An explicit sub-Weyl bound for $\zeta(1/2 + it)$
Dhir Patel, Andrew Yang
TL;DR
This paper establishes an explicit sub-Weyl bound for the Riemann zeta function on the critical line, providing the sharpest known bounds for large t values, which advances understanding of its growth behavior.
Contribution
It presents the first explicit sub-Weyl bound for ta(1/2 + it), improving the known upper bounds on its magnitude for large t.
Findings
Proves ta(1/2 + it) 66.7 t^{27/164} for t 3
Provides the sharpest bounds for ta(1/2 + it) for t ^{61}
Enhances understanding of the growth rate of ta on the critical line.
Abstract
In this article we prove an explicit sub-Weyl bound for the Riemann zeta function on the critical line . In particular, we show that for . Combined, our results form the sharpest known bounds on for .
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Taxonomy
TopicsAnalytic and geometric function theory · Analytic Number Theory Research · Graph theory and applications
An explicit sub-Weyl bound for
Dhir Patel and Andrew Yang
Abstract.
In this article we prove an explicit sub-Weyl bound for the Riemann zeta function on the critical line . In particular, we show that for . Combined, our results form the sharpest known bounds on for .
Key words and phrases:
Van der Corput estimate, exponential sums, sub-Weyl bound, Riemann zeta function.
2010 Mathematics Subject Classification:
Primary 11L07, 11M06
1. Introduction
An important open problem in analytic number theory is the growth rate of the Riemann zeta-function on the critical line as . The well-known Lindelöf Hypothesis asserts that for any . Among the consequences of the hypothesis are many profound results for prime number distributions. Although the Lindelöf Hypothesis is currently unproven, much effort have been expended to bound the zeta-function on the critical line, culminating in the current best-known bound of for any , due to Bourgain [Bou16].
In this article we are concerned with explicit bounds on . Such explicit bounds have been used to derive zero-free regions [For02, MTY22, Yan23], zero-density estimates [KLN18] and bounds on the argument of on the critical line [Tru14, HSW21]. Recently, these results have in turn been used to obtain explicit theorems about prime distributions [KL14, CH21, Bro+21, CHL22, CHJ22, JY22, FKS22, FKS22a], so there is substantial motivation to sharpen such bounds as much as possible. Nevertheless, known explicit bounds on currently lag far behind the asymptotically sharpest-known results. Only two types of explicit subconvexity results are known — the first being the classical van der Corput estimate of the form for for some absolute constants and . Such bounds are sometimes known as Weyl estimates because the exponent of was first achieved via the Weyl–Littlewood–Hardy method. The sharpest estimate of this type is due to [HPY22], who built on the work of [CG04, Tru15, Hia16] to prove
[TABLE]
A second type of explicit bound, known sometimes as sub-Weyl estimates, was first made explicit by Patel [Pat21], who showed
[TABLE]
Note that . In particular, (1.2) is the best-known explicit bound for the zeta-function on the critical line for .
In this work we improve (1.2). Our main result is
Theorem 1.1**.**
For , we have
[TABLE]
Theorem 1.1 represents the sharpest known bound on for . In §3.1 we show that still sharper bounds are possible for smaller . Together, our results form the best known bound for . Therefore, (1.1) remains sharper at , the verification height of the Riemann Hypothesis [PT21]. This is significant since bounds for near such values of are used in multiple explicit results [HSW21, KLN18, For02].
On the other hand, sharp bounds on for larger values of are useful for deriving explicit zero-free regions [For02, MTY22, Yan23], for improved bounds on [Tru14, HSW21], for refinements to Turing’s method [Tru11, Tru16], and for asymptotically improved zero-density estimates [Tit86, Thm. 9.18]. For instance, following the method of [KLN18], we may use Theorem 1.1 to prove an explicit zero-density result of the form
[TABLE]
where is the number of zeroes of with and .
