2024 Explicit estimates for the Riemann zeta function close to the 1-line

We provide explicit upper bounds of the order log t/ log log t for | ζ ′ ( s ) /ζ ( s ) | and | 1 /ζ ( s ) | when σ is close to 1. These improve existing bounds for ζ ( s ) on the 1-line.

Trudgian [18] gave explicit upper bounds of the order O(log t) for both functions in (1) (see also [10,Theorem 6.7]).These estimates are especially notable because they are valid inside the critical strip (0 < Re(s) < 1), but only within known zero-free regions.Because of this, the constants had to be sufficiently large to bound the functions as ζ(s) → 0, which would happen as s approaches the boundary of the zero-free region.As such, the problem of improving these explicit constants is extremely non-trivial, and explicit estimates heavily depend on explicit zero-free regions.
The recent work of Yang [20] establishes a Littlewood-type zero-free region (see Lemma 4), which is the largest known region for exp(209) ≤ t ≤ exp(5•10 5 ).With this, Hiary, Leong and Yang [6] gave bounds on |ζ ′ (s)/ζ(s)| and |1/ζ(s)| of the order log t/ log log t on the 1-line, which are the first explicit bounds of this type.The goal of this article is to improve and extend their results inside the critical strip.

Main results
The following four corollaries are the new estimates we prove in Sections 4 and 5.They are the first explicit bounds of this order inside the critical strip, and also improve current bounds at σ = 1.On the 1-line, Corollary 2 improves [6, Theorem 2], and Corollary 4 improves [6, Theorem 2] for t ≥ 3, and [1, Proposition A.2] for t ≥ 222116.
Throughout this article, we denote the height to which the Riemann Hypothesis has been verified [12] by H := 3 000 175 332 800.
Corollary 1.For t ≥ t 0 ≥ e e and σ ≥ 1 − log log t W log t , where (W, R 1 ) = (22, 5471) is valid for t 0 = e e , and other values are given in Table 3.
Corollary 2. For t ≥ t 0 and σ ≥ 1, we have where K 1 = 113.3 is valid for t 0 = 500, and other values are given in Table 4.
Corollary 3.For t ≥ t 0 ≥ e e and σ ≥ 1 − log log t W log t , we have where (W, R 2 ) = (22, 3438) is valid for t 0 = 500, and other values are given in Table 5.
Corollary 4. For t ≥ t 0 ≥ e e and σ ≥ 1 we have where K 2 = 107.7 is valid for t 0 = 500, and other values are given in Table 6.
Furthermore, with the restriction σ = 1, the range of t can be extended.In other words, for t ≥ 3 we have ≤ 107.7 log t log log t .

