Explicit zero-free regions for the Riemann zeta-function

We prove that the Riemann zeta-function $\zeta(\sigma + it)$ has no zeros in the region $\sigma \geq 1 - 1/(55.241(\log|t|)^{2/3} (\log\log |t|)^{1/3})$ for $|t|\geq 3$. In addition, we improve the constant in the classical zero-free region, showing that the zeta-function has no zeros in the region $\sigma \geq 1 - 1/(5.558691\log|t|)$ for $|t|\geq 2$. We also provide new bounds that are useful for intermediate values of $|t|$. Combined, our results improve the largest known zero-free region within the critical strip for $3\cdot10^{12} \leq |t|\leq \exp(64.1)$ and $|t| \geq \exp(1000)$.


Introduction
Let ζ(s) denote the Riemann zeta-function, where s = σ + it is a complex variable.All non-trivial zeros of ζ(s) lie in the critical strip with 0 < σ < 1. Determining regions in the critical strip that are devoid of zeros of ζ(s) is of great interest in number theory.Such regions take the shape σ ≥ 1 − 1/f (|t|) for some function f (t) tending to infinity with t.The so-called classical zero-free region has f (t) = R 0 log t, where R 0 is a positive constant.An asymptotically larger region, proved by Korobov [17] and Vinogradov [30], has f (t) = R 1 (log t) 2/3 (log log t) 1/3 , for some constant R 1 > 0.
Considerable effort has been made to make these results explicit, and we briefly recall the sharpest zero-free regions known.For the Korobov-Vinogradov region, in 2000 Ford [9] (see also [10] for minor corrections) established two explicit bounds, one holding for essentially all t, and a larger one holding for sufficiently large t.In the former case, Ford proved that there are no zeros of ζ(s) in the region σ ≥ 1 − 1 57.54(log|t|) 2/3 (log log|t|) 1/3 , |t| ≥ 3. (1.1) In the latter case, Ford showed that there are no zeros when t is sufficiently large and σ ≥ 1 − 1 49.13(log|t|) 2/3 (log log|t|) 1/3 . (1.2) This was recently slightly improved by Nielsen [19], who showed that there are no zeros with σ ≥ 1 − 1 49.08(log|t|) 2/3 (log log|t|) 1/3  (1.3) for sufficiently large |t|.For the classical region, in 2015 the first two authors [18] proved that the region is devoid of zeros of the zeta-function.This region is wider than that of (1.1) for all |t| ≤ exp(10151.5).
We improve (1.1), (1.3) and (1.4) in this article.For the first of these, we establish the following theorem.(1.5) Our improvement in Theorem 1.1 over (1.1) is a result of several ingredients.
• We employ a trigonometric polynomial of large degree: this produces our largest improvement.• We optimize the choice of certain parameters to account for secondary error terms.• We use improved intermediate zero-free regions to cover medium-sized t values where the argument for the asymptotic region does not perform as well.
• We employ sharper explicit estimates on the growth of the zeta-function in the critical strip.• We use a new height up to which the Riemann hypothesis (RH) has been proved.It appears substantially more difficult to improve the constant in the asymptotic region (1.3).We identify three potential avenues of improvement: using a better smoothing function, finding a more favorable trigonometric polynomial, and improving the constant B in Richert's bound-see (3.1).In [12] and [32], improvements related to the first approach have been largely explored.In this work, we exploit the second method to produce the following improvement.Theorem 1.2.For sufficiently large |t|, there are no zeros of ζ(σ + it) with σ ≥ 1 − 1 48.1588(log|t|) 2/3 (log log|t|) 1/3 . (1.6) We remark that Nielsen's result (1.3) improved (1.2) by replacing a particular trigonometric polynomial that was used in Ford's argument.Ford used a polynomial of degree 4, while Nielsen adopted one of degree 5 (see Section 3.3).We use polynomials of substantially larger degree here for our improvements.Also, we mention that Khale [16] recently established analogues of Theorems 1.1 and 1.2 for Dirichlet L-functions.While some of the ingredients we employ in this article do not appear to be suitable for use with Dirichlet L-functions, we note that the use of higher-degree trigonometric polynomials may lead to some improvements in [16].
