This reference contains the best result of this kind currently known for $\mu(n)$:

Tadej Kotnik and Herman te Riele The Mertens Conjecture revisited. Algorithmic number theory, 156--167, Lecture Notes in Comput. Sci., 4076, Springer, Berlin, 2006.

They prove that $\limsup_{x \rightarrow +\infty}M(x)/\sqrt{x} \geq 1.218$ and that $\liminf_{x \rightarrow +\infty}M(x)/\sqrt{x} \leq -1.229$. Here

$ M(x) = \sum_{n \leq x}\mu(n) $

is the conventional notation for the summatory function of the Möbius function. Their proof is a mixture of analytic number theory and large scale computations. They also have a survey of what is known and what is conjectured about the size of $M(x)$.

Now on to the Liouville function $\lambda(n)$ and its summatory function $L(x)$. The latter is very closely connected with $M(x)$, for

$ \sum_{n \leq \sqrt{x}}\mu(n)L\left(\frac{x}{n^2}\right) = \sum_{n \leq \sqrt{x}}\mu(n)\sum_{m \leq x/n^2}\lambda(m) = \sum_{N \leq x}\sum_{mn^2 = N}\mu(n)\lambda(m) = \sum_{N \leq x}\mu(N) = M(x). $

Thus

$ |M(x)| \leq \sum_{n \leq \sqrt{x}}|L\left(\frac{x}{n^2}\right)| $

and so the assumption

$ L(x) = o\left(\frac{\sqrt{x}}{\log^{1 + \epsilon}(x)}\right) $

for example, leads to a contradiction with the Kotnik-te Riele result (or earlier results) for any $\epsilon > 0$.

My guess is that if one looks up the old (pre-computer) results on $|M(x)|$ from below, one might prove that $\limsup_{x \rightarrow +\infty}|L(x)|/\sqrt{x} > 0$. This may even be in the literature somewhere.

Alternatively

$ \sum_{n = 1}^{\infty}\lambda(n)n^{-s} = \prod_{p}\sum_{k=0}^{\infty}\lambda(p^k)p^{-ks} = \prod_{p}\sum_{k=0}^{\infty}(-1)^kp^{-ks} = \prod_{p}(1 + p^{-s})^{-1} = \frac{\zeta(2s)}{\zeta(s)} $

by the Euler product formula, and from here on it is more elementary than it was with $M(x)$ in the argument that David Speyer gave, because we don't need the zeros on the critical line. For $\zeta(2s)$ has a pole at $s = 1/2$ which is not canceled by a pole of $\zeta(s)$ at the same point. Thus $L(x) = O(x^{\alpha})$ is impossible with $\alpha < 1/2$ by partial summation.

For multiplicative functions of modulus $1$, the situation is much less clear. For simplicity, assume that $f(n)$ is a totally multiplicative function (multiplicative and $f(p^k) = f(p)^k$) with $|c_p| = 1$ where $c_p = f(p)$. The Liouville function is the case $c_p \equiv -1$. Then

$ \sum_{n = 1}^{\infty}f(n)n^{-s} = \prod_{p}\frac{1}{1 - c_pp^{-s}} $

by the Euler product formula. The basic principle is that if $A(x) = O(x^{\alpha})$, then the Dirichlet series on the left hand side is convergent in the half plane $\sigma > \alpha$, so the sum is holomorphic in that half plane. If we can find a singularity $s_0$ of the product on the right hand side with $\mathrm{Re}(s_0) = \sigma_0$, that tells us that $A(x) = O(x^{\alpha})$ with $\alpha < \sigma_0$ is impossible. The bad thing now is that the product may diverge at a point without having a singularity there, because the product may diverge to zero, and a holomorphic function may be zero at a point without being singular there.

But it is straightforward to show that if $\mathrm{Re}(c_p) \geq \delta > 0$ for all $p$, then $A(x) = O(x^{\alpha})$ is impossible for any $\alpha < 1$, by showing that the series $ \sum_{p}c_pp^{-\sigma} $ goes to infinity as $\sigma \rightarrow 1^{+}$.

This reference contains the best result of this kind currently known for $\mu(n)$:

Tadej Kotnik and Herman te Riele The Mertens Conjecture revisited. Algorithmic number theory, 156--167, Lecture Notes in Comput. Sci., 4076, Springer, Berlin, 2006.

They prove that $\limsup_{x \rightarrow +\infty}M(x)/\sqrt{x} \geq 1.218$ and that $\liminf_{x \rightarrow +\infty}M(x)/\sqrt{x} \leq -1.229$. Here

$ M(x) = \sum_{n \leq x}\mu(n) $

is the conventional notation for the summatory function of the Möbius function. Their proof is a mixture of analytic number theory and large scale computations. They also have a survey of what is known and what is conjectured about the size of $M(x)$.