Some preliminaries
This section contains the lemmas and theorems used to prove our main results.The first two lemmas, which form the basis of our method, are built upon a theorem of Hadamard, Borel and Carathéodory, sometimes also referred to as the Borel-Carathéodory theorem.This theorem enables us to deduce an upper bound for the modulus of a function and its derivatives on a circle, using bounds for its real part on a larger concentric circle.
If in addition f (s 0 ) = 0, then for any r ∈ (0, R) and any s such that |s−s 0 | ≤ r, Lemma 5] Let s 0 be a complex number, and let r and α be positive numbers (possibly depending on s 0 ) such that α < 1/2.Suppose that the function f (s) is analytic in a region containing the disc |s−s 0 | ≤ r.Suppose further that there is a number A 1 independent of s such that Then, for any s in the disc |s − s 0 | ≤ αr we have where ρ runs through the zeros of f (s) in the disc |s − s 0 | ≤ 1 2 r, counted with multiplicity.
Lemma 2. [6, Lemma 6] Let s and s 0 be complex numbers with real parts σ and σ 0 , respectively.Let r, α, β, A 1 and A 2 be positive numbers, possibly depending on s 0 , such that α < 1/2 and β < 1. Suppose that the function f (s) satisfies the conditions of Lemma 1 with r, α and A 1 , and that Suppose, in addition, that f (s) = 0 for any s in both the disc |s − s 0 | ≤ r and the right half-plane σ ≥ σ 0 − αr.Then, for any s in the disc |s − s 0 | ≤ αβr, Theorem 2. Let k ≥ 3 be an integer and σ Also define Then for every κ > 0, all t ≥ t 0 , and where B = 1 + 8/3ω 1 and We give some admissible values of A κ , B, κ, ω for σ k ≤ σ ≤ 1 + κ and t ≥ t 0 ≥ 3, where A κ is defined in (2).
From here we can use the proof of Titchmarsh's Theorem 5.17 [15].Let where ω 1 is chosen such that we have k ≥ 3 for all t ≥ t 0 .This k implies with ω 2 as defined in the statement of the lemma.
The bound in (4) indicates that since (3) holds for σ ≥ σ k , it also holds for The expression for k also implies becomes where A κ is in (2).For specific t 0 , we can find a κ that minimises A κ , then find the smallest ω 1 that permits our choice of k for all t ≥ t 0 .It is then possible to compute B and ω 2 .This allows us to compute the values in Table 1 mentioned in the statement of the lemma.
We can compare this result to that of Hiary, Leong and Yang [6, Lemma 7], derived using a Richert-type bound for ζ(s) by Ford in [4].For instance, in the range t ≥ e e and 1 − 0.309(log log t) where Proof.This is derived in a similar way to Lemma 3 of [17].Trudgian's lemma is an adaption of a theorem from Rademacher on a generalisation of the Phragmén-Lindelöf theorem [13, §33].The Phragmén-Lindelöf theorem [13, §29] is applied to a function F (s), to which Trudgian adds a factor of log(Q+s), to use the theorem for bounds on f (s) that have a factor of log |Q + s|.
We will make a similar change to Rademacher's theorem.However, instead of a factor of log(Q + s), we will include a factor of log(Q + s)/ log log(Q + s) to form the function , where E is defined in [17,Lem. 2].We also include the restriction Q + a > e to ensure that F (s) is holomorphic in the strip a ≤ σ ≤ b.The proof then proceeds as in [17, Lemma 3] and [13, §32]. where Admissible values of Z are given in Table 2 for specific t 0 and optimized σ 1 .Proof.This is similar to the proof of Lemma 3. We take f (s) : Note that f (s) is entire in this strip, and satisfies the growth condition given by Rademacher or [17, Lemma 3].
We will need two bounds on |ζ(s)| to verify the condition in (5): one for σ = 1 and one for σ = σ 1 .The latter is relatively simple to derive, as we can use |ζ(σ 1 + it)| ≤ ζ(σ 1 ), which is true for all t and computable.For the other bound we use Theorem 3.
That is, we use Taking σ = 1 + δ := 1 + d log log t/ log t in the above gives These two factors can be bounded with Lemmas 5 and 7. First consider 1 < σ ≤ min{σ Applying these bounds in Lemmas 5 and 7, and then (9), we find Second, to obtain a bound over all σ > 1, we simply repeat the process with the trivial bound ζ(σ) ≤ σ/(σ − 1) in place of Lemma 5, as it is decreasing for all σ > 1.However, it is worth noting that this bound is not as sharp as Lemma 5 for σ close to 1.