1.1. Approach and exponential sums
As with all existing explicit bounds on , Theorem 1.1 relies on upper bounds on particular types of exponential sums, obtained via van der Corput’s method of exponent pairs (for an exposition, see [GK91]). Roughly stated, let be a suitably well-defined and sufficiently smooth function satisfying for some . If , and
[TABLE]
then is an exponent pair. For instance, from the trivial bound we see that is an exponent pair. The motivation for studying exponent pairs is highlighted by the result that if is an exponent pair with , then
[TABLE]
see e.g. Phillips [Phi33].
The van der Corput method estimates by iteratively transforming it into simpler exponential sums, via two processes. The process, also known as Weyl-differencing, expresses in terms of , where is a function of lower order than (and is hence easier to control). By applying the process, we obtain that if is an exponent pair, then so is
[TABLE]
The process, also known as Poisson summation, expresses in terms of another exponential sum that is typically shorter. Using the process, if is an exponent pair, then so is
[TABLE]
Favourable exponential pairs and, by extension, good estimates of , can be obtained by beginning with the trivial exponent pair, then chaining together multiple applications of the and processes. The simplest van der Corput bound, such as (1.1), is obtained from the exponent pair . On the other hand, bounds such as (1.2) and Theorem 1.1 can be obtained using .
1.2. Explicit exponent pairs
Both the and processes have been made explicit. For the process, we have the following lemma, due to [Yan23] which builds on the work of [CG04, PT15].
Lemma 1.2** ([Yan23] Lem. 2.3).**
Let be real-valued and defined on , for some integers . For all integers , we have
[TABLE]
where .
A general explicit version of the process was proved111We note here that in this general explicit version of process derived by Karatsuba and Korolev, one of the “lower” order term, , in their assertion could grow larger than the main-term given by the sum, if is small. in Karatsuba and Korolev [KK07], which relied on controlling the first four derivatives of the phase function . Patel [Pat21, Thm. 2.31] proved the following explicit Poisson summation formula, which only relied on the first three derivatives.
Lemma 1.3** ([Pat21] Thm. 2.31).**
Let be three times differentiable. Let be decreasing in and , . Further, let be defined by
[TABLE]
Furthermore suppose that and . Then
[TABLE]
where
[TABLE]
and is the Euler-Mascheroni constant.
In practical application of van der Corput’s method, we employ two tricks that frequently appear in the literature [CG04, PT15, Hia16, Pat21, HPY22, Yan23]. First, in the process it is often helpful to replace the Poisson summation step with a second-derivative test that uses the Kuzmin–Landau lemma. This substitution preserves the original goal of shortening the exponential sum under consideration, without generating problematic secondary error terms that typically arise when applying Poisson summation. Second, to minimise tedium we typically apply an block as a single operation instead of separate operations. The following lemma, due to [Yan23], incorporates both of these modifications, which we will make extensive use of in this work.
Lemma 1.4** (Explicit th derivative test).**
Let be integers with . Let be equipped with continuous derivatives, with monotonic, and suppose that for all and some and . Then, for all , we have
[TABLE]
where , and
[TABLE]
[TABLE]
and for are defined recusively via
[TABLE]
[TABLE]
[TABLE]
Proof.
Follows by combining [Yan23, Lem. 2.4] and [Yan23, Lem. 2.5] ∎
1.3. Sources of improvement
We briefly review the main sources of improvement of Theorem 1.1 over (1.2), in case similar methods may be applied in other settings. Our first source of improvement originates from an interval-based argument as follows. An intermediary result in the argument of Patel [Pat21] produces a bound of the form
[TABLE]
where is a bounded function that is decreasing in and increasing in . This immediately implies the bound for , where
[TABLE]
However, in our application is typically large unless we take to be very large, which defeats the purpose of obtaining an explicit bound holding for all . Instead, we may proceed as follows. For , we apply
[TABLE]
and so, for all ,
[TABLE]
The central idea is to choose the ’s so that is never too large. For instance, we take to be sufficiently large so that
[TABLE]
is of an acceptable size. For more details and computation, see §3.1.
A second source of improvement comes from using the improved th derivative test (Lemma 1.4) which makes use of the trivial bound to increase its sharpness. For details, see [Yan23, §2].