Now on to the Liouville function $\lambda(n)$ and its summatory function $L(x)$. The latter is very closely connected with $M(x)$, for

$ \sum_{n \leq \sqrt{x}}\mu(n)L\left(\frac{x}{n^2}\right) = \sum_{n \leq \sqrt{x}}\mu(n)\sum_{m \leq x/n^2}\lambda(m) = \sum_{N \leq x}\sum_{mn^2 = N}\mu(n)\lambda(m) = \sum_{N \leq x}\mu(N) = M(x). $

Thus

$ |M(x)| \leq \sum_{n \leq \sqrt{x}}|L\left(\frac{x}{n^2}\right)| $

and so the assumption

$ L(x) = o\left(\frac{\sqrt{x}}{\log^{1 + \epsilon}(x)}\right) $

for example, leads to a contradiction with the Kotnik-te Riele result (or earlier results) for any $\epsilon > 0$.

My guess is that if one looks up the old (pre-computer) results on $|M(x)|$ from below, one might prove that $\limsup_{x \rightarrow +\infty}|L(x)|/\sqrt{x} > 0$. This may even be in the literature somewhere.

Alternatively

$ \sum_{n = 1}^{\infty}\lambda(n)n^{-s} = \prod_{p}\sum_{k=0}^{\infty}\lambda(p^k)p^{-ks} = \prod_{p}\sum_{k=0}^{\infty}(-1)^kp^{-ks} = \prod_{p}(1 + p^{-s})^{-1} = \frac{\zeta(2s)}{\zeta(s)} $

by the Euler product formula, and from here on it is more elementary than it was with $M(x)$ in the argument that David Speyer gave, because we don't need the zeros on the critical line. For $\zeta(2s)$ has a pole at $s = 1/2$ which is not canceled by a pole of $\zeta(s)$ at the same point. Thus $L(x) = O(x^{\alpha})$ is impossible with $\alpha < 1/2$ by partial summation.

For multiplicative functions of modulus $1$, the situation is much less clear. For simplicity, assume that $f(n)$ is a totally multiplicative function (multiplicative and $f(p^k) = f(p)^k$) with $|c_p| = 1$ where $c_p = f(p)$. The Liouville function is the case $c_p \equiv -1$. Then

$ \sum_{n = 1}^{\infty}f(n)n^{-s} = \prod_{p}\frac{1}{1 - c_pp^{-s}}{\quad},{\quad}A(x) = \sum_{n \leq x}f(n) $

by the Euler product formula. The basic principle is that if $A(x) = O(x^{\alpha})$, then the Dirichlet series on the left hand side is convergent in the half plane $\sigma > \alpha$, so the sum is holomorphic in that half plane. If we can find a singularity $s_0$ of the product on the right hand side with $\mathrm{Re}(s_0) = \sigma_0$, that tells us that $A(x) = O(x^{\alpha})$ with $\alpha < \sigma_0$ is impossible. The bad thing now is that the product may diverge at a point without having a singularity there, because the product may diverge to zero, and a holomorphic function may be zero at a point without being singular there.

But it is straightforward to show that if $\mathrm{Re}(c_p) \geq \delta > 0$ for all $p$, then $A(x) = O(x^{\alpha})$ is impossible for any $\alpha < 1$, by showing that the series $ \sum_{p}c_pp^{-\sigma} $ goes to infinity as $\sigma \rightarrow 1^{+}$.

The following reference contains the best result of this kind currently known for $\mu(n)$:

Tadej Kotnik and Herman te Riele The Mertens Conjecture revisited. Algorithmic number theory, 156--167, Lecture Notes in Comput. Sci., 4076, Springer, Berlin, 2006.

They prove that $\limsup_{x \rightarrow +\infty}M(x)/\sqrt{x} \geq 1.218$ and that $\liminf_{x \rightarrow +\infty}M(x)/\sqrt{x} \leq -1.229$. Here

$ M(x) = \sum_{n \leq x}\mu(n) $

is the conventional notation for the summatory function of the Möbius function. Their proof is a mixture of analytic number theory and large scale computations. They also have a survey of what is known and what is conjectured about the size of $M(x)$.

Now on to the Liouville function $\lambda(n)$ and its summatory function $L(x)$. The latter is very closely connected with $M(x)$, for

$ \sum_{n \leq \sqrt{x}}\mu(n)L\left(\frac{x}{n^2}\right) = \sum_{n \leq \sqrt{x}}\mu(n)\sum_{m \leq x/n^2}\lambda(m) = \sum_{N \leq x}\sum_{mn^2 = N}\mu(n)\lambda(m) = \sum_{N \leq x}\mu(N) = M(x). $

Thus

$ |M(x)| \leq \sum_{n \leq \sqrt{x}}|L\left(\frac{x}{n^2}\right)| $

and so the assumption

$ L(x) = o\left(\frac{\sqrt{x}}{\log^{1 + \epsilon}(x)}\right) $

for example, leads to a contradiction with the Kotnik-te Riele result (or earlier results) for any $\epsilon > 0$.