For the classical region, in [18] the inequality (1.4) was established by extending some work of Kadiri [15] by constructing a more favorable trigonometric polynomial, by optimizing some analytic arguments, and by employing the verification of RH up to 3.06 • 10 10 from [22].It was also recorded in [18] that if RH were verified for |t| ≤ 3 • 10 11 , then the constant in (1.7) could be replaced by 5.5666305, and this is now permissible due to the verification performed in [24].We take the opportunity here to record a further improvement for the classical region, using two ideas.First, RH was verified for |t| ≤ 3 • 10 12 in [24].Second, in 2014, Jang and Kwon [13] derived another improvement to Kadiri's result by different means, involving the replacement of a particular smoothing function employed in [15], and we incorporate this as well.We prove the following theorem.We remark that the region (1.7) is wider than that of (1.5) for |t| < exp(8928).
In order to obtain a good constant in Theorem 1.1, it is necessary to deduce another zero-free region to cover medium-sized values of t, in addition to (1.7).We obtain such a zero-free region by using bounds on the growth rate of the zetafunction on the half-line and establish the following theorem.where J(t) = 1 6 log t + log log t + log 3.By combining these results, we may summarize the largest known zero-free region for the Riemann zeta-function within the critical strip for each height t.For |t| ≤ 3 • 10 12 , all zeros are known to lie on the critical line.The region of Theorem 1.3 provides the best known bound for 3 • 10 12 < |t| ≤ exp(64.1),then for exp(64.1)< |t| ≤ exp(1000) the expression (1.8) produces the widest region.After this, for exp(1000) < |t| ≤ exp(52238), Theorem 1.4 is best, and for |t| > exp(52238) the result of Theorem 1.1 produces the widest known region.
This article is organized in the following manner.Section 2 lists some immediate applications of our results.Section 3 reviews some results we require from the literature.Section 4 contains a number of lemmas that we require for the proofs of our main theorems.Section 5 contains the proof of Theorem 1.1.Section 6 describes the proof of Theorem 1.4, and Section 7 has the proof of Theorem 1.2.Section 8 summarizes our computations for determining the constants in Theorems 1.1 and 1.2.Section 9 describes the improvement in the classical region for Theorem 1.3, and Section 10 suggests some potential future work.
Another application is explicit bounds on the error term of the prime number theorem-see Johnston and Yang [14], who developed work of Platt and Trudgian [25] particularly with reference to the Korobov-Vinogradov zero-free region.Other related recent works such as [2,3,5,7] required explicit zero-free regions to obtain precise estimates.
As per Ford [8, p. 566], one can use Theorem 1.2 to improve on the error term in the prime number theorem.From the work of Pintz [21], a zero-free region like that in (1.6) with c in place of 48.1588 shows that π(x) − li(x) x exp{−d(log x) 3/5 (log log x) −1/5 }, where . With c = 48.1588from Theorem 1.2, we obtain d ≥ 0.2123, improving on the current best value of 0.2098.

Preliminaries
To assist in our argument we review the following results from the literature.
3.1.Bounds on ζ(s) near the 1-line.The underlying philosophy in furnishing zero-free regions is to estimate ζ(s) in a region close to the line σ = 1.Richert's theorem [27] (see also [29, p. 135]), accomplishes this by proving that there exist positive constants A and B for which Ford [8] obtains a relatively small value of B while maintaining a completely explicit value of A: in (3.1) one can take A = 76.2 and B = 4.45.We remark in passing that the advances made in [26] may be applied to Ford's paper [8], and should lead to a slight reduction in the value of B in (3.1).We also require bounds on ζ(s) slightly to the right of σ = 1.A result recorded in Bastien and Rogalski [1], originally due to O. Ramaré, states that for σ > 1, we have where γ is Euler's constant.Using the Maclaurin series for e x one sees that the approximation in (3.2) is very good.

