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The Liouville Lambda function, denoted by λ(''n'') and named after
Joseph Liouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérès ...
, is an important arithmetic function. Its value is +1 if ''n'' is the product of an even number of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s, and −1 if it is the product of an odd number of primes. Explicitly, the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
states that any 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 o ...
''n'' can be represented uniquely as a product of powers of primes:   n = p_1^\cdots p_k^   where ''p''1 < ''p''2 < ... < ''p''''k'' are primes and the ''aj'' are positive integers. (1 is given by the empty product.) The
prime omega function In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. Thereby \omega(n) (little omega) counts each ''distinct'' prime factor, whereas the related function \Omega(n) (big omega) ...
s count the number of primes, with (Ω) or without (ω) multiplicity: :''ω''(''n'') = ''k'', :Ω(''n'') = ''a''1 + ''a''2 + ... + ''a''''k''. λ(''n'') is defined by the formula :\lambda(n) = (-1)^. . λ is
completely multiplicative In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime ...
since Ω(''n'') is completely
additive Additive may refer to: Mathematics * Additive function, a function in number theory * Additive map, a function that preserves the addition operation * Additive set-functionn see Sigma additivity * Additive category, a preadditive category with fi ...
, i.e.: Ω(''ab'') = Ω(''a'') + Ω(''b''). Since 1 has no prime factors, Ω(1) = 0 so λ(1) = 1. It is related to 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 of ...
''μ''(''n''). Write ''n'' as ''n'' = ''a''2''b'' where ''b'' is squarefree, i.e., ''ω''(''b'') = Ω(''b''). Then :\lambda(n) = \mu(b). The sum of the Liouville function over the
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s of ''n'' is the characteristic function of the
squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length ...
: : \sum_\lambda(d) = \begin 1 & \textn\text \\ 0 & \text \end
Möbius inversion Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Pa ...
of this formula yields :\lambda(n) = \sum_ \mu\left(\frac\right). The
Dirichlet inverse In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet. Definition If f , g : \mathbb\to\mathbb are two arithmetic fun ...
of Liouville function is the absolute value of the Möbius function, \lambda^(n) = , \mu(n), = \mu^2(n), the characteristic function of the squarefree integers. We also have that \lambda(n) \mu(n) = \mu^2(n).


Series

The
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 ana ...
for the Liouville function is related to 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) > ...
by :\frac = \sum_^\infty \frac. Also: :\sum\limits_^ \frac=-\zeta(2)=-\frac. The
Lambert series In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form :S(q)=\sum_^\infty a_n \frac . It can be resumed formally by expanding the denominator: :S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty b_m ...
for the Liouville function is :\sum_^\infty \frac = \sum_^\infty q^ = \frac\left(\vartheta_3(q)-1\right), where \vartheta_3(q) is the
Jacobi theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field the ...
.


Conjectures on weighted summatory functions

The
Pólya conjecture In number theory, the Pólya conjecture (or Pólya's conjecture) stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an ''odd'' number of prime factors. The conjecture was set forth by the Hungarian mathe ...
is a conjecture made by
George Pólya George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental ...
in 1919. Defining : L(n) = \sum_^n \lambda(k) , the conjecture states that L(n)\leq 0 for ''n'' > 1. This turned out to be false. The smallest counter-example is ''n'' = 906150257, found by Minoru Tanaka in 1980. It has since been shown that ''L''(''n'') > 0.0618672 for infinitely many positive integers ''n'', while it can also be shown via the same methods that ''L''(''n'') < -1.3892783 for infinitely many positive integers ''n''. For any \varepsilon > 0, assuming the Riemann hypothesis, we have that the summatory function L(x) \equiv L_0(x) is bounded by :L(x) = O\left(\sqrt \exp\left(C \cdot \log^(x) \left(\log\log x\right)^\right)\right), where the C > 0 is some absolute limiting constant. Define the related sum : T(n) = \sum_^n \frac. It was open for some time whether ''T''(''n'') ≥ 0 for sufficiently big ''n'' ≥ ''n''0 (this conjecture is occasionally–though incorrectly–attributed to
Pál Turán Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting ...
). This was then disproved by , who showed that ''T''(''n'') takes negative values infinitely often. A confirmation of this positivity conjecture would have led to a proof of the Riemann hypothesis, as was shown by
Pál Turán Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting ...
.


Generalizations

More generally, we can consider the weighted summatory functions over the Liouville function defined for any \alpha \in \mathbb as follows for positive integers ''x'' where (as above) we have the special cases L(x) := L_0(x) and T(x) = L_1(x) :L_(x) := \sum_ \frac. These \alpha^-weighted summatory functions are related to the
Mertens function In number theory, the Mertens function is defined for all positive integers ''n'' as : M(n) = \sum_^n \mu(k), where \mu(k) is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive re ...
, or weighted summatory functions of the
Moebius function Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * P ...
. In fact, we have that the so-termed non-weighted, or ordinary function L(x) precisely corresponds to the sum :L(x) = \sum_ M\left(\frac\right) = \sum_ \sum_ \mu(n). Moreover, these functions satisfy similar bounding asymptotic relations. For example, whenever 0 \leq \alpha \leq \frac, we see that there exists an absolute constant C_ > 0 such that :L_(x) = O\left(x^\exp\left(-C_ \frac\right)\right). By an application of
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, ...
, or equivalently by a key (inverse)
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 ...
, we have that :\frac = s \cdot \int_1^ \frac dx, which then can be inverted via the inverse transform to show that for x > 1, T \geq 1 and 0 \leq \alpha < \frac :L_(x) = \frac \int_^ \frac \cdot \frac ds + E_(x) + R_(x, T), where we can take \sigma_0 := 1-\alpha+1 / \log(x), and with the remainder terms defined such that E_(x) = O(x^) and R_(x, T) \rightarrow 0 as T \rightarrow \infty. In particular, if we assume that the Riemann hypothesis (RH) is true and that all of the non-trivial zeros, denoted by \rho = \frac + \imath\gamma, of 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) > ...
are simple, then for any 0 \leq \alpha < \frac and x \geq 1 there exists an infinite sequence of \_ which satisfies that v \leq T_v \leq v+1 for all ''v'' such that :L_(x) = \frac + \sum_ \frac \cdot \frac + E_(x) + R_(x, T_v) + I_(x), where for any increasingly small 0 < \varepsilon < \frac-\alpha we define :I_(x) := \frac \int_^ \frac \cdot \frac ds, and where the remainder term :R_(x, T) \ll x^ + \frac + \frac, which of course tends to ''0'' as T \rightarrow \infty. These exact analytic formula expansions again share similar properties to those corresponding to the weighted
Mertens function In number theory, the Mertens function is defined for all positive integers ''n'' as : M(n) = \sum_^n \mu(k), where \mu(k) is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive re ...
cases. Additionally, since \zeta(1/2) < 0 we have another similarity in the form of L_(x) to M(x) in so much as the dominant leading term in the previous formulas predicts a negative bias in the values of these functions over the positive natural numbers ''x''.


References

* * * * * * {{DEFAULTSORT:Liouville Function Multiplicative functions