My guess is that if one looks up the old (pre-computer) results on $|M(x)|$ from below, one might prove that $\limsup_{x \rightarrow +\infty}|L(x)|/\sqrt{x} > 0$. This may even be in the literature somewhere.

Alternatively

$ \sum_{n = 1}^{\infty}\lambda(n)n^{-s} = \prod_{p}\sum_{k=0}^{\infty}\lambda(p^k)p^{-ks} = \prod_{p}\sum_{k=0}^{\infty}(-1)^kp^{-ks} = \prod_{p}(1 + p^{-s})^{-1} = \frac{\zeta(2s)}{\zeta(s)} $

by the Euler product formula, and from here on it is more elementary than it was with $M(x)$ in the standard argument that David Speyer gave, because we don't need the zeros on the critical line. For $\zeta(2s)$ has a pole at $s = 1/2$ which is not canceled by a pole of $\zeta(s)$ at the same point. Thus $L(x) = O(x^{\alpha})$ is impossible with $\alpha < 1/2$ by partial summation.

This reference contains the best result of this kind currently known for $\mu(n)$:

Tadej Kotnik and Herman te Riele The Mertens Conjecture revisited. Algorithmic number theory, 156--167, Lecture Notes in Comput. Sci., 4076, Springer, Berlin, 2006.

They prove that $\limsup_{x \rightarrow +\infty}M(x)/\sqrt{x} \geq 1.218$ and that $\liminf_{x \rightarrow +\infty}M(x)/\sqrt{x} \leq -1.229$. Here

$ M(x) = \sum_{n \leq x}\mu(n) $

is the conventional notation for the summatory function of the Möbius function. Their proof is a mixture of analytic number theory and large scale computations. They also have a survey of what is known and what is conjectured about the size of $M(x)$.

Now on to the Liouville function $\lambda(n)$ and its summatory function $L(x)$. The latter is very closely connected with $M(x)$, for

$ \sum_{n \leq \sqrt{x}}\mu(n)L\left(\frac{x}{n^2}\right) = \sum_{n \leq \sqrt{x}}\mu(n)\sum_{m \leq x/n^2}\lambda(m) = \sum_{N \leq x}\sum_{mn^2 = N}\mu(n)\lambda(m) = \sum_{N \leq x}\mu(N) = M(x). $

Thus

$ |M(x)| \leq \sum_{n \leq \sqrt{x}}|L\left(\frac{x}{n^2}\right)| $

and so the assumption

$ L(x) = o\left(\frac{\sqrt{x}}{\log^{1 + \epsilon}(x)}\right) $

for example, leads to a contradiction with the Kotnik-te Riele result (or earlier results) for any $\epsilon > 0$.

My guess is that if one looks up the old (pre-computer) results on $|M(x)|$ from below, one might prove that $\limsup_{x \rightarrow +\infty}|L(x)|/\sqrt{x} > 0$. This may even be in the literature somewhere.

Alternatively

$ \sum_{n = 1}^{\infty}\lambda(n)n^{-s} = \prod_{p}\sum_{k=0}^{\infty}\lambda(p^k)p^{-ks} = \prod_{p}\sum_{k=0}^{\infty}(-1)^kp^{-ks} = \prod_{p}(1 + p^{-s})^{-1} = \frac{\zeta(2s)}{\zeta(s)} $

by the Euler product formula, and from here on it is more elementary than it was with $M(x)$ in the argument that David Speyer gave, because we don't need the zeros on the critical line. For $\zeta(2s)$ has a pole at $s = 1/2$ which is not canceled by a pole of $\zeta(s)$ at the same point. Thus $L(x) = O(x^{\alpha})$ is impossible with $\alpha < 1/2$ by partial summation.

For multiplicative functions of modulus $1$, the situation is much less clear. For simplicity, assume that $f(n)$ is a totally multiplicative function (multiplicative and $f(p^k) = f(p)^k$) with $|c_p| = 1$ where $c_p = f(p)$. The Liouville function is the case $c_p \equiv -1$. Then

$ \sum_{n = 1}^{\infty}f(n)n^{-s} = \prod_{p}\frac{1}{1 - c_pp^{-s}} $

by the Euler product formula. The basic principle is that if $A(x) = O(x^{\alpha})$, then the Dirichlet series on the left hand side is convergent in the half plane $\sigma > \alpha$, so the sum is holomorphic in that half plane. If we can find a singularity $s_0$ of the product on the right hand side with $\mathrm{Re}(s_0) = \sigma_0$, that tells us that $A(x) = O(x^{\alpha})$ with $\alpha < \sigma_0$ is impossible. The bad thing now is that the product may diverge at a point without having a singularity there, because the product may diverge to zero, and a holomorphic function may be zero at a point without being singular there.

But it is straightforward to show that if $\mathrm{Re}(c_p) \geq \delta > 0$ for all $p$, then $A(x) = O(x^{\alpha})$ is impossible for any $\alpha < 1$, by showing that the series $ \sum_{p}c_pp^{-\sigma} $ goes to infinity as $\sigma \rightarrow 1^{+}$.