Using sieve methods similar to those used in prime counting, the Mertens function has been computed for all integers up to an increasing range of x.

人名	年	上限
Mertens	1897	104
von Sterneck	1897	1.5×105
von Sterneck	1901	5×105
von Sterneck	1912	5×106
Neubauer	1963	108
Cohen and Dress	1979	7.8×109
Dress	1993	1012
Lioen and van de Lune	1994	1013
Kotnik and van de Lune	2003	1014
Hurst	2016	1016

最大xまでのすべての整数値に対するMertens関数の計算時間は、O(x log log x)である。 Combinatorial based algorithms can compute isolated values of M(x) in O(x^2/3(log log x)^1/3) time, and faster non-combinatorial methods are also known.

10の累乗でM（x）の値については、A084237を参照してください。

関連項目

• ヘロンの公式
• Liouville関数

参考文献

• Edwards, Harold (1974). Riemann's Zeta Function. Mineola, New York: Dover. ISBN 0-486-41740-9 
• Mertens, F. (1897). “"Über eine zahlentheoretische Funktion", Akademie Wissenschaftlicher Wien Mathematik-Naturlich”. Kleine Sitzungsber, IIa 106: 761–830. 
• Odlyzko, A. M.; te Riele, Herman (1985). “Disproof of the Mertens Conjecture”. Journal für die reine und angewandte Mathematik 357: 138–160. http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf. 
• Weisstein, Eric W. "Mertens function". MathWorld (English).
• Sloane, N.J.A. (ed.). "Sequence A002321 (Mertens's function)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• Deléglise, M. and Rivat, J. "Computing the Summation of the Möbius Function." Experiment. Math. 5, 291-295, 1996. https://projecteuclid.org/euclid.em/1047565447
• Hurst, Greg (2016). "Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture". arXiv:1610.08551 [math.NT]。
• Nathan Ng, "The distribution of the summatory function of the Möbius function", Proc. London Math. Soc. (3) 89 (2004) 361-389. http://www.cs.uleth.ca/~nathanng/RESEARCH/mobius2b.pdf

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