Explicit zero density estimate near unity

Let $\zeta(s)$ be the Riemann zeta function and $\rho=\beta+i\gamma$ a non-trivial zero of $\zeta$, with $0<\operatorname{Re}(\rho)<1$. Given $1/2<\sigma<1$, $T>0$, we define the quantity

which counts the number of non-trivial zeros of $\zeta$ with real part greater than the fixed value $\sigma$. Finding upper bounds for $N(\sigma,T)$, commonly known as zero-density estimates, is a problem that has always attracted much interest in Number Theory. If the Riemann Hypothesis is true, then we have $N(\sigma,T)=0$ for every $1/2<\sigma<1$, since all the non-trivial zeros of $\zeta$ would lie on the half-line $\sigma=1/2$. In 2021, Platt and Trudgian [PT21] verified that the Riemann Hypothesis is true up to height $3\cdot 10^{12}$ and, consequently, $N(\sigma,T)=0$ for every $1/2<\sigma<1$, $T\leq 3\cdot 10^{12}$. This allow us to restrict the range of values of $T$ for which we aim to estimate $N(\sigma,T)$ to $T>3\cdot 10^{12}$.
Different methods have been used by several mathematicians throughout the years, depending on the range for $\sigma$ inside the right half of the critical strip in which we are working. In 1937 Ingham [Ing37] proved that, assuming that $\zeta\left(\frac{1}{2}+it\right)\ll t^{c+\epsilon}$, one has $N(\sigma,T)\ll T^{(2+4c)(1-\sigma)}(\log T)^{5}$. In particular, the Lindelöf Hypothesis $\zeta\left(\frac{1}{2}+it\right)\ll t^{\epsilon}$ implies $N(\sigma,T)\ll T^{2(1-\sigma)+\epsilon}$, known as the Density Hypothesis. Ingham’s type estimate has been made explicit by Ramaré [Ram16] and improved in 2018 by Kadiri, Lumley and Ng [KLN18]. Ingham’s method is particular powerful when $\sigma$ is close to the half-line, as the upper bound for $\zeta(\frac{1}{2}+it)$ is involved.
When $\sigma$ is really close to $1$ and $T$ sufficiently large, estimates of the type

become sharper. Indeed, for $\sigma$ sufficiently close to $1$, and hence $T$ sufficiently large, Richert’s bound [Ric67] $|\zeta(\sigma+it)|\leq A|t|^{B(1-\sigma)^{3/2}}(\log|t|)^{2/3}$ is the sharpest known upper bound for $\zeta$. However, contrary to Ingham’s type zero-density estimate, the estimate (1.1) has never been made explicit. The best non-explicit zero-density estimate of this form is due to Heath-Brown [HB17] (actually, this result has been recently slightly improved by Trudgian and Yang [TY23], as they found new optimal exponent pairs; see also [Pin23] for more recent results), in which he used a slightly different bound for $\zeta$, i.e. $\zeta(\sigma+it)\ll t^{B(1-\sigma)^{3/2}+\epsilon}$, where the exponent $B$ turns out to be much smaller ($B<\frac{1}{2}$) than the currently best-known value for $B$ in Richert’s bound, that is $B=4.43795$ [Bel23]. However, differently from Richert’s bound that is fully explicit, the bound used by Heath-Brown cannot be used to get an explicit version for the zero-density estimate of the form (1.1), since the factor $t^{\epsilon}$ cannot be made explicit (see [Ivi03, Mon71] for previous non-explicit results).

The aim of this paper is to provide the first explicit zero-density estimate for $\zeta$ of the form (1.1). Combining Ivić’s zero-detection method [Ivi03] with Richert’s bound and an explicit zero-free region for $\zeta$, we will prove the following result.

Actually, it is possible to deduce a much simpler upper bound for $N(\sigma,T)$ compared to Theorem 1.1, at the expense of a slightly worse exponent for the factor $(\log T)$.

We observe that the exponent $10393/900=11.547\dots$ for the factor $\log T$ is less than $11.548$, which improves the best-known exponent equal to $15$ due to Ivic [Ivi03]. Furthermore, the value $10393/900$ is the near-optimal one that one can get using Ivić’s zero-detection method. Indeed, a factor $(\log T)^{3}$ comes from the relation for the divisor function in Theorem 2.4, where the powers of the logarithm are already the optimal ones. Then, Ivić’s zero-detection method produces $5$ more powers of $\log T$, as we will see later, that cannot be reduced, when using this method. Finally, another factor $\log T$ comes from the number of zeros in Theorem 2.2. Hence, all these contributions give already $9$ powers that cannot be removed. The remaining ones come from convergence arguments throughout the proof, where we already tried to optimize our choices as much as possible.

We recall some results that will be useful for the proof of Theorem 1.2.
First of all we give an overview of the currently best-known explicit zero-free regions for the Riemann zeta function. As we already mention, Platt and Trudgian [PT21] verified that the Riemann Hypothesis is true up to height $3\cdot 10^{12}$. Hence, in the proof of Theorem 1.2 we will work with $T\geq 3\cdot 10^{12}$. Now, we list the best-known zero-free regions for $\zeta$. All of them hold for every $|T|\geq 3$, but we indicate the range of values for $|T|$ for which each of them is the widest one.