3.2.
Bounds on ζ(s) on the half-line.We recall the recent sub-Weyl bound due to Patel [20]: for |t| ≥ 3, we have This is required in the proof of Theorem 1.4 in Section 6.

Trigonometric polynomials.
For any K ≥ 2, let where b k are constants such that b k ≥ 0, b 1 > b 0 and P K (x) ≥ 0 for all real x.We refer to P K (x) as a Kth degree non-negative trigonometric polynomial.Choosing a favorable P K plays an integral part in determining the size of the zero-free region: see [18] for a detailed history of this problem.
In [9], Ford employed the degree 4 polynomial while Nielsen [19] adopted the degree 5 function We employ simulated annealing in a large-scale search to determine favorable trigonometric polynomials of higher degree.We utilize two polynomials in this article.Theorem 1.1 relies on a polynomial P 40 (x) with degree 40 having while Theorem 1.2 employs a polynomial P 46 (x) with degree 46 where We show the full polynomials in Tables 1 and 2 in Section 8, and provide details there on the process used to find these polynomials as well as a justification that they are non-negative.for all T ≥ e.The bound in (3.8) can be improved in the region where RH is known to hold [24], and for some intermediate values of T as well using [23].Nevertheless, the bound in (3.8) is the best available at present for T ≥ 10 410 .While that bound could be reduced using (3.3), it suffices for our purposes.

Required lemmas
Before proceeding to the proof of Theorem 1.1 in the next section, we review some useful lemmas.Throughout, let where K is the degree of the trigonometric polynomial under consideration.Let b 0 , b 1 and b be as in (3.6) or (3.7).In addition, assume throughout that (3.1) holds for some A, B > 0. Unless otherwise stated, our results will remain valid for all A, B > 0 for which (3.1) holds.Since ζ(s) = ζ(s), it suffices to consider only t > 0 throughout.
As in Ford [9], the main tool used to pass from upper bounds on ζ(s) to a zero-free region is the "zero-detector", which expresses − ζ ζ (s) as an integral involving ζ(s) over two vertical lines on either side of σ = 1 (plus a small error term).Concretely, we have the following lemma.Lemma 4.1 (Zero-detector for ζ).Let s = 1 + it with t = 0 and ρ run through the non-trivial zeros of ζ(s).For all η > 0, except for a set of Lebesgue measure 0, we have where in the sum, each zero is counted with multiplicity.
Proof.Follows from taking ζ(s) and As is common practice, instead of working directly with Lemma 4.1, we consider a "mollified" version, presented in Lemma 4.2.The choice of the smoothing function f significantly influences the eventual zero-free region constant.As in Ford [9], we base our choice of f on Lemma 7.5 of Heath-Brown [12].Jang and Kwon [13] obtained improvements for the classical zero-free region by choosing a different mollifier described in Xylouris [32], however we find that such a choice of f did not produce significant improvements here for the Korobov-Vinogradov zero-free region.
We construct the smoothing function the same way as Ford [9], which we briefly review here for completeness.Given a qualifying trigonometric polynomial P (x) = K k=0 b k cos(kx), let θ = θ(b 0 , b 1 ) be the unique solution to the equation For the choice of b 0 , b 1 in (3.6), we compute and for (3.7) we find θ = 1.13269369969232 . . . .Define where, in particular, We then choose the smoothing function to be where λ is a positive parameter to be fixed later.Note that f (u) ≥ 0 since g(u) ≥ 0. We will primarily require properties about the Laplace transform of this smoothing function.Let denote the Laplace transform of f (u), and similarly let W (z) be the Laplace transform of w(u).The function W (z) has a closed formula, given in Ford [9], which we state here for convenience: where In particular, via a direct substitution, we have This will be useful later in the proof of Lemma 6.1.Meanwhile, for R ≥ 3, and using (4.5), we have where This allows us to bound F 0 (z) := F (z) − f (0)/z, via the identity which is a consequence of (4.4).We obtain, as per Ford, where The motivation for bounding |F 0 (z)| is shown in the next lemma.
