Liouville function
The Liouville function \lambda(n) is a completely multiplicative arithmetic function defined on the positive integers n by \lambda(n) = (-1)^{\Omega(n)}, where \Omega(n) denotes the total number of prime factors of n counted with multiplicity (and \Omega(1) = 0, so \lambda(1) = 1). Introduced by the French mathematician Joseph Liouville (1809–1882) in his foundational work on arithmetic identities and quadratic forms during the 1830s and 1840s, the function encodes the parity of the exponent sum in the prime factorization of n, taking the value $1if\Omega(n)is even and-1$ if odd.[1] The Dirichlet series associated with the Liouville function is \sum_{n=1}^\infty \lambda(n) n^{-s} = \zeta(2s)/\zeta(s) for \Re(s) > 1, where \zeta(s) is the Riemann zeta function; this identity highlights its deep ties to analytic number theory and the distribution of primes. The function is completely multiplicative, meaning \lambda(mn) = \lambda(m)\lambda(n) for all positive integers m and n, and it appears in various identities involving divisor functions and theta series, such as the Lambert series expansion \sum_{n=1}^\infty \frac{\lambda(n) x^n}{1 - x^n} = \sum_{n=-\infty}^\infty x^{n^2} for |x| < 1.[1] A central object of study is the summatory function L(x) = \sum_{n \leq x} \lambda(n), whose asymptotic behavior is intimately linked to the Riemann hypothesis (RH): RH holds if and only if L(x) = O(x^{1/2 + \varepsilon}) for every \varepsilon > 0, as shown by Edmund Landau in his 1899 thesis exploring prime number distribution. (Note: This equivalence stems from the prime number theorem and zero-free regions of \zeta(s).) Early conjectures, like George Pólya's 1919 assertion that L(n) \leq 0 for all n \geq 2, were disproved in 1958 by R. M. Haselgrove (who showed the existence of counterexamples), with the first explicit counterexample at n = 906,180,359 identified by R. Sherman Lehman in 1960 and the smallest at n = 906,150,257 found by Minoru Tanaka in 1980; these developments underscore the function's role in probing the oscillatory nature of prime-related sums.[2]Definition and Basic Properties
Definition
The Liouville function \lambda is a completely multiplicative arithmetic function defined for positive integers n in terms of the prime factorization n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} by \lambda(n) = (-1)^{a_1 + a_2 + \cdots + a_k}, where the exponent a_1 + a_2 + \cdots + a_k = \Omega(n) denotes the total number of prime factors of n counted with multiplicity.[3] This definition assigns \lambda(n) = 1 if \Omega(n) is even and \lambda(n) = -1 if \Omega(n) is odd.[3] As a completely multiplicative function, \lambda satisfies \lambda(mn) = \lambda(m)\lambda(n) for all positive integers m and n, regardless of whether m and n are coprime.[3] This property follows directly from the multiplicative nature of \Omega, since \Omega(mn) = \Omega(m) + \Omega(n).[3] For example, \lambda(1) = 1 since $1 has no prime factors (\Omega(1) = 0, an even number). For a prime p, \lambda(p) = -1 (\Omega(p) = 1). For p^2, \lambda(p^2) = 1 (\Omega(p^2) = 2). For distinct primes pandq, \lambda(pq) = 1 (\Omega(pq) = 2$).[3] The Liouville function is named after the French mathematician Joseph Liouville (1809–1882), who introduced it in his work on arithmetic identities and quadratic forms during his number-theoretic investigations from 1858 to 1865.[1] Unlike the Möbius function \mu(n), which depends only on the number of distinct prime factors, \lambda(n) accounts for multiplicity.[3]Multiplicativity and Values
The Liouville function \lambda(n) is completely multiplicative, meaning that \lambda(mn) = \lambda(m) \lambda(n) for all positive integers m and n. This property follows directly from its definition in terms of the total number of prime factors. Specifically, since the function \Omega(n), which counts the prime factors of n with multiplicity, is completely additive—satisfying \Omega(mn) = \Omega(m) + \Omega(n) for all m, n—it follows that \lambda(mn) = (-1)^{\Omega(mn)} = (-1)^{\Omega(m) + \Omega(n)} = (-1)^{\Omega(m)} (-1)^{\Omega(n)} = \lambda(m) \lambda(n).[4] The explicit relation \lambda(n) = (-1)^{\Omega(n)} ties the function directly to the prime factorization of n, where \Omega(1) = 0 and thus \lambda(1) = [1](/page/1). For prime powers, \lambda(p^k) = (-1)^k, alternating between -1 for odd exponents and [1](/page/1) for even exponents. This connection highlights how \lambda(n) encodes the parity of the total exponent sum in the prime factorization of n.[4] To illustrate, the values of \lambda(n) for small n up to 20 reveal patterns tied to the parity of \Omega(n):| n | Prime factorization | \Omega(n) | \lambda(n) |
|---|---|---|---|
| 1 | 1 | 0 | 1 |
| 2 | $2 | 1 | -1 |
| 3 | $3 | 1 | -1 |
| 4 | $2^2 | 2 | 1 |
| 5 | $5 | 1 | -1 |
| 6 | $2 \cdot 3 | 2 | 1 |
| 7 | $7 | 1 | -1 |
| 8 | $2^3 | 3 | -1 |
| 9 | $3^2 | 2 | 1 |
| 10 | $2 \cdot 5 | 2 | 1 |
| 11 | $11 | 1 | -1 |
| 12 | $2^2 \cdot 3 | 3 | -1 |
| 13 | $13 | 1 | -1 |
| 14 | $2 \cdot 7 | 2 | 1 |
| 15 | $3 \cdot 5 | 2 | 1 |
| 16 | $2^4 | 4 | 1 |
| 17 | $17 | 1 | -1 |
| 18 | $2 \cdot 3^2 | 3 | -1 |
| 19 | $19 | 1 | -1 |
| 20 | $2^2 \cdot 5 | 3 | -1 |