# Asymptotic behavior of the summatory Liouville function

EDIT: This is a major revision in light of sources provided by @ErickWong's comment.

The summatory Liouville function (OEIS sequence A002819) is $$L(n)=\sum_{1\le k\le n}(-1)^{\Omega(k)}\quad (n=1,2,3,\ldots)$$ where $$\Omega(k)$$ is the number of prime factors of $$k$$ counted with multiplicity. Thus $$L(n)$$ is the sum of the "parities" ($$+1$$ if even, $$-1$$ if odd) of the first $$n$$ values of $$\Omega.$$

Here is a plot of the first $$10^9$$ values of $$L$$, including overlays of some approximating curves: The overlays are as follows:

• A "central" curve (gray) $${\sqrt{n}\over\zeta({1\over 2})}$$ whose functional form is suggested by Tanaka's 1980 proof (mentioned here and proved here) that $$\ L(n)-{\sqrt{n}\over\zeta({1\over 2})}\$$ changes sign infinitely often as $$n\to\infty.$$

• A "sinusoid" (red) which oscillates about the central curve with increasing "wavelength" and amplitude: $$\hat{L}(n) = {\sqrt{n}\over\zeta({1\over 2})}\left( 1 + \sqrt{1\over 2}\,\sin\left(t_1\ln(n) + 2.75\right) \right)$$ where $$\zeta$$ is the Riemann zeta function and $$t_1$$ is the imaginary part of the first nontrivial zero of $$\zeta$$. Thus $$\zeta({1\over 2})=-0.684765236\ldots$$ and $$t_1=14.1347251\dots.$$ The occurrence of $$t_1$$ in the argument of the sine function is suggested by the "summary description" of a Wikipedia plot here. I've adjusted the parameters of the sinusoid, including amplitude and phase, to obtain a visual fit.

Q1: The mentioned "summary description" cites this paper ("On Differences of Zeta Values"), which proves an asymptotically oscillatory behavior of the coefficients in a certain series representation of Riemann zeta. How does this lead to a zero of Riemann zeta appearing in $$\hat{L}?$$

Q2: Can it be shown that regular oscillatory behavior of $$L(n)$$ persists for arbitrarily large $$n$$?

NB: This behavior says nothing about how often $$L(n)>0$$ may occur (such as the "spike" in the plot at $$n\approx 9\cdot 10^8$$); i.e., it may be independent of whether $$L$$ changes sign infinitely often.

• I do not know about your specific question, but you will find more references by searching for “Liouville lambda”. For instance, Wikipedia compiles some well-known results here en.m.wikipedia.org/wiki/Liouville_function – Erick Wong Sep 30 '19 at 3:01
• @ErickWong - Thank you very much -- I hadn't known quite what to search for! – r.e.s. Sep 30 '19 at 6:19
• Your "central" curve (gray) and "sinusoid" (red) are related to the explicit formula $L_o(x)=\frac{\sqrt{x}}{\zeta\left(\frac{1}{2}\right)}+\sum\limits_{k=1}^\infty\left(\frac{\zeta\left(2\,\rho_k\right)}{\rho_k\,\zeta'\left(\rho_k\right)}\,x^{\rho_k}+\frac{\zeta\left(2\,\rho_{-k}\right)}{\rho_{-k}\zeta'\left(\rho_{-k}\right)}\,x^{\rho_{-k}}\right)+I(x)$. – Steven Clark Oct 2 '19 at 23:29
• Try plotting $f(x)=\frac{\sqrt{x}}{\zeta\left(\frac{1}{2}\right)}+\left(\frac{\zeta\left(2\,\rho_1\right)}{\rho_1\zeta'\left(\rho_1\right)}x^{\rho_1}+\frac{\zeta\left(2\,\rho_{-1}\right)}{\rho_{-1}\zeta'\left(\rho_{-1}\right)}x^{\rho_{-1}}\right)$ where $\rho_1=\frac{1}{2}+i\,14.1347$ and $\rho_{-1}=\frac{1}{2}-i\,14.1347$ is the first pair of non-trivial zeta zeros. – Steven Clark Oct 2 '19 at 23:29
• $L_o(x)$ above is the explicit formula for $L(x)=\sum\limits_{n\le x}\lambda(n)$ where $\lambda(n)=(-1)^{\Omega(n)}$. – Steven Clark Oct 2 '19 at 23:39

The summatory Liouville function is defined as follows where $$\Omega(n)$$ is the number of prime factors of n counted with multiplicity.

(1) $$\quad L(x)=\sum\limits_{n\le x}\lambda(n)\,,\qquad\lambda(n)=(-1)^{\Omega(n)}$$

The OP's "central" curve (gray) and "sinusoid" (red) are related to the explicit formula for $$L(x)$$ illustrated in formula (2) below (see THE EXPLICIT FORMULA FOR $$L_o(x)$$ By A. Y. FAWAZ).