• •

  For $3\cdot 10^{12}<|T|\leq e^{46.2}$ the largest zero-free region[MTY22] is the classical one:

  $$\sigma\geq 1-\frac{1}{5.558691\log|T|}$$

  (2.1)

• •

  For $e^{46.2}<|T|\leq e^{170.2}$ the currently sharpest known zero-free region is

  $$\sigma>1-\frac{0.04962-0.0196/(J(|T|)+1.15)}{J(|T|)+0.685+0.155\log\log|T|},$$

  (2.2)

  where $J(T):=\frac{1}{6}\log T+\log\log T+\log 0.618$. As per [Yan23], this result is obtained by substituting Theorem 1.1 of [HPY22] in Theorem 3 of [For22] and observing that $J(T)<\frac{1}{4}\log T+1.8521$ for $T\geq 3$.

• •

  For $e^{170.2}<|T|\leq e^{481958}$, Littlewood zero-free region [Yan23] becomes the largest one¹¹1The constant $21.432$ in [Yan23] has been updated to $21.233$ (communicated by the author, who has sent a proof of the revised Littlewood zero-free region).:

  $$\sigma\geq 1-\frac{\log\log|T|}{21.233\log|T|}.$$

  (2.3)

• •

  Finally, for $|T|>e^{481958}$, Korobov–Vinogradov zero-free region [Bel23] is the widest one²²2The value $54.004$ in [Bel23] can be improved to $53.989$ due to the improved estimate (2.3).:

  $$\sigma\geq 1-\frac{1}{53.989\log^{2/3}|T|(\log\log|T|)^{1/3}}.$$

  (2.4)

The method used to detect the Korobov–Vinogradov zero-free region involves Richert’s bound $|\zeta(\sigma+it)|\leq A|t|^{B(1-\sigma)^{3/2}}(\log|t|)^{2/3}$, which will also be an essential tool in our proof of Theorem 1.2, as we already mentioned. Below, we state the best-known estimate of this type (see [For02] for previous results).

Related to the zeros of $\zeta$, we state the best-known explicit estimate for the quantity $N(T)$, which is the number of zeros $\rho=\beta+i\gamma$ of $\zeta(s)$ with $0\leq\gamma\leq T$.

A powerful tool that is widely used in Ivić’s zero-detection method is the Halász–Montgomery inequality we recall below.

Then, the following result on the divisor function will be useful.

In particular, for our purpose, when $x=10^{85}$ we have

Finally, we recall an explicit estimate for the $\Gamma$ function ([Olv74], p.294).

We will follow Ivić’s zero-detection method ([Ivi03], $\S 11$) using near-optimal choices. As in [Ivi03], we start considering the following relation derived by a well-known Mellin Transform:

Then, we consider the function

where $X=X(T)$ and $Y=Y(T)$ are parameters we will choose later. Furthermore, by our later choices, we will have

By the elementary relation

we see that each zero $\rho=\beta+i\gamma$ of $\zeta(s)$ counted by $N(\sigma,T)$ satisfies

with

As per [Ivi03], for a fixed zero $\rho=\beta+i\gamma$, we move the line of integration to $\operatorname{Re}w=\alpha-\beta<0$ for some suitable $\frac{1}{2}\leq\alpha\leq 1$ . As per Ivić [Ivi03], the optimal choice for $\alpha$ is $\alpha=5\sigma-4$. Hence, since in the hypothesis of Theorem 1.2 we assumed $\sigma\in[0.98,1)$, it follows that $\alpha\geq 0.9$. If $|\gamma|>2\log T$, the poles we find are at $w=0$ and $w=1-\rho$. Hence, using the residue theorem, the relation (3.2) becomes

Furthermore, we split both the integral and the sum in (3.3) into the following terms:

and

Finally, we define the following quantities:

Using the above notation, the relation (3.3) can be rewritten as

Also,

As we will see in Lemma 3.1, the quantity $D\rightarrow 0$ as $Y\rightarrow\infty$, hence for $Y\rightarrow+\infty$ one has $B-A\rightarrow 1$. It follows that at least one between the quantities $A$ and $B$ must be $\gg 1$. More precisely, given $0<c<1$, we have $|B|\geq c$ or, if $|B|<c$, then $|A|\geq 1-\epsilon-c$, where

Putting $c=1-\epsilon-c$, the optimal bound for both $|A|$ and $|B|$ is

where we recall that $Y\geq X\geq 10^{85}$.
It follows that each $\rho_{r}=\beta_{r}+i\gamma_{r}$, $\beta_{r}\geq\sigma$ counted by $N(\sigma,T)$ satisfies at least one of the following conditions:

or

We recall that the coefficients $a(n)$ in (3.6) satisfy $|a(n)|\leq|d(n)|$.
The number of zeros $\rho$ satisfying (3.8) is

where we used Theorem 2.2 and the fact that $T>3\cdot 10^{12}$.
Then, we denote with $R_{1}$ the number of zeros satisfying (3.6) and with $R_{2}$ the number of zeros satisfying (3.7) so that the imaginary parts of these zeros differ from each other by at least $2\log^{1.4}T$. We have

Before proceeding with the estimate for $R_{1}$ and $R_{2}$, we prove that $D\rightarrow 0$ as $Y\rightarrow\infty$, as we mentioned before, where $D$ is defined in (3.5).