A third source of improvement arises from applying a sharpened version of the Poisson summation formula (i.e. using Lemma 2.3 in place of Lemma 1.3). In our application, the error terms introduced in estimating the stationary phase approximation to an exponential sum can be significant.
Lastly, in bounding long exponential sums, we make use of geometrically-sized intervals so that there are subintervals instead of subintervals, for some fixed . Since the method of proof is unable to detect cancellation between terms of two different subintervals, having less divisions is beneficial.
2. Improved Poisson summation formulae
In this section we prove a sharpened version of Lemma 1.3, which is useful since error terms arising from Poisson summation formulae are significant in our application. The main result of this section (Lemma 2.3) is an explicit van der Corput process.
We begin by recalling some useful results, starting with bounds on exponential integrals. If is continuous and for , then by Rogers [Rog05, Lem. 3]
[TABLE]
In addition, a corollary of a result due to Kershner [Ker35, Ker38] is that if is continuous and for , then
[TABLE]
The constant 1.343 is sharp (up to rounding) and has an exact representation in terms of Fresnel integrals, however for our purposes the decimal approximation is sufficient.222Using an arbitrary-precision numerical integration package, we find that the variables and appearing in the main result of [Ker35] appear to be and instead of the stated values and respectively. The same result was also proved with a constant of in Titchmarsh [Tit86, Lem. 4.4] and in [Rog05, Eqn. (3)].
Throughout, we let denote the digamma function, defined as the logarithmic derivative of the gamma function, i.e. . We briefly recall that for , we have
[TABLE]
The digamma function has the following series representation, valid for all (see e.g. [AS13, §6.3.16])
[TABLE]
Finally, we recall the following upper bound on harmonic numbers. For , we have
[TABLE]
where, here and throughout, means . The equality is due to [MV73] and the inequality follows from .
We begin by approximating an exponential sum with a sum of exponential integrals in Lemma 2.1 below, which makes explicit a result of van der Corput [Cor21]. As a small technical detail, the range of the second sum in the below lemma is typically taken to be for arbitrary — see e.g. [Tit86, Lem. 4.7]. In our presentation, we fix , which greatly simplifies the arguments that follow. This result may be compared to [Pat21, Lem. 2.26].
Lemma 2.1**.**
Let be real valued, with a continuous and steadily decreasing derivative in , and let . Then
[TABLE]
Proof.
Assume without loss of generality that , for otherwise we may replace with for a suitable integer . Using Euler–Maclaurin summation (see [Tit86, Eqn. (2.12)]), we have
[TABLE]
where
[TABLE]
so that . Meanwhile, for all non-integer , we have
[TABLE]
so that
[TABLE]
say. We have
[TABLE]
and similarly
[TABLE]
By the second mean-value theorem, there exists such that
[TABLE]
However
[TABLE]
so
[TABLE]
Therefore,
[TABLE]
First, since , we have and
[TABLE]
where the first inequality follows from (2.4) for and via a separate evaluation for . Similarly, since ,
[TABLE]
hence
[TABLE]
Now consider . Let and
[TABLE]
We have
[TABLE]
If , then . Furthermore note that for all , and that there is one integer in , say . Therefore, since and by (2.7),
[TABLE]
If , then , and hence the first term on the RHS is at most . Meanwhile using the same argument as (2.7), the sum on the RHS is bounded by . On the other hand if , then by the same argument used in (2.7)
[TABLE]
In either case the RHS of (2.10) is at most . Similarly, writing , so that for all (since ), and using ,
[TABLE]
Thus
[TABLE]
We now divide our argument into the following two cases.
Case 1:
Then, and is an empty sum. Then, we have (vacuously)
[TABLE]
since the sum on the LHS is empty.
