Deducing a lower bound for the number of sign changes

Question

Let $Z(t)$ denote Hardy's Z function (just treat it as any continuous real function defined on positive real numbers). It is known that for all large $T$ there exists absolute constants $A,B>0$ such that

$$ \int_T^{2T}|Z(t)|\mathrm dt\ge AT\tag1 $$

$$ \left|\int_T^{2T}Z(t)\mathrm dt\right|\le BT^{3/4}\tag2 $$

In §2.1 of Ivić's The Theory of Hardy's Z-Function, the author states without justification that (for large $T$)

This argument actually shows that there is always a zero of $Z(t)$ in $[T,T+CT^{3/4}]$ for suitable $C>0$.

Then, the author says that one can deduce from this that there exists $K>0$ such that $N_0(T)>KT^{1/4}$ for $T$ large enough, where $N_0(T)$ denotes the number of zeros of $Z(t)$ on $[0,T]$.

However, I am only able to deduce an inferior estimate: Suppose $Z(t)$ does not change sign in $[T,2T]$ for all large $T$. Then

$$ AT\le\int_T^{2T}|Z(t)|\mathrm dt\le BT^{3/4} $$

which leads to a contradiction. This means there exists a constant $M>0$ such that $N_0(T)\ge M\log T$ for all sufficiently large $T$.

I wonder how the author is able to reduce the range of potential zero into $[T,T+CT^{3/4}]$ and obtain such a strong lower bound.

Answer 1

Let $1/4<a<1$ say.

Now $\int_T^{T+CT^a}|Z(t)|dt \ge |\int_T^{T+CT^a}\zeta(1/2+it)|dt=|(\int_{1/2+iT}^{2+iT}+\int_{2+iT}^{2+i(T+CT^a)}+\int_{2+i(T+CT^a)}^{1/2+i(T+CT^a)})\zeta(s)ds|$

We have that $|\int_{1/2+iT}^{2+iT}\zeta(s)ds|+|\int_{2+i(T+CT^a)}^{1/2+i(T+CT^a)}\zeta(s)ds| = O(T^{1/4})$

(from the convexity estimate of $\zeta$ - and we can do even better using sharper available estimates )

and $|\int_{2+iT}^{2+i(T+CT^a)}\zeta(s)ds|=CT^a+O(1)$ from the Dirichlet series expansion, giving:

$\int_T^{T+CT^a}|Z(t)|dt \ge CT^a+ O(T^{1/4})$

On the other hand,the proof in Ivic shows that $|\int_T^{T+CT^a}Z(t)dt|= O(T^{3/4})$ (where the $O$ doesn't depend on $C,a$ as long as say $CT^a<T$), so it follows that for $a \ge 3/4$ and for $C$ large enough (hence for $T$ large enough so $CT^a<T$ say) we have that $\int_T^{T+CT^a}|Z(t)|dt> |\int_T^{T+CT^a}Z(t)dt|$ implying the existence of a zero there

Answer 2

To generalize Conrad's argument, one is motivated to define

$$ I(T,\Delta)=\left|\int_T^{T+\Delta}Z(t)\mathrm dt\right| $$

and

$$ J(T,\Delta)=\int_T^{T+\Delta}|Z(t)|\mathrm dt=\int_T^{T+\Delta}\left|\zeta\left(\frac12+it\right)\right|\mathrm dt $$

Undeniably, it is possible to attack them via approximate functional equation, but I personally found the Phragmén-Lindelöf corollary a simpler result to use (see chapter 4 of Titchmarsh's The theory of the Riemann zeta-function):

$$ \zeta(\sigma+it)\ll_\varepsilon|t|^{(1-\sigma)/2+\varepsilon} $$

Applying this and a few other approximation lemmas in chapter 4 of the same book, one obtains that when $\Delta\le T$,

$$ I(T,\Delta)\ll_\delta T^{3/4+\delta}\tag1 $$

for all $\delta>0$. In addition, we have

$$ J(T,\Delta)\gg\Delta\tag2 $$

whenever $\Delta\gg_\varepsilon T^{1/4+\varepsilon}$. This indicates that if we were plug $\Delta=T^{3/4+\eta}$ into (2) and set $\delta=\varepsilon/2$ in (1), then we can obtain the following density estimate:

For every $\eta>0$, there exists $T_0=T_0(\eta)>0$ such that there is always a zero of $Z(t)$ lying within $[T,T+T^{3/4+\eta}]$.

Though weaker than the original question's density estimate, this result still allows us to give a rough lower bound for $N_0(T)$:

$$ N_0(T)\gg_\eta T^{1/4-\eta}\tag3 $$

In §10.5 of Titchmarsh's book, the author studied the $\Delta=T$ case and showed that there is always a zero in $[T,2T]$, and it is not difficult to port these arguments to deduce (3). I have written a detailed account on this in Chinese available on Zhihu.