Bounding |ζ ′ (s)/ζ(s)|
In this section, we use the lemmas in the previous section to bound |ζ ′ (s)/ζ(s)|.We follow the argument of [18], which gives an explicit version of results in [16, §3].This has been done in [6] to obtain bounds for |ζ ′ (s)/ζ(s)| and |1/ζ(s)| on the 1-line, and we now extend these results to hold within a zero-free region.The method is identical to that of [6], but uses sharper bounds on ζ, and a more careful handling of constants to obtain some savings.Given this, we will only give an outline of how to obtain such bounds, defining the functions with the same notation, but omitting details.The reader is referred to [6] for a full elucidation on the proof, especially on justification of certain parameter choices.
As before, we construct concentric discs, centred just to the right of the line σ = 1 + it, and extend them slightly to the left and into the critical strip.We aim to apply Lemmas 1 and 2 with f (s) = ζ(s).
Let t 0 ≥ e e and t ′ ≥ t 0 be constants.The centre of the concentric discs will be denoted by where d is a real positive constant to be chosen later.Notice that δ t ′ is decreasing for t ′ > e e , and is at most d/e.Also, let r and ǫ ≤ 1 be positive parameters chosen such that σ ′ + r ≤ 1 + ǫ, which will be satisfied if Let s = σ + it be a complex number.Aiming to apply Lemma 1 in the disc |s − s ′ | ≤ r, we first seek a valid A 1 in that disc.Similarly, for Lemma 2, we seek a valid A 2 at the disc centre s = s ′ .In applying Lemma 2, we also need to ensure that the non-vanishing condition on f (s) = ζ(s) is met, which we do with a zero-free region.
First, let where ω 2 is defined in Lemma 3.Then, by the same lemma, for each t ∈ [t ′ − r, t ′ + r] and any σ ∈ [σ 1,t ′ , 1 + κ], we have with A κ and B given in Table 1.
For our purposes, we choose a radius of To ensure the constraint (10) is met for all t ′ ≥ t 0 , we need as (log log t ′ ) 2 / log t ′ reaches a maximum of 4/e 2 at t ′ = t e := e e 2 .This choice of r also requires Lemma 3 to be applicable for σ up to 1 + δ t ′ + r, which corresponds to This will be satisfied for all t 0 ≥ e e for κ ≥ 4 e 2 ω 2 + 2d log log t 0 , and the values of κ in Table 1 satisfy this condition.
To fulfil the bounding condition of Lemma 1, we can take either lower bound for |ζ(s ′ )| from Lemma 8, and use a constant where d and t ′ are always taken such that 1 + δ t ′ ≤ (1 + γ)/γ.With this, by ( 12) and ( 13), throughout the disc |s − s ′ | ≤ r we have Noting that A 3 is decreasing in t ′ ≥ t 0 , we obtain that throughout |s − s ′ | ≤ r, where Next, we fulfill the zero-free condition of Lemma 2 by way of the zero-free region in Lemma 4, where for t ≥ 3, This leads to the following choice of where to keep c 1 ≤ c 0 , and we require to satisfy the constraint α < 1/2 in Lemma 1.
To determine A 2 , we utilise [5, Lemma 70.1], which states that for σ > 1, Taking s = s ′ in this inequality leads to Finally, we turn to the choice of β in Lemma 2. From here on, we differ from [6] as we want the circles to be able to extend into the zero-free region.For any results from Lemma 2 to be meaningful to us, we want the left-most point of |s − s ′ | ≤ αβr to lie between the zero-free region and not exceed the 1-line.In other words, we want σ ′ − αβr ≤ 1, which means β must satisfy Up to this point, Lemma 2 holds for For larger σ ′ , we will just use (21).To finish, observe that 1/(1 − 2α) 2 is decreasing in t ′ , so we can substitute t ′ → t to arrive at the following lemma.

Computations
We can compute admissible R 1 in Corollary 1 using Lemma 9.
Proof of Corollary 1.Although we can use Lemma 9 with t 0 = e e , we can reduce the size of Q 1 by computing it for t 0 = H, the height to which the Riemann Hypothesis has been verified in [12], and combining it with a result for e e ≤ t 0 ≤ t < H.Over the latter range, we know there are no zeros of ζ(s) with σ > 1/2.We can therefore use a wider zero-free region than in Lemma 4 when verifying the condition in Lemma 2. Since 1 2 < 1 − (e/2) log log t log t for all t > 0, we can replace c 0 with any c RH < e/2 in ( 17) as long as it fulfils (20).Given (19), we see that ( 20) is satisfied by since log(t 0 + ǫ)/ log t 0 > 1 for all t 0 > 0. When optimising over d to calculate Q 1 for t ≥ H, we consistently found an optimal d < 0.05.Hence, we will assume d < 0.05, which, along with ǫ ≤ 1 and t 0 = e e , gives us that the right-hand side of ( 23 Although a larger c RH would mean we have a larger zero-free region, taking it too large can have an adverse effect on the parameter α.Smaller c RH reduces α, which means a better bound in Lemma 2, and hence in Lemma 9.However, larger c RH means we can take smaller β to satisfy the expression for W , which leads to a smaller value for λ 1 in Lemma 9. Starting by fixing a value for W , we compute Q 1 in the range t ≥ H by optimising over ǫ, β, and d, subject to the constraints of Lemma 9, and using c 0 as in (17).We similarly obtain Q 1 in the range t 0 ≤ t ≤ H, but instead optimise over ǫ, β, d < 0.05, and c RH ≤ 1/8.22.We also require the result to hold for which leads to replacing the condition in (22) with Finally we take the maximum of the two results, which is an admissible Q 1 over all t ≥ t 0 .This value is R 1 in Corollary 1.In Table 3, we opt to use the smallest t 0 that makes the Q 1 over t 0 ≤ t ≤ H the smallest of the two Q 1 .For c RH we use 12 for all entries up to and including W = 31; for W = 35 we use c RH = 10.5, and for all larger W we use c RH = 9.
In addition, we have Corollary 2, especially relevant for σ = 1.
Proof of Corollary 2. The process of obtaining the bound for σ ≥ 1 is the same as that of [6, Theorem 2], but with the corresponding changes in A max , B, and C 1 due to Lemma 3. Note that the method is essentially the same as our Corollary 1, but we take β = d/(c 1 + d).The rest of the proof follows mutatis mutandis.
As per Corollary 1, the computations are split into cases t 0 ≤ t ≤ H and t ≥ H, and we take the larger of the two resulting constants.An optimised value of c RH is used for each t 0 < H, and we note that both entries in

