Bounding the log-derivative of the zeta-function

Assuming the Riemann hypothesis we establish explicit bounds for the modulus of the log-derivative of Riemann’s zeta-function in the critical strip.


Introduction
Let ζpsq be the Riemann zeta-function. In this paper we are interested in its log-derivative Λpnq n s pRe s ą 1q and its growth behaviour in the strip 1{2 ă Re s ă 1 (above Λpnq is the von Mangoldt function). Let ρ denote the zeros of ζpsq in the critical strip. The Riemann hypothesis (RH) states that the zeros are aligned: ρ " 1 2`i γ with γ P R. Assuming RH, a classical estimate for the log-derivative of ζpsq (see [12,Theorem 14.5]) establishes that ζ 1 ζ pσ`itq " O`plog tq 2´2σ˘, uniformly in 1 2`δ ď σ ď 1´δ, for any fixed δ ą 0. The purpose of this paper is to establish this bound in explicit form.
We believe that λ 0 is simply a by-product of our proof, although it is curious that such a number appears. It turns out that when pσ´1{2q log log t is too small, our main technique delivers a bound of the form A σ plog tq{log log t, however the calculations are lengthy and convoluted, and this it not the purpose of this note. Moreover, a conjecture of Ki [10], related to the distribution of the zeros of ζ 1 psq, states that the bound Opplog tq 2´2σ q still holds in the range σ ě 1{2`c{ log t, but this lies outside of what this technique can accomplish. Theorem 1 is derived by combining Theorem 2 and estimates for the real part of the log-derivative of ζpsq obtained in [4,Theorem 2]:ˇˇˇR uniformly in the range (1.1) (in fact λ 0`c can be replaced by just c).
The main technique to prove these theorems revolves in bounding a certain sum over the ordinates of zeta-zeros ÿ where f is some explicit real function that varies according to the problem of study.
The key idea is to replace f by explicit bandlimited majorants and minorants that are in turn admissible for the Guinand-Weil explicit formula (Proposition 5). From there estimating the sum is usually easier. This bandlimited approximation idea originates in the works of Beurling and Selberg (see [14,Introduction]), and was first employed in this form by Goldston and Gonek [8], and Chandee and Soundararajan [6], but many others after them (see [1,2,3,4,7] to name a few). In our specific case, f " f a as in (2.1), which has zero mass and therefore is not in the scope of the machinery developed in [5], nor its close relatives (the constructions in [5] are regarded as the most general thus far and have been used widely). