Liouville function

The Liouville function , named after Joseph Liouville , is a multiplicative number theoretic function . It is denoted by the Greek letter and is defined as follows: ${\displaystyle \lambda }$

${\displaystyle \lambda (n)=(-1)^{\Omega (n)},\,}$

denotes the order of , i.e. the number of its (not necessarily different) prime factors . ${\displaystyle \Omega (n)}$ ${\displaystyle n}$

You also define and . ${\displaystyle \lambda (0)=0}$ ${\displaystyle \lambda (1)=1}$

The first values ​​(starting at ) are ${\displaystyle n=1}$

1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, - 1, ... ( OEIS , A008836)

properties

It applies

${\displaystyle \sum _{d|n}\lambda (d)={\begin{cases}1,\qquad \mathrm {wenn} \;n\;\mathrm {eine\;Quadratzahl\;ist} \\0,\qquad \mathrm {sonst.} \end{cases}}}$

The Liouville function is related to the Möbius function by ${\displaystyle \mu }$

${\displaystyle \lambda (n)=\sum _{d^{2}|n}\mu \left({\frac {n}{d^{2}}}\right).}$

Rows

The Dirichlet series of the Liouville function can be expressed by the Riemann zeta function : ${\displaystyle \zeta }$

${\displaystyle \sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}={\frac {\zeta (2s)}{\zeta (s)}}.}$

Your Lambert series is given by

${\displaystyle \sum _{n=1}^{\infty }{\frac {\lambda (n)q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }q^{n^{2}}={\frac {1}{2}}(\vartheta _{3}(q)-1),}$

where denotes the Jacobian theta function . ${\displaystyle \vartheta _{3}}$

to hum

Graph from to n = 10,000 ${\displaystyle L_{n}}$

Graph from to ${\displaystyle L_{n}}$ ${\displaystyle 10^{7}}$

Be it

${\displaystyle L(n)=\sum _{k=1}^{n}\lambda (k).}$

The Pólya conjecture says that it is - as the graphics on the right suggest - always

${\displaystyle L(n)\leq 0.}$

This assumption has now been refuted; is the smallest counterexample . However, it is not yet known whether its sign changes infinitely often. ${\displaystyle n=906150257}$ ${\displaystyle L}$

A related sum is

${\displaystyle M(n)=\sum _{k=1}^{n}{\frac {\lambda (k)}{k}}.}$

For this it was assumed that it is always positive for sufficiently large ones; this was refuted in 1958 by the English mathematician Colin Brian Haselgrove , where he showed that infinitely often takes negative values. A proof of the conjecture would have resulted in the correctness of the Riemann conjecture . ${\displaystyle n}$ ${\displaystyle M}$

Chowla conjecture

A conjecture by Sarvadaman Chowla states that for different natural numbers the following applies: ${\displaystyle k}$ ${\displaystyle h_{1},\ldots ,h_{k}}$

${\displaystyle \sum _{1\leq n\leq x}\lambda (n+h_{1})\cdot \cdot \cdot \lambda (n+h_{k})=o(x)}$

(that is, the sum vanishes asymptotically with , see Landau symbols ). The guess is open to . In 2015, Kaisa Matomäki , Maksym Radziwill and Terence Tao made progress on an averaged version of the conjecture. The conjecture can also be formulated for the Möbius function instead of the Liouville function. ${\displaystyle x\to \infty }$ ${\displaystyle k\geq 2}$

Another formulation of the conjecture is that the pattern of the values ​​of for a randomly chosen natural number and any asymptotically for is uniformly distributed. ${\displaystyle \lambda (n),\cdot \cdot \cdot ,\lambda (n+k)}$ ${\displaystyle n\leq x}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle x\to \infty }$

Web links

• Eric W. Weisstein : Liouville Function . In: MathWorld (English).
• AF Lavrik: Liouville function. In: Online Encyclopedia of Mathematics . (English)
• Kimberly Lloyd: Liouville function. On: PlanetMath .org. (English)

Individual evidence

1. ↑ A008836 Liouville's function lambda (n) = (-1) ^ k, where k is number of primes dividing n (counted with multiplicity). ( English ) The OEIS Foundation. Retrieved July 16, 2019.
2. ↑ See episodes A026424 and A028260 .
3. ↑ Kimberly Lloyd: Liouville function. On: PlanetMath .org. (English)
4. ↑ AF Lavrik: function Liouville. In: Online Encyclopedia of Mathematics . (English)
5. ^ Russell Sherman Lehman: On Liouville's Function. (PDF; 824 kB) In: Mathematics of Compution ⟨American Mathematical Society⟩ 14 (1960), No. 72, pp. 311–320.
6. ↑ Eric W. Weisstein : Polya Conjecture . In: MathWorld (English).
7. ^ Colin Brian Haselgrove: A disproof of a conjecture of Polya . In: Mathematika ⟨London Mathematical Society⟩ 5 (1958), No. 2, pp. 141–145.
8. ^ Hisanobu Shinya: On an arithmetical approach to the Riemann hypothesis . In: arxiv : 0906.4155 (June 23, 2009).
9. ^ Sarvadaman Chowla: The Riemann Hypothesis and Hilbert's tenth problem , Gordon and Breach 1965
10. ↑ K. Matomäki, M. Radziwill, Terence Tao : An averaged form of Chowla's conjecture, Algebra & Number Theory, Volume 9, 2015, pp. 2167-2196, Arxiv
11. ^ Sign patterns of Liouville and Mobius functions, Blog Terry Tao