Case 2:
Then, and we let
[TABLE]
where
[TABLE]
Therefore,
[TABLE]
where in the last inequality we have used (2.5) if , and a direct evaluation if . It follows that in this case, from combining (2.9) and (2.13), that
[TABLE]
Therefore, in either case, by collecting (2.6), (2.8), (2.11), (2.12) and (2.14) we have
[TABLE]
where
[TABLE]
To complete the argument we note that the assumption that implies that ,333This inequality can be readily sharpened (the constant of 2 is chosen for cosmetic purposes). In any case, for our application the constant term makes no difference to the final result. and that sums over are equivalent to sums over . ∎
Next, we require a lemma related to the principle of stationary phase, which approximates an exponential integral. The traditional presentation of this result (see e.g. [Tit86, Lem. 4.6]) has a main error term of size , where , are respectively the orders of the second and third derivative of the phase function. In the following lemma, we make explicit an argument of Phillips [Phi33] to bound this error term to , which is smaller in our application. We also record that Heath-Brown [HB83] has shown that under suitable conditions, the main error term may be removed completely. However, since the error term is already of an acceptable size, we do not pursue such an optimisation here.
Lemma 2.2**.**
Let be real and three times differentiable, satisfying ,
[TABLE]
Furthermore, suppose for some . Then,
[TABLE]
Proof.
Suppose first that , for some fixed to be chosen later. Let
[TABLE]
say. Since , for all we have by the mean-value theorem
[TABLE]
for some . Similarly, for all . Via (2.1), we have
[TABLE]
Let
[TABLE]
so that, for all there exists some such that
[TABLE]
Hence
[TABLE]
However,
[TABLE]
so that
[TABLE]
We will now bound the modulus of
[TABLE]
First, suppose that for some arbitrary constant to be chosen later. Then via the trivial bound, we have
[TABLE]
Assume now that . Then
[TABLE]
say. First, consider . Using the trivial bound, we have
[TABLE]
Next, consider . Letting , and integrating by parts, we have
[TABLE]
However,
[TABLE]
and for some , we have
[TABLE]
Furthermore, we use the identity
[TABLE]
to obtain using (2.17) that
[TABLE]
This implies that, by combining (2.23) and (2.24),
[TABLE]
and thus
[TABLE]
Meanwhile, once again using (2.24), and the triangle inequality,
[TABLE]
and so, collecting (2.22), (2.25) and (2.26), and using ,
[TABLE]
We bound in the same way. Therefore, collecting (2.21) and (2.27) we find
[TABLE]
in the case where . However, since the bound (2.28) is strictly greater than (2.20), we conclude (2.28) in fact holds for all . Combining this with (2.16), (2.18), (2.19), we find that
[TABLE]
However, since is arbitrary, we take the limit as and choose
[TABLE]
to balance the two terms on the RHS. This choice gives
[TABLE]
If , then we instead bound using
[TABLE]
which follows from (2.1) since for all , we have , as and . Similarly, if , we instead bound using
[TABLE]
The desired result follows from combining (2.29), (2.30) and (2.31). ∎
Lemma 2.3** (Improved Poisson summation formula).**
Let be three times differentiable. Let be decreasing in and , . For all integer , let be defined by . Furthermore suppose that and . Then
[TABLE]
Proof.
We use Lemma 2.2 to obtain
[TABLE]
Since there is at most one integer each in the intervals and , and , we have
[TABLE]
Next, since , we have ,
[TABLE]
Finally, consider . For all , we have and . Furthermore, the th smallest integer in the interval is bounded below by . Therefore,
[TABLE]
where is the digamma function. Similarly
[TABLE]
Therefore, using for ,
[TABLE]
Finally, since the intervals and contain at most one integer each, and using (2.2),
[TABLE]
The desired result follows upon applying Lemma 2.1 and collecting (2.32), (2.33), (2.34), (2.35) and (2.36). ∎
3. Proof of Theorem 1.1
This section contains the proof of our main result. We derive an upper bound on using the Riemann–Siegel formula, which allows us to express in terms of an exponential sum of length . This enables us to readily apply the techniques of the previous sections to produce a non-trivial estimate of . The main ingredients this step are the explicit , and processes, given by Lemma 1.2, 2.3 and 1.4 respectively.
To begin, we recall the following result, due to Hiary [Hia16], which is a consequence of the Riemann–Siegel formula.
Lemma 3.1**.**
For all ,
[TABLE]
Proof.