(2) $$\quad L_o(x)=\frac{\sqrt{x}}{\zeta\left(\frac{1}{2}\right)}+\sum\limits_{k=1}^\infty\left(\frac{\zeta\left(2\,\rho_k\right)}{\rho_k\,\zeta'\left(\rho_k\right)}\,x^{\rho_k}+\frac{\zeta\left(2\,\rho_{-k}\right)}{\rho_{-k}\zeta'\left(\rho_{-k}\right)}\,x^{\rho_{-k}}\right)+I(x)\,,\quad x>0$$

Consider the function $$f(x)$$ defined below which represents the primary growth term plus the contribution of the first non-trivial zeta zero pair $$\rho_1=\frac{1}{2}+i\,14.1347$$ and $$\rho_{-1}=\frac{1}{2}-i\,14.1347$$.

(3) $$\quad f(x)=\frac{\sqrt{x}}{\zeta\left(\frac{1}{2}\right)}+\left(\frac{\zeta\left(2\,\rho_1\right)}{\rho_1\zeta'\left(\rho_1\right)}x^{\rho_1}+\frac{\zeta\left(2\,\rho_{-1}\right)}{\rho_{-1}\zeta'\left(\rho_{-1}\right)}x^{\rho_{-1}}\right)$$

Figure (1) below illustrates $$L(x)$$ in blue, the primary growth term $$\frac{\sqrt{x}}{\zeta\left(\frac{1}{2}\right)}$$ in gray, and $$f(x)$$ in red. Figure (1): Illustration of $$L(x)$$ (blue), $$\frac{\sqrt{x}}{\zeta\left(\frac{1}{2}\right)}$$ (gray), and $$f(x)$$ (red)

Assuming the Riemann Hypothesis (RH):

• $$\rho_k=\frac{1}{2}+i\,\Im(\rho_k)$$,
• $$\rho_{-k}=\frac{1}{2}-i\,\Im(\rho_k)$$,
• $$x^{\rho_k}=\sqrt{x}\left(\cos\left(\Im\left(\rho_k\right)\log x\right)+i\,\sin\left(\Im\left(\rho_k\right)\log x\right)\right)$$, and
• $$x^{\rho_{-k}}=\sqrt{x}\left(\cos\left(\Im\left(\rho_k\right)\log x\right)-i\,\sin\left(\Im\left(\rho_k\right)\log x\right)\right)$$.

This can be combined with $$\frac{\zeta\left(2\,\rho_{-k}\right)}{\rho_{-k}\zeta'\left(\rho_{-k}\right)}=\left(\frac{\zeta\left(2\,\rho_k\right)}{\rho_k\zeta'\left(\rho_k\right)}\right){}^*$$ (where $$*$$ indicates the complex conjugate) to derive the following result which is strictly real.

(4) $$\quad\frac{\zeta\left(2\,\rho_k\right)}{\rho_k\,\zeta'\left(\rho_k\right)}\,x^{\rho_k}+\frac{\zeta\left(2\,\rho_{-k}\right)}{\rho_{-k}\,\zeta'\left(\rho_{-k}\right)}\,x^{\rho_{-k}}=\\$$ $$\qquad 2\sqrt{x}\left(\Re\left(\frac{\zeta\left(2\,\rho_k\right)}{\rho_k\,\zeta'\left(\rho_k\right)}\right)\cos\left(\Im\left(\rho_k\right)\log x\right)-\Im\left(\frac{\zeta\left(2\,\rho_k\right)}{\rho_k\,\zeta'\left(\rho_k\right)}\right)\sin\left(\Im\left(\rho_k\right)\log x\right)\right)\quad\text{(Assuming RH)}$$

In mathematics the natural logarithm of $$x$$ is commonly referred to as $$\log(x)$$ instead of $$\ln(x)$$, and $$\log_b(x)$$ refers to the base $$b$$ logarithm of $$x$$. This is consistent with the notation generated by Mathematica when formulas are copied as Latex. Also, $$\rho_k$$ is returned by the Mathematica function $$ZetaZero[k]$$.

All the explicit formulas come from the inverse Mellin transform + residue theorem $$\frac{\zeta(2s)}{\zeta(s)}=s\int_1^\infty (\sum_{n\le x}(-1)^{\Omega(n)})x^{-s-1}dx$$ $$\sum_{n\le x}(-1)^{\Omega(n)} = \frac{1}{2i\pi} \int_{2-i\infty}^{2+i\infty} \frac{\zeta(2s)}{\zeta(s)} \frac{x^s}{s} ds=\sum Res(\frac{\zeta(2s)}{\zeta(s)} \frac{x^s}{s})$$ except that the convergence of that sum over the residues (over the trivial and non-trivial zeros) is quite unclear, thus either we add a regulator (looking at $$\int_1^x \sum_{n\le y}(-1)^{\Omega(n)}dy$$), or we approximate the integral by $$\int_{2-iT}^{2+iT}$$ for well chosen value of $$T$$ such that we can "traverse the critical strip" to express it as a sum over the first few non-trivial zeros.