Lemma 4.2.Let f be a non-negative, compactly supported real function with a continuous derivative and an absolutely convergent Laplace transform F (z) for z > 0, where and write F 0 (z) := F (z) − f (0) z .Suppose further that for 0 < η ≤ 3/2 we have for some absolute constant D.
We remark that some negligible improvements are possible if one takes t ≥ t 0 for some t 0 > 1000.These do not affect our final results, given the number of decimal places to which they are stated.
We seek to determine an upper bound on linear combinations of the right side of (4.8).We briefly outline our approach, which follows Ford [9], while incorporating some improvements from Section 3. The first integral, is taken on a vertical line inside the critical strip, and it can be bounded using Lemma 4.3 below and (3.1).This term is by far the most significant, and highlights the sensitivity of the resulting zero-free region to the constants A and B appearing in (3.1).The contour of the second integral lies outside the critical strip, and for this we employ Ford's trick of combining log|ζ(•)| terms, combined with (3.2).This term is the subject of Lemma 4.4.Next, the sum is bounded with the aid of N (t, v), the number of zeros ρ with |1 + it − ρ| ≤ v.This is discussed in Lemmas 4.5 and 4.6.In Lemma 4.7, we combine all these results with the trigonometric polynomial (3.6) to establish an inequality involving the real and imaginary parts of a zero.Lemma 4.3 (Ford [9]).Suppose that, for fixed 1 2 ≤ σ < 1 and t ≥ 3, we have where X, Y, Z are positive constants and Proof.See [9,Lemma 3.4].
Lemma 4.4.Let K > 1 and b j be the coefficients of a non-negative trigonometric polynomial of degree K. Furthermore let Proof.This is an immediate generalization of [9, Lemma 5.1] to degree K polynomials.
Lemma 4.5.Let 0 < η ≤ 1/4 and t ≥ 100.Then Proof.Same as in [9, Lemma 4.2]. 1 Using η in place of R in Ford's treatment, we replace the second inequality in (4.1) of [9] with and the result immediately follows from this.
Proof.We proceed in the same manner as [9, Lemma 4.3], except we use (3.8) in place of a classical inequality of Rosser [28].In addition, we propagate the new constant (4.10) in the bound for N (t, η) through to this bound, where we obtain Equipped with these lemmas, we are now prepared to form an inequality involving the real and imaginary parts of a zero ρ = β + it.Lemma 4.7.Let 0 < η ≤ 1/4 and R ≥ 3 be constants.Suppose β + it is a zero satisfying t ≥ 10000 and 1 − β ≤ η/2.Further, suppose that there are no zeros in the rectangle where λ is a constant satisfying 0 < λ ≤ min {1 − β, η/(R + 1)}.If b 0 , b 1 and b are constants associated with a degree K non-negative trigonometric polynomial, then To form an explicit zero-free region for ζ(s), we apply Lemma 4.7 with η as a fixed function of t.The rate at which η → 0 as t → ∞ determines the shape and width of the zero-free region.We choose, as in [9], for some constant E > 0. Ultimately, we take in order to attain the constant of 55.241 appearing in Theorem 1.1.This choice replaces that of appearing in Ford [9], where b 0 and b are constants from the trigonometric polynomial in (3.6).Ford's choice minimizes the asymptotic zero-free region constant, whereas we choose E to minimize the zero-free region constant holding for all t ≥ 3. The difference between our choices reflects the presence of a secondary error term that decreases as E increases, and which is significant for small values of t.

Proof of Theorem 1.1
We divide our argument into four sections depending on the size of t.Throughout, let ν(t) := 1 55.241(log t) 2/3 (log log t) 1/3 .First, for 3 ≤ t ≤ H := 3 • 10 12 , we use the rigorous verification of RH up to height H in [24].Next, for H ≤ t < exp(8928), the desired result follows immediately from Theorem 1.