See Hiary [Hia16, Lem. 2.1] and also Gabcke [Gab79]. ∎
We use a similar approach as [Tit86, Thm. 5.18] to evaluate the main sum on the RHS of (3.1). We divide the sum into three subsums and bound each individually. Firstly, for , we use the triangle inequality and the trivial bound. Secondly, for we use Lemma 1.4 with . Lastly, for , we use the process (Lemma 3.4 below).
We remark that the last subsum, taken over , is by far the most significant. In fact, the second sum can be bounded to be which is . Additionally, we have the freedom make the first subsum as small as we please, by appropriately choosing the boundary between the first and second subsums. Therefore, in what follows we will expend the most effort in bounding the third subsum.
To begin, let , and , , be scaling parameters to be chosen later. Define
[TABLE]
where
[TABLE]
Note that this choice of guarantees that
[TABLE]
Similarly, let
[TABLE]
[TABLE]
so that . These parameters are chosen so that .
We thus divide
[TABLE]
say. The next few lemmas are used to bound each of the three subsums.
Lemma 3.2**.**
For and , we have
[TABLE]
Proof.
Recall that , so that, by the triangle inequality and the trivial bound,
[TABLE]
for all . Here, the second inequality follows from the convexity of and Jensen’s inequality, since
[TABLE]
∎
Lemma 3.3**.**
Suppose , and . Then
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
Proof.
With as defined in (3.5), we have
[TABLE]
since
[TABLE]
Applying Lemma 1.4 with , , , and
[TABLE]
we obtain, for any ,
[TABLE]
where
[TABLE]
Next, by partial summation
[TABLE]
so that, combined with
[TABLE]
we obtain
[TABLE]
The result follows from . ∎
Lemma 3.4**.**
Let and , be arbitrary constants. Suppose satisfy and . Then
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
and are functions defined in Lemma 1.4.
Proof.
Let
[TABLE]
so that
[TABLE]
Also, let
[TABLE]
[TABLE]
Note that since , we have . Furthermore, let be such that , i.e.
[TABLE]
Finally, define
[TABLE]
3.0.1. Applying the process
We begin by considering the exponential sum
[TABLE]
which is an intermediary sum encountered prior to applying the final process. We bound this sum using the th derivative test, which corresponds to the process. Via a direct computation, we have
[TABLE]
where . For all , we have , since
[TABLE]
as implies , and is increasing for .444In fact a much sharper inequality can be applied here, since we ultimately take . However, such optimisations do not appear to affect the final result. Therefore,
[TABLE]
Furthermore,
[TABLE]
so that
[TABLE]
Combining (3.8), (3.9) and (3.10), for , we have
[TABLE]
where
[TABLE]
Meanwhile, by the mean-value theorem we have for some , and so by directly computing we have
[TABLE]
We apply Lemma 1.4 with , , and to obtain, using (3.11) and (3.12),
[TABLE]
for any , where
[TABLE]
and
[TABLE]
[TABLE]
Note that the leading term of (3.13) corresponds to the exponent pair.
3.0.2. Applying the process
Equipped with a bound for , we apply the process (Lemma 2.2) with . The end result of this subsection is an explicit exponent pair. To do this we first require a few intermediary results. To begin, note that
[TABLE]
Here, the second inequality follows from the arithmetic-geometric means inequality. Therefore, by partial summation and using (3.13),
[TABLE]
where
[TABLE]
Additionally, note that
[TABLE]
and thus
[TABLE]
Similarly,
[TABLE]
[TABLE]
Applying Lemma 2.2 and using (3.14), (3.15) and (3.16), we finally obtain
[TABLE]
where, from Lemma 2.3, the error term satisfies
[TABLE]
Setting
[TABLE]
and substituting (3.15) and (3.16), we have
[TABLE]
Furthermore, since for all , we have, using (3.12),
[TABLE]
This implies, from , that
[TABLE]
Combining (3.17), (3.18) and (3.19), we have
[TABLE]
where
[TABLE]
[TABLE]
3.0.3. Applying the process
To complete the proof we apply the process once more to obtain the exponent pair . To do so we rely on the following inequality, which can be found in e.g. [Pat21]:
[TABLE]
for all integers . Applying this formula, and using (3.20), we obtain
[TABLE]
We use this in Lemma 1.2, together with , to obtain
[TABLE]
We choose for some to be chosen later, so that
[TABLE]
Now, with defined in the lemma statement, we have
[TABLE]
and hence by assumption, . Observe that the following inequalities hold
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Using the above inequalities, we obtain
[TABLE]
Taking square roots of both sides, applying , and substituting the values of , , , and , the desired result follows. ∎
Lemma 3.5**.**
Let , and be constants jointly satisfying the conditions of Lemma 3.4. Furthermore assume that . Then
[TABLE]
where
[TABLE]
[TABLE]
and , , , and are as defined in Lemma 3.4. Furthermore, for all ,
[TABLE]
where
[TABLE]
[TABLE]
Proof.
With as defined in (3.2), we have
[TABLE]
say. Therefore, we may apply Lemma 3.4, with , , and to obtain, via partial summation,
[TABLE]
and thus
[TABLE]
Since
[TABLE]
we have, for any and ,
[TABLE]
and, again for and ,
[TABLE]
Substituting these into (3.27), we obtain the estimate
[TABLE]
which gives (3.23) from , and forms the main bound for for in finite intervals . To obtain a bound holding for all , we use
[TABLE]
to continue the argument from (3.28) and (3.29) to obtain, for ,
[TABLE]
[TABLE]
Equation (3.25) then follows from substituting these estimates into (3.27) and using . ∎
3.1. Computations
For each row of Table LABEL:coefficientstable, we substitute the relevant parameter values into Lemma 3.2, 3.3 and 3.5 to verify, in each case, that
[TABLE]
Upon inspection, we have in each case, which proves Theorem 1.1 for . These parameters are found via a stochastic optimisation routine so are not necessarily globally optimal, however they suffice for justifying an upper bound on the constant factor in Theorem 1.1.
In addition, by taking , , , , , , and in Lemma 3.2, 3.3 and (3.25), we obtain
[TABLE]
Note that in the application of Lemma 3.3, we take the limit as . This implies Theorem 1.1 for .
For small values of , we use the following bound
[TABLE]
proved in §3.4 of [HPY22]. This estimate covers the range . Finally, for , we use the classical van der Corput estimate (1.1). This completes the proof of Theorem 1.1. Lastly, we note that the bounds in Table LABEL:coefficientstable improve on both (3.30) and (1.2) for all , and is thus the sharpest known bound on in this range.
4. Conclusion and future work
Theorem 1.1 represents the first of many successively sharper sub-Weyl bounds of the form obtainable from van der Corput’s method. The next few values of , due to [Phi33], [Tit42], [Min49], [Han63] and [Kol82] respectively, are
[TABLE]
The first result, , can be obtained via the exponent pair
[TABLE]
and can thus be made explicit using a similar but longer version of the arguments presented in this paper. Exponents starting from , however, rely on estimates of higher-dimensional exponential sums. For example, in the two-dimensional case, the function encountered in the process (Lemma 1.2) can be treated as a function of two variables, and . Such a sum can be estimated using two-dimensional analogs of the and processes.
The main obstacles to computing an explicit version of such results are difficulties with the two-dimensional Poisson summation formula. In the two-dimensional analog of the process, the factor appearing in Lemma 2.3 is replaced by the Hessian of , defined by
[TABLE]
However, if vanishes within the rectangle of summation, as is the case when bounding , it can be difficult to control the transformed sum. Successful implementations [Tit35, Min49] of two-dimensional exponent pairs rely on elaborate arguments to isolate problematic regions within the summation rectangle, and applying the trivial bound in those regions instead. Explicit versions of higher-dimensional Poisson summation formulae will be investigated in a future article.
Acknowledgements
We would like to thank Timothy S. Trudgian and Ghaith A. Hiary for their continuous support and helpful suggestions throughout the writing of this paper.
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