- Source: Liouville 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 primes.
Explicitly, the fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: n = p1a1 ⋯ pkak, where p1 < p2 < ... < pk are primes and the aj are positive integers. (1 is given by the empty product.) The prime omega functions count the number of primes, with (Ω) or without (ω) multiplicity:
ω
(
n
)
=
k
,
{\displaystyle \omega (n)=k,}
Ω
(
n
)
=
a
1
+
a
2
+
⋯
+
a
k
.
{\displaystyle \Omega (n)=a_{1}+a_{2}+\cdots +a_{k}.}
λ(n) is defined by the formula
λ
(
n
)
=
(
−
1
)
Ω
(
n
)
{\displaystyle \lambda (n)=(-1)^{\Omega (n)}}
(sequence A008836 in the OEIS).
λ is completely multiplicative since Ω(n) is completely additive, 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 μ(n). Write n as n = a2b, where b is squarefree, i.e., ω(b) = Ω(b). Then
λ
(
n
)
=
μ
(
b
)
.
{\displaystyle \lambda (n)=\mu (b).}
The sum of the Liouville function over the divisors of n is the characteristic function of the squares:
∑
d
|
n
λ
(
d
)
=
{
1
if
n
is a perfect square,
0
otherwise.
{\displaystyle \sum _{d|n}\lambda (d)={\begin{cases}1&{\text{if }}n{\text{ is a perfect square,}}\\0&{\text{otherwise.}}\end{cases}}}
Möbius inversion of this formula yields
λ
(
n
)
=
∑
d
2
|
n
μ
(
n
d
2
)
.
{\displaystyle \lambda (n)=\sum _{d^{2}|n}\mu \left({\frac {n}{d^{2}}}\right).}
The Dirichlet inverse of Liouville function is the absolute value of the Möbius function, λ–1(n) = |μ(n)| = μ2(n), the characteristic function of the squarefree integers. We also have that λ(n) = μ2(n).
Series
The Dirichlet series for the Liouville function is related to the Riemann zeta function by
ζ
(
2
s
)
ζ
(
s
)
=
∑
n
=
1
∞
λ
(
n
)
n
s
.
{\displaystyle {\frac {\zeta (2s)}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}.}
Also:
∑
n
=
1
∞
λ
(
n
)
ln
n
n
=
−
ζ
(
2
)
=
−
π
2
6
.
{\displaystyle \sum \limits _{n=1}^{\infty }{\frac {\lambda (n)\ln n}{n}}=-\zeta (2)=-{\frac {\pi ^{2}}{6}}.}
The Lambert series for the Liouville function is
∑
n
=
1
∞
λ
(
n
)
q
n
1
−
q
n
=
∑
n
=
1
∞
q
n
2
=
1
2
(
ϑ
3
(
q
)
−
1
)
,
{\displaystyle \sum _{n=1}^{\infty }{\frac {\lambda (n)q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }q^{n^{2}}={\frac {1}{2}}\left(\vartheta _{3}(q)-1\right),}
where
ϑ
3
(
q
)
{\displaystyle \vartheta _{3}(q)}
is the Jacobi theta function.
Conjectures on weighted summatory functions
The Pólya problem is a question raised made by George Pólya in 1919. Defining
L
(
n
)
=
∑
k
=
1
n
λ
(
k
)
{\displaystyle L(n)=\sum _{k=1}^{n}\lambda (k)}
(sequence A002819 in the OEIS),
the problem asks whether
L
(
n
)
≤
0
{\displaystyle L(n)\leq 0}
for n > 1. The answer turns out to be no. The smallest counter-example is n = 906150257, found by Minoru Tanaka in 1980. It has since been shown that L(n) > 0.0618672√n for infinitely many positive integers n, while it can also be shown via the same methods that L(n) < -1.3892783√n for infinitely many positive integers n.
For any
ε
>
0
{\displaystyle \varepsilon >0}
, assuming the Riemann hypothesis, we have that the summatory function
L
(
x
)
≡
L
0
(
x
)
{\displaystyle L(x)\equiv L_{0}(x)}
is bounded by
L
(
x
)
=
O
(
x
exp
(
C
⋅
log
1
/
2
(
x
)
(
log
log
x
)
5
/
2
+
ε
)
)
,
{\displaystyle L(x)=O\left({\sqrt {x}}\exp \left(C\cdot \log ^{1/2}(x)\left(\log \log x\right)^{5/2+\varepsilon }\right)\right),}
where the
C
>
0
{\displaystyle C>0}
is some absolute limiting constant.
Define the related sum
T
(
n
)
=
∑
k
=
1
n
λ
(
k
)
k
.
{\displaystyle T(n)=\sum _{k=1}^{n}{\frac {\lambda (k)}{k}}.}
It was open for some time whether T(n) ≥ 0 for sufficiently big n ≥ n0 (this conjecture is occasionally–though incorrectly–attributed to Pál Turán). This was then disproved by Haselgrove (1958), 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.
= Generalizations
=More generally, we can consider the weighted summatory functions over the Liouville function defined for any
α
∈
R
{\displaystyle \alpha \in \mathbb {R} }
as follows for positive integers x where (as above) we have the special cases
L
(
x
)
:=
L
0
(
x
)
{\displaystyle L(x):=L_{0}(x)}
and
T
(
x
)
=
L
1
(
x
)
{\displaystyle T(x)=L_{1}(x)}
L
α
(
x
)
:=
∑
n
≤
x
λ
(
n
)
n
α
.
{\displaystyle L_{\alpha }(x):=\sum _{n\leq x}{\frac {\lambda (n)}{n^{\alpha }}}.}
These
α
−
1
{\displaystyle \alpha ^{-1}}
-weighted summatory functions are related to the Mertens function, or weighted summatory functions of the Moebius function. In fact, we have that the so-termed non-weighted, or ordinary function
L
(
x
)
{\displaystyle L(x)}
precisely corresponds to the sum
L
(
x
)
=
∑
d
2
≤
x
M
(
x
d
2
)
=
∑
d
2
≤
x
∑
n
≤
x
d
2
μ
(
n
)
.
{\displaystyle L(x)=\sum _{d^{2}\leq x}M\left({\frac {x}{d^{2}}}\right)=\sum _{d^{2}\leq x}\sum _{n\leq {\frac {x}{d^{2}}}}\mu (n).}
Moreover, these functions satisfy similar bounding asymptotic relations. For example, whenever
0
≤
α
≤
1
2
{\displaystyle 0\leq \alpha \leq {\frac {1}{2}}}
, we see that there exists an absolute constant
C
α
>
0
{\displaystyle C_{\alpha }>0}
such that
L
α
(
x
)
=
O
(
x
1
−
α
exp
(
−
C
α
(
log
x
)
3
/
5
(
log
log
x
)
1
/
5
)
)
.
{\displaystyle L_{\alpha }(x)=O\left(x^{1-\alpha }\exp \left(-C_{\alpha }{\frac {(\log x)^{3/5}}{(\log \log x)^{1/5}}}\right)\right).}
By an application of Perron's formula, or equivalently by a key (inverse) Mellin transform, we have that
ζ
(
2
α
+
2
s
)
ζ
(
α
+
s
)
=
s
⋅
∫
1
∞
L
α
(
x
)
x
s
+
1
d
x
,
{\displaystyle {\frac {\zeta (2\alpha +2s)}{\zeta (\alpha +s)}}=s\cdot \int _{1}^{\infty }{\frac {L_{\alpha }(x)}{x^{s+1}}}dx,}
which then can be inverted via the inverse transform to show that for
x
>
1
{\displaystyle x>1}
,
T
≥
1
{\displaystyle T\geq 1}
and
0
≤
α
<
1
2
{\displaystyle 0\leq \alpha <{\frac {1}{2}}}
L
α
(
x
)
=
1
2
π
ı
∫
σ
0
−
ı
T
σ
0
+
ı
T
ζ
(
2
α
+
2
s
)
ζ
(
α
+
s
)
⋅
x
s
s
d
s
+
E
α
(
x
)
+
R
α
(
x
,
T
)
,
{\displaystyle L_{\alpha }(x)={\frac {1}{2\pi \imath }}\int _{\sigma _{0}-\imath T}^{\sigma _{0}+\imath T}{\frac {\zeta (2\alpha +2s)}{\zeta (\alpha +s)}}\cdot {\frac {x^{s}}{s}}ds+E_{\alpha }(x)+R_{\alpha }(x,T),}
where we can take
σ
0
:=
1
−
α
+
1
/
log
(
x
)
{\displaystyle \sigma _{0}:=1-\alpha +1/\log(x)}
, and with the remainder terms defined such that
E
α
(
x
)
=
O
(
x
−
α
)
{\displaystyle E_{\alpha }(x)=O(x^{-\alpha })}
and
R
α
(
x
,
T
)
→
0
{\displaystyle R_{\alpha }(x,T)\rightarrow 0}
as