Bounding |1/ζ(s)|
Moving from a bound on the logarithmic derivative of ζ(s) to one on the reciprocal of ζ(s) is done in the same way as [6, Lemma 10].In essence, a bound on the latter depends on a bound for the former in some range of Re(s).We are able to make some savings by using multiple bounds for |ζ ′ (s)/ζ(s)| across the desired range, using corresponding pairs of (W, R 1 ) from Tables 3 and 4.
Lemma 10.Let t 0 ≥ e e and d 1 > 0 be constants.Let {W j : 1 ≤ j ≤ J} be a sequence of increasing real numbers where (W j , R 1,j ) is a pair for which t ≥ t 0 and Then, for where with any σ 1 ≥ 1 + d 1 log log t 0 / log t 0 , γ denoting Euler's constant, K 1 defined in Corollary 2, Z defined in Lemma 7, and V, V 1 defined in (15) and Lemma 8, respectively.
Corollary 3 follows from Lemma 10 by computing Q 2 for specific choices of t 0 and W j , and optimizing over d 1 and σ 1 .These computed values of Q 2 are labelled R 2 .
Proof of Corollary 3.After choosing t 0 , we aim to minimise Q 2 by optimising over d 1 > 0 and σ 1 .This requires choosing J, which corresponds to the number of W j values.Using more W j will give a better result, but the improvements eventually become negligible, as Q 2 is largely determined by the initial few W j .
For computations, we only used values for W j from Table 3 (Corollary 1).This meant that using W = 70, for instance, would only be valid for t ≥ 500.On the plus side, larger t 0 meant we could also use smaller K 1 from Corollary 2. With these factors in mind, we chose to compute two results for a selection of W , one for the largest possible range, and the other for the smallest achievable constant.This meant that computing Q 2 for t 0 = e e and W ≤ 23, for instance, only used the entries of Table 3 for t 0 = e e .Computing Q 2 for t 0 = 500, however, allowed us to use all available W greater than the one chosen.
For W = 21.24,we can take R 2 = Q 2 (0.0031, 6.52, e e ) = 44910.Other values of R 2 are listed in Table 5 for the specified t 0 , and we note that all results for t 0 < 500 had the same approximate optimal d 1 = 0.0031 alongside the stated σ 1 , and the results for t 0 = 500 all had d 1 = 0.0067 and σ 1 = 12.35.If we only consider the case σ ≥ 1 it is possible to reduce R 2 .This is done in Corollary 4. This result is especially useful for bounds on the line s = 1 + it.

W
Proof of Corollary 4. The proof follows the same method as Lemma 10, but in the case σ ≤ 1 + δ 1 , we only need the following bound in (25), This simplifies Q 2 (d 1 , σ 1 , t 0 ), replacing the third case in (24) with We can now fix t 0 and optimise over d 1 using the values of K 1 in Table 4.
For the second assertion of the corollary, Carneiro, Chirre, Helfgott, and Mejía-Cordero verify, using interval arithmetic in the proof of Proposition A.2 [1]