Define the function and let Z(β, t). (5.2) If M ≥ M 1 , then we are done, since if β + it is a zero with t ≥ T 0 , then and B 2/3 /M 1 < 55.241.Assume for a contradiction that M < M 1 . (5.3) We will show that this implies M ≥ M 1 .Under assumption (5.3), there is a zero β + it such that t ≥ T 0 and for which Z(β, t) is arbitrarily close to M .In particular, we may take a zero satisfying There are no zeros of ζ(s) in the rectangular region This is because if a zero β + it exists in that region, then log t ≤ L 1 (t ) and log log t ≤ L 2 (t ), so so that Z(β , t ) ≥ M 1 > M , a contradiction.Note that the third inequality is verified by a numerical computation.On the other hand if t ∈ [T 0 , Kt + 1], then by (5.2) we also arrive at a contradiction.Thus any zero β + it must satisfy thereby proving claim (5.5).
Next, we choose η = EB −2/3 (L 2 /L 1 ) 2/3 , as in (4.11), with E = 1.8821259.This choice gives, for all t ≥ T 0 , where the last inequality follows from K = 40 and t ≥ T 0 .Furthermore, using Z(β, t) < M 1 we have that for all t ≥ T 0 , Finally, by substituting the values of E, T 0 and K, we have where We proceed by bounding each of the terms T 1 , T 2 and T 3 .First, using (5.1) we find where, since t ≥ T 0 , Next, using (3.2) and (5.9), we have where (5.12) Next, we have where, since t ≥ T 0 and L 1 (T 0 ) > log(KT 0 ), we may take Last, we have where, since t ≥ T 0 , Combining (5.10), (5.12), (5.13), (5.11) and (5.14), we conclude 2/3 5.392 where A short Mathematica computation is used to verify that Y (t) is decreasing for t ≥ T 0 . 2 Therefore, Y (t) ≤ Y (T 0 ) < 0.4110503.However, from the definition of λ 2 Our monotonicity argument is used to overcome a small issue in Ford's [9] treatment (after corrections in [10]).In (8.9), the following inequality was used: This should be reversed if t is sufficiently large, as both sides of the inequality are negative.
together with t ≥ T 0 , we have hence we have arrived at the desired contradiction.

Proof of Theorem 1.4
The proof of this theorem is similar to that of Theorem 1.1.Instead of bounding ζ(s) on σ = 1±η(t), we use upper bounds on ζ(s) on the lines σ = 1/2 and σ = 3/2, i.e., η is fixed at 1/2.The main disadvantage of this new scheme is that the best known bounds of ζ(1/2 + it) are of order t θ+ for some fixed θ > 0, which means that the resulting zero-free region will only have width O(1/ log t).Nevertheless, the resulting zero-free region has a better asymptotic constant than (1.4), so we use this result to cover the range exp(8928) ≤ t ≤ exp(52238).
Throughout this section, let us define J(t) := 27 164 log t + log 307.098.( This is the logarithm of the sub-Weyl bound in (3.3).First, we require the following lemma, which corresponds to Lemma 4.7 for this zero-free region.
Define Y (β, t) implicitly via the equation As t → ∞, we have Y (β, t) → −∞, and M := max t≥t0 Y (β, t) is well-defined and finite.If M ≤ 1.3686 then we are done, so assume that M > 1.3686, i.e., that there exists a zero ρ = β + it such that Y (β, t) = M > 1.3686.We pursue a contradiction argument as before.By our assumption, there is a rectangular region near ρ that is zero-free, specifically ζ(s) = 0 for all s satisfying Combining this with the definition of Y (β, t), we obtain Applying Lemma 6.1, we obtain and hence, substituting the values of b 0 and b, and using (6.12) with Y (β, t) > 1.3686, we obtain Observe that the contradiction argument in the proof of Theorem 1.1 relies on the inequality which appears in (5.15),where Y 1 (t), Y (t) → 0 as t → ∞.If we choose E as in (4.12), then (7.1) is satisfied for any so long as we take T 0 sufficiently large.Also, as in [9, §1], by appealing to zerodensity theorems, the number of zeros of ζ(s) in the rectangle 3/4 ≤ s ≤ 1, ) for some δ > 0, so for sufficiently large T 0 , most such rectangles are free of zeros.Therefore, following the argument in Section 5, there are no zeros in the region for sufficiently large T 0 .Using the polynomial P 46 (x) with values shown in (3.7), we conclude by noting that cos 2 θ 4 3 Remark.The value of 0.05617776 improves on 0.05507 appearing in [9] and 0.055127 appearing in [19].

