Mertens function
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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, the Mertens function is defined for all positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s ''n'' as : M(n) = \sum_^n \mu(k), where \mu(k) is the
Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
. The function is named in honour of Franz Mertens. This definition can be extended to positive
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
as follows: : M(x) = M(\lfloor x \rfloor). Less formally, M(x) is the count of
square-free integer In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...
s up to ''x'' that have an even number of prime factors, minus the count of those that have an odd number. The first 143 ''M''(''n'') values are The Mertens function slowly grows in positive and negative directions both on average and in peak value, oscillating in an apparently chaotic manner passing through zero when ''n'' has the values :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, ... . Because the Möbius function only takes the values −1, 0, and +1, the Mertens function moves slowly, and there is no ''x'' such that , ''M''(''x''), > ''x''. The
Mertens conjecture In mathematics, the Mertens conjecture is the statement that the Mertens function M(n) is bounded by \pm\sqrt. Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 ...
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 Andrew Michael Odlyzko (Andrzej Odłyżko) (born 23 July 1949) is a Polish-American mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute. He began his career in ...
and
Herman te Riele Hermanus Johannes Joseph te Riele (born 5 January 1947) is a Dutch mathematician at CWI in Amsterdam with a specialization in computational number theory. He is known for proving the correctness of the Riemann hypothesis for the first 1.5 billio ...
. However, the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
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 \sqrt, this puts a rather tight bound on its rate of growth. Here, ''O'' refers to
big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
. The true rate of growth of ''M''(''x'') is not known. An unpublished conjecture of Steve Gonek states that : 0 < \limsup_ \frac < \infty. Probabilistic evidence towards this conjecture is given by Nathan Ng. In particular, Ng gives a conditional proof that the function e^ M(e^y) has a limiting distribution \nu on \mathbb. That is, for all bounded
Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
functions f on the reals we have that : \lim_ \frac \int_0^Y f\big(e^ M(e^y)\big) \,dy = \int_^\infty f(x) \,d\nu(x).


Representations


As an integral

Using the
Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Eu ...
, one finds that : \frac = \prod_p (1 - p^) = \sum_^\infty \frac, where \zeta(s) is the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, and the product is taken over primes. Then, using this
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analyti ...
with
Perron's formula In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform. Statement Let \ be an arithmetic function, a ...
, one obtains : \frac \int_^ \frac \, ds = M(x), where ''c'' > 1. Conversely, one has the
Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used i ...
: \frac = s \int_1^\infty \frac\,dx, which holds for \operatorname(s) > 1. A curious relation given by Mertens himself involving the second
Chebyshev function In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function or is given by :\vartheta(x)=\sum_ \ln p where \ln denotes the natural logarithm, ...
is : \psi(x) = M\left( \frac \right) \log 2 + M \left( \frac \right) \log 3 + M \left( \frac\right ) \log 4 + \cdots. Assuming that the Riemann zeta function has no multiple non-trivial zeros, one has the "exact formula" by the
residue theorem In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well ...
: : M(x) = \sum_\rho \frac - 2 + \sum_^\infty \frac.
Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
conjectured that the Mertens function satisfied the approximate functional-differential equation : \frac - \sum_^N \frac D_t^ y \left(\frac\right) + x \int_0^x \frac \, du = x^ H(\log x), where ''H''(''x'') is the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
, ''B'' are
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s, 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 the Riemann zeta function in the form : \sum_^\infty \frac g(\log n) = \sum_\gamma \frac + 2 \sum_^\infty \frac \int_^\infty g(x) e^ \, dx, where the first sum on the right-hand side is taken over the non-trivial zeros of the Riemann zeta function, and (''g'', ''h'') are related by the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
, such that : 2 \pi g(x) = \int_^\infty h(u) e^ \, du.


As a sum over Farey sequences

Another formula for the Mertens function is : M(n) = -1 + \sum_ e^, where \mathcal_n is the
Farey sequence In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ''n'', arranged in ord ...
of order ''n''. This formula is used in the proof of the Franel–Landau theorem.


As a determinant

''M''(''n'') is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the ''n'' × ''n'' Redheffer matrix, a (0, 1) matrix in which ''a''''ij'' is 1 if either ''j'' is 1 or ''i'' divides ''j''.


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

: M(x) = 1 - \sum_ 1 + \underset 1 - \underset 1 + \underset 1 - \cdots This formulation expanding the Mertens function suggests asymptotic bounds obtained by considering the Piltz divisor problem, which generalizes the
Dirichlet divisor problem Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
of computing asymptotic estimates for the summatory function of the
divisor function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...
.


Calculation

Neither of the methods mentioned previously leads to practical algorithms to calculate the Mertens function. 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''. The Mertens function for all integer values up to ''x'' may be computed in time. Combinatorial algorithms can compute isolated values of ''M''(''x'') in time, and faster non-combinatorial methods are also known. See for values of ''M''(''x'') at powers of 10.


Known upper bounds

Ng notes that the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
(RH) is equivalent to :M(x) = O\left(\sqrt \exp\left(\frac\right)\right), for some positive constant C>0. Other upper bounds have been obtained by Maier, Montgomery, and Soundarajan assuming the RH including : \begin , M(x), & \ll \sqrt \exp\left(C_2 \cdot (\log x)^\right) \\ , M(x), & \ll \sqrt \exp\left(\sqrt (\log\log x)^\right). \end Other explicit upper bounds are given by Kotnik as : \begin , M(x), & < \frac,\ \text x>2160535 \\ , M(x), & < \frac,\ \text x>685. \end


See also

*
Perron's formula In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetic function, by means of an inverse Mellin transform. Statement Let \ be an arithmetic function, a ...
*
Liouville's function The Liouville Lambda function, denoted by λ(''n'') and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if ''n'' is the product of an even number of prime numbers, and −1 if it is the product of an odd number of ...


Notes


References

* * * * * *Deléglise, M. and Rivat, J. "Computing the Summation of the Möbius Function." Experiment. Math. 5, 291-295, 1996
Computing the summation of the Möbius function
* *Nathan Ng, "The distribution of the summatory function of the Möbius function", Proc. London Math. Soc. (3) 89 (2004) 361-389

{{DEFAULTSORT:Mertens Function Arithmetic functions