Discussion
Using the methods of this paper, the main obstruction to obtaining better bounds on |ζ ′ (s)/ζ(s)| and |1/ζ(s)| is the width of the zero-free region.Lemmas 1 and 2 rely on the Borel-Carathéodory theorem (Theorem 1), which, in the setting of a general function f , already has the best possible constant (see [9, §5] for a discussion).Thus, to get improvements, one would have to input additional information on the specific function used.For example, in Lemmas 1 and 2 we use information on the zero-free region of ζ(s), so one could utilise more information to do better.
Because we sought bounds of the form log t/ log log t rather than log t, we were restricted by the bound on |ζ(s)| in the region σ ≥ 1 − ω 2 (log log t) 2 / log t.The width of this region hampers us in a similar fashion to the width of the zero-free region, so improving ω 2 is another possibility.
We conclude with a discussion on the trigonometric inequality (8)  Notice that Z in Lemma 7 depends on, and also tends to, the constant in Theorem 3 (as t 0 gets large).Unfortunately, an increase or decrease in this constant has a minimal impact on our main results, since our main constants are quite large.Thus, even though Lemma 8 offers a bound better than trivial, one would be better served seeking improvements from other avenues like the ones mentioned at the beginning of this section, for more substantial savings.
Another interesting point is that the conditions for trigonometric inequalities like (8) to give good bounds on |1/ζ(s)| for σ > 1 differ from the conditions needed for good zero-free regions.For instance, when dealing with the latter, one requires a non-negative trigonometric polynomial N n=0 a n cos(nθ) = a 0 + a 1 cos(θ) + . . .+ a N cos(N θ), where a 1 > a 0 , with each coefficient a n non-negative, to state a few conditions (see [8], [11], for more information).However, for |1/ζ(s)| the criteria is to have the sum of all coefficients in the polynomial n |a n | ≤ 2a 1 , with a 1 > 0. This condition is to ensure the overall bound is of the desired order.To illustrate this, if we had used a polynomial in the proof of Lemma 8 with n |a n | = 2a 1 +ǫ, for any ǫ > 0, then we would have ended up with a factor of (log t/ log log t) 1+(ǫ/a 1 ) instead of log t/ log log t.
The natural question to ask is if one could do better with a different choice of inequality than (8) when bounding |1/ζ(s)|.This is equivalent to asking for (28) with 0 < a 0 /a 1 < 3/4, while at the same time satisfying our new criteria.We briefly did a search for such polynomials of higher degree, to no avail.Although there have been many trigonometric polynomials found in the arena of refining zero-free regions, none of these fit our stated new criteria while also improving on 3/4.We hypothesise that no better polynomials exist.Certainly for a degree-2 polynomial of the form (x + y cos θ) 2 = x 2 + y 2 2 + 2xy cos θ + y 2 2 cos 2θ (which is the prequel to (8)), it is easily seen that to fulfil our new criteria, one needs (x − y) 2 ≤ 0. Thus the only option is x = y, and so ( 8) is the best possible in this situation.

1 IntroductionΛ
In the study of the Riemann zeta function ζ(s) and related functions, one often requires upper bounds for the logarithmic derivative of ζ(s) and the reciprocal of ζ(s).The Chebyshev function ψ(x) = n≤x Λ(n), where Λ(n) is the von Mangoldt function, or M (x) = n≤x µ(n), where µ(n) is the Möbius function are particularly relevant examples of where these estimates are useful.To estimate either with Perron's formula, we use the fact that n≥1

Theorem 1 . [ 6 ,
Theorem 4][Borel-Carathéodory] Let s 0 be a complex number, and let R be a positive number possibly dependent on s 0 .Suppose that the function f (s) is analytic in a region containing the disc |s − s 0 | ≤ R. Let M denote the maximum of Re f (s) on the boundary |s − s 0 | = R.Then, for any r ∈ (0, R) and any s such that |s − s 0 | ≤ r,

Lemma 6 .
Let a, b, Q, and k be real numbers such that Q + a > e, and let f (s) be a holomorphic function in the strip a ≤ Re(s) ≤ b such that it satisfies the growth condition |f (s)| < C exp e k|t| for some C > 0 and 0 < k < π/(b − a).Suppose further that 1/(2 k −2) log t.

Table 1 :
Admissible values of parameters in Lemma 3.

Table 2 :
Values for Z in Lemma 7 for specific t 0 after optimizing over σ 1 .

Table 4
are determined by the value of Q 1 for t 0 ≤ t ≤ H.

Table 4 :
Values for K 1 in Corollary 2, based on (α, d, ǫ) from Lemma 9, where α 1 is an upper bound on α, and each entry is valid for t ≥ t 0 .

Table 5 :
Values of R 2 for specific W in Corollary 3, for t ≥ t 0 .

Table 6 :
Values of K 2 for specific t 0 in Corollary 4.