Computations
The work of Ford [9] relied in part on the selection of a trigonometric polynomial P K (x) with certain properties.Recall that and we require that each b k ≥ 0, that b 1 > b 0 , and that P K (x) ≥ 0 for all real x.Any such polynomial gives rise to an asymptotic zero-free region of the Riemann zeta-function having the form when |t| is sufficiently large, and a value for the constant R 2 can be computed using the polynomial.Let θ be the unique solution in (0, π/2) and as in (3.6) and (3. Using (7.2) we may take In [9] and [19] respectively, the polynomials (3.4) and (3.5) with degrees K = 4 and 5 were used.Here we employ a heuristic optimization technique to determine good polynomials P K (x) with K > 4 that produce an improved constant R 2 .
As in [18], we apply the technique of simulated annealing to find favorable trigonometric polynomials P K (x) having the required properties.Suppose we have selected a degree K.We must guarantee that any candidate polynomial P K (x) have the property that P K (x) ≥ 0 for all real x.For this, instead of manipulating the coefficients b k of P K (x) directly, instead we maintain a list of coefficients c 0 = 1, c 1 , c 2 , . . ., c K .We set c k e ikx and let Thus, b 0 = 1 and for k > 0 the coefficient b k is the kth autocorrelation of the sequence of coefficients c k , suitably scaled: With this construction, we are assured that P K (x) ≥ 0 for all x.
Given K and a positive real number H, our procedure begins by setting c 0 = 1 and selecting each c k with 1 ≤ k ≤ K uniformly at random from the interval [0, H], and then it computes the associated coefficients b k .Then P K (x) ≥ 0 and each b k ≥ 0 by construction, and we need only check if b 1 > b 0 .If this does not hold we simply pick a new list of values c k and restart.We typically find a qualifying polynomial in short order, and we compute its associated value R 2 using (8.2).We then employ simulated annealing to search for better polynomials from this starting point.
In this process, we maintain a current maximum step size S and temperature T .Given these values, we perform an adjustment to our current polynomial for a certain number of iterations N .In each iteration, we select a positive integer k < K at random, and a random real value s ∈ [−S, S], then add s to c k and perform the O(K) operations required to update the b k values.If any b k < 0, or if b 1 ≤ b 0 , then we reject this adjustment and return c k to its prior value.Otherwise, we compute the value R 2 for the adjusted polynomial, using Newton's method to determine θ in (8.1).If the new value is smaller than our prior value, then we keep this adjustment and move to the next iteration.If the new value is larger than our prior value, then we keep the adjustment with probability depending on the current temperature T , otherwise we reject it and return c k to its prior value.The threshold probability is e −∆R2/T , so that the likelihood of accepting a larger value for R 2 is smaller when the change in R 2 is greater, but we are more likely to accept larger adjustments when the temperature T is higher.For a fixed value of S, our method executes N iterations for each value of a decreasing sequence of temperature values T , culminating with the effective selection T = 0, so where only improvements to R 2 are allowed.We repeat this for several values of S, which decay exponentially.