T
→
∞
{\displaystyle 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
ρ
=
1
2
+
ı
γ
{\displaystyle \rho ={\frac {1}{2}}+\imath \gamma }
, of the Riemann zeta function are simple, then for any
0
≤
α
<
1
2
{\displaystyle 0\leq \alpha <{\frac {1}{2}}}
and
x
≥
1
{\displaystyle x\geq 1}
there exists an infinite sequence of
{
T
v
}
v
≥
1
{\displaystyle \{T_{v}\}_{v\geq 1}}
which satisfies that
v
≤
T
v
≤
v
+
1
{\displaystyle v\leq T_{v}\leq v+1}
for all v such that
L
α
(
x
)
=
x
1
/
2
−
α
(
1
−
2
α
)
ζ
(
1
/
2
)
+
∑
|
γ
|
<
T
v
ζ
(
2
ρ
)
ζ
′
(
ρ
)
⋅
x
ρ
−
α
(
ρ
−
α
)
+
E
α
(
x
)
+
R
α
(
x
,
T
v
)
+
I
α
(
x
)
,
{\displaystyle L_{\alpha }(x)={\frac {x^{1/2-\alpha }}{(1-2\alpha )\zeta (1/2)}}+\sum _{|\gamma |
where for any increasingly small
0
<
ε
<
1
2
−
α
{\displaystyle 0<\varepsilon <{\frac {1}{2}}-\alpha }
we define
I
α
(
x
)
:=
1
2
π
ı
⋅
x
α
∫
ε
+
α
−
ı
∞
ε
+
α
+
ı
∞
ζ
(
2
s
)
ζ
(
s
)
⋅
x
s
(
s
−
α
)
d
s
,
{\displaystyle I_{\alpha }(x):={\frac {1}{2\pi \imath \cdot x^{\alpha }}}\int _{\varepsilon +\alpha -\imath \infty }^{\varepsilon +\alpha +\imath \infty }{\frac {\zeta (2s)}{\zeta (s)}}\cdot {\frac {x^{s}}{(s-\alpha )}}ds,}
and where the remainder term
R
α
(
x
,
T
)
≪
x
−
α
+
x
1
−
α
log
(
x
)
T
+
x
1
−
α
T
1
−
ε
log
(
x
)
,
{\displaystyle R_{\alpha }(x,T)\ll x^{-\alpha }+{\frac {x^{1-\alpha }\log(x)}{T}}+{\frac {x^{1-\alpha }}{T^{1-\varepsilon }\log(x)}},}
which of course tends to 0 as
T
→
∞
{\displaystyle T\rightarrow \infty }
. These exact analytic formula expansions again share similar properties to those corresponding to the weighted Mertens function cases. Additionally, since
ζ
(
1
/
2
)
<
0
{\displaystyle \zeta (1/2)<0}
we have another similarity in the form of
L
α
(
x
)
{\displaystyle L_{\alpha }(x)}
to
M
(
x
)
{\displaystyle 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
Pólya, G. (1919). "Verschiedene Bemerkungen zur Zahlentheorie". Jahresbericht der Deutschen Mathematiker-Vereinigung. 28: 31–40.
Haselgrove, C. Brian (1958). "A disproof of a conjecture of Pólya". Mathematika. 5 (2): 141–145. doi:10.1112/S0025579300001480. ISSN 0025-5793. MR 0104638. Zbl 0085.27102.
Lehman, R. (1960). "On Liouville's function". Mathematics of Computation. 14 (72): 311–320. doi:10.1090/S0025-5718-1960-0120198-5. MR 0120198.
Tanaka, Minoru (1980). "A Numerical Investigation on Cumulative Sum of the Liouville Function". Tokyo Journal of Mathematics. 3 (1): 187–189. doi:10.3836/tjm/1270216093. MR 0584557.
Weisstein, Eric W. "Liouville Function". MathWorld.
A.F. Lavrik (2001) [1994], "Liouville function", Encyclopedia of Mathematics, EMS Press
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- Sturm–Liouville theory
- Elementary function
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- Joseph Liouville
- Liouville's theorem (complex analysis)
- Pólya conjecture
- List of mathematical functions
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