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Liouville Function

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LiouvilleLambda

The function

lambda(n)=(-1)^(Omega(n)),
(1)

where Omega(n) is the number of not necessarily distinct prime factors of n, with Omega(1)=0. The values of lambda(n) for n=1, 2, ... are 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, -1, ... (OEIS A008836). The values of n such that lambda(n)=-1 are 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, ... (OEIS A026424), while then values such that lambda(n)=+1 are 1, 4, 6, 9, 10, 14, 15, 16, 21, 22, 24, ... (OEIS A028260).

The Liouville function is implemented in the Wolfram Language as LiouvilleLambda[n].

The Liouville function is connected with the Riemann zeta function by the equation

(zeta(2s))/(zeta(s))=sum_(n=1)^infty(lambda(n))/(n^s)
(2)

(Lehman 1960). It has the Lambert series

sum_(n=1)^(infty)(lambda(n)x^n)/(1-x^n)=sum_(n=1)^(infty)x^(n^2)
(3)
=1/2[theta_3(q)-1],
(4)

where theta_3(q)=theta_3(0,q) is a Jacobi theta function.

LiouvilleL

Consider the summatory function

L(n)=sum_(k=1)^nlambda(k),
(5)

the values of which for n=1, 2, ... are 1, 0, -1, 0, -1, 0, -1, -2, -1, 0, -1, -2, -3, -2, -1, 0, -1, -2, -3, -4, ... (OEIS A002819).

Lehman (1960) gives the formulas

L(x)=sum_(m=1)^(x/w)mu(m){|_sqrt(x/m)_|-sum_(k=1)^(v-1)lambda(k)(|_x/(km)_|-|_x/(mv)_|)}-sum_(l=x/w-1)^(x/v)L(x/l)sum_(m|l; m=1)^(x/w)mu(m)
(6)

and

L(x)=sum_(k=1)^gM(x/(k^2))+sum_(l=1)^(x/g^2)mu(l)|_sqrt(x/l)_|-M(x/(g^2))|_sqrt(x/(g^2))_|,
(7)

where k, l, and m are variables ranging over the positive integers, mu(n) is the Möbius function, M(x) is Mertens function, and v, w, and x are positive real numbers with v<w<x.

The conjecture that L(n) satisfies L(n)<=0 for n>=2 is called the Pólya conjecture and has been proved to be false. L(n) is positive for n=1, but not for any other n for a long time. In fact, the first n for which L(n)=0 are for n=2, 4, 6, 10, 16, 26, 40, 96, 586, 906150256, ... (OEIS A028488), and n=906150257 is the first counterexample to the Pólya conjecture (Tanaka 1980). However, it is unknown if L(x) changes sign infinitely often (Tanaka 1980).

The values of L(10^n) for n=0, 1, 2, ... are 1, 0, -2, -14, -94, -288, -530, -842, -3884, ... (OEIS A090410).

See also

Pólya Conjecture, Prime Factor, Riemann Zeta Function

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References

Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, p. 37, 1976.Fawaz, A. Y. "The Explicit Formula for L_0(x)." Proc. London Math. Soc. 1, 86-103, 1951.Gupta, H. "On a Table of Values of L(n)." Proc. Indian Acad. Sci. Sec. A 12, 407-409, 1940.Gupta, H. "A Table of Values of Liouville's Function L(n)." Res. Bull. East Panjab University, No. 3, 45-55, Feb. 1950.Lehman, R. S. "On Liouville's Function." Math. Comput. 14, 311-320, 1960.Ramanujan, S. "Irregular Numbers." J. Indian Math. Soc. 5, 105-106, 1913. Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., pp. 20-21, 2000.Ribenboim, P. Algebraic Numbers. New York: Wiley, p. 44, 1972.Roberts, J. The Lure of the Integers. Washington, DC: Math. Assoc. Amer., p. 279, 1992.Sloane, N. J. A. Sequences A002819/M0042, A008836, A026424, A028260, A028488, and A090410 in "The On-Line Encyclopedia of Integer Sequences."Tanaka, M. "A Numerical Investigation on Cumulative Sum of the Liouville Function." Tokyo J. Math. 3, 187-189, 1980.

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Liouville Function

Cite this as:

Weisstein, Eric W. "Liouville Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LiouvilleFunction.html

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