We used this procedure to search for favorable polynomials with degree from K = 10 to K = 72.In each case we typically selected H ∈ [100, 200], step values S decreasing gradually from 50 or 60 and slowly decreasing to approximately 2, using about twelve temperature values T , and selecting N near 8000.We found many polynomials with R 2 < 48.18, and our best polynomial has degree 46 and is recorded in Table 1.It produces R 2 = 48.1587921551117,which we employ in the statement of Theorem 1.2. Figure 1 For the result in Theorem 1.1, we employed two similar procedures.First, we amended the objective function to compute a value R 1 in the zero-free region for all |t| ≥ 3.However, each iteration of this computation was much slower, so our computations here were limited.Indeed, we restricted our searches to degrees K ≤ 28 in this case due to the greater computational complexities.Second, we used the objective function for the asymptotic constant, and reset our parameters to allow the step size to decrease more rapidly while greatly increasing the number of iterations N per round (taking N between 10 5 and 10 6 ), as well as the number of total number searches performed per degree selection.This allowed for a much larger search, and we tested degrees K ≤ 55 with this strategy.Each search recorded polynomials optimized for the asymptotic constant R 2 , then these polynomials were tested to determine their R 1 value for Theorem 1.1.One degree-40 polynomial found with this procedure had an asymptotic constant of R 2 ≈ 48.162, which is inferior to the polynomial displayed in Table 1, but it had the best value for R 1 .We used   this polynomial as our initial state in a further annealing procedure that optimized for R 1 to determine an additional small improvement.Our final polynomial is listed in Table 2.This polynomial P 40 (x) produced the value R 1 = 55.241used for Theorem 1.1.Figure 1(a) displays a plot of this polynomial over [π/2, π].

The classical zero-free region
In 2005, Kadiri [15] established the value R 0 = 5.69693 in the classical zerofree region of the Riemann zeta-function by means of a clever iterative procedure.This method relied in part on a particular smoothing function f (z) having certain properties: it was required that f ∈ C 2 [0, 1], with compact support, and having a Laplace transform F (z) that is non-negative on the positive real axis.Kadiri noted that Heath-Brown's work on Linnik's theorem [12] developed four families of such functions, and that these were well-adapted for application to the problem of the classical zero-free region.We denote these four families of functions by f λ,θ (ηt) for 1 ≤ i ≤ 4, where λ > 0, θ, and η are real parameters, and h (i) λ,θ (u) are certain functions.Kadiri employed the fourth of these, where and one requires π/2 < θ < π.Kadiri set λ = 1 and selected θ = 1.848 in her analysis.
In 2014, Jang and Kwon [13] applied all four families of functions f (i) η,λ,θ (u) inherited from Heath-Brown's list to this problem, and optimized over λ and θ in each case.They found that some of Heath-Brown's other functions performed slightly better than (9.1), and obtained a better value for R 0 .Most of their improvement was due to the use of a larger height T 0 for which RH had been verified: they used T 0 = 3.06 • 10 10 while Kadiri employed the best value known at the time, approximately 3.3 • 10 9 .However, investigation of other auxiliary functions allowed them to reduce their R 0 value further.They found that the four functions h (i) λ,θ (u) produced in turn 5.68372, 5.68483, 5.68484, and 5.68486 with the new T 0 value, so among these their best result arose from h (1) λ,θ , which is defined by and one requires 0 < θ < π/2.Jang and Kwon chose λ = 1.03669 and θ = 1.13537, selecting these values in concert with their choice of a non-negative trigonometric polynomial, in order to optimize the constant with this function using Kadiri's method.As in [15], Jang and Kwon selected a favorable non-negative trigonometric polynomial of degree 4. Jang and Kwon also employed a function h (5) from Xylouris [31], which out-performed h λ,θ just slightly, producing R 0 = 5.68371.This was the final value established in [13].
Independent of [13], in 2015 the first two authors [18] determined an improved value for the constant R 0 in the classical region by amending Kadiri's method in different ways, and showed that R 0 = 5.573412 is permissible.A small part of that improvement arose by employing the larger value for T 0 = 3.06 • 10 10 .Most of the gain resulted from two other changes: optimizing over a particular error term, and investigating admissible non-negative trigonometric polynomials of larger degree.A polynomial of degree 16 was constructed there by using simulated annealing with an appropriate objective function, and the constant R 0 = 5.574312 was computed using h (4) 1,θ as in [15], with an appropriate value of θ.We take the opportunity here to combine the ideas from [13] and [18], together with the recent work [25] establishing RH to the height T 0 = 3 • 10 12 , to record an improved value for the constant R 0 in the classical zero-free region of the zetafunction.We employ the admissible non-negative trigonometric polynomial of degree 16 from [18, Table 5], the auxiliary function h λ,θ (u) from (9.2), and the new value for T 0 .We set λ = 1 since adjusting this value did produce any further gains of significance, and choose θ = 1.13489.We follow the method detailed in [18], with the following adjustments owing to the use of (9.2) rather than (9.1): • Since θ is now restricted to (0, π/2), we set d 1 (θ) = 2θ cot θ.
We refer the reader to [18] for full details on the method.Using this strategy, after seven iterations we compute the value R 0 = 5.5586904517 and establish Theorem 1.3.The successive values of R 0 determined after each iteration are displayed in Table 3, together with the values of other parameters that arise during the calculation-we include these to mirror the data shown in [18, Table 3].We remark that with h 1,θ we produced a slightly larger constant with the method, R 0 = 5.5608403, so swapping the smoothing function in this way allowed us to reduce the value of R 0 by approximately an additional 2.15 • 10 −3 .This is about twice the marginal gain that reported in [13] for the same swap of auxiliary functions.We did not investigate the function of Xylouris, as this had a very small marginal benefit in [13].

Future work
The approach taken in the proof of Theorem 1.1 may be summarized as follows.If a zero-free region σ ≥ 1 − ν(t) can be established over a small finite region, say for t ∈ [T 0 − 1, T 0 ), then under appropriate conditions the same zero-free region holds for all t ≥ T 0 .This suggests an inductive argument may be used-given a sequence of suitable functions ν 1 (t), ν 2 (t), . . ., ν N (t), we may use the zero-free region σ ≥ 1 − ν j (t) to show that there are no zeros in a small finite region, which then implies the next zero-free region σ ≥ 1 − ν j+1 (t), and so on.If in addition the first zero-free region can be established unconditionally, this produces a method of iteratively constructing a zero-free region as a union of small zero-free regions.
One possible choice for ν j (t) is given by ν j (t) := 1 r j (log t) φj (log log t) 1−φj , t ≥ T (j) 0 , for some φ j > 2/3, which can be easily established by choosing in place of (4.11), then following the rest of the proof of Theorem 1.1.If r j is small enough, then the resulting zero-free region will be sharper than Theorem 1.1 over some finite interval t ∈ [T (j) 1 , T . By combining multiple such results, we create an envelope of zero-free regions whose union covers the interval [3, T 0 ] for some large T 0 .We then use the same argument as Theorem 1.1 to cover the range [T 0 , ∞).In particular, this allows us to take T 0 much larger than is otherwise possible, which reduces the size of R 1 , the zero-free region constant.
To attain a non-trivial result via this method, we need to take r j small enough that ν j (T (j) 0 ) > ν j−1 (T (j) 0 − 1), so there is a small region T 0 − 1 ≤ t ≤ T 0 , ν j (t) ≤ σ ≤ ν j−1 (t), (10.1) in which zeros may exist, hence invalidating the inductive argument.Therefore, if we have tools to exclude the possibility of zeros in small, finite regions at known locations within the critical strip, immediate improvements to Theorem 1.1 are possible.By judiciously choosing φ j , we find that using 355 such regions suffice to improve the constant of Theorem 1.1 to 52.74, provided that no zeros exist in regions of the form (10.1) for each j.Conventional arguments, such as raw computation or zero-density estimates, are currently insufficient to completely exclude zeros in these regions due to their large height, ranging from t ≈ exp(40000) to t ≈ exp(5 • 10 7 ).However, if a new method was developed to exclude zeros in small finite regions, then immediate improvements to Theorem 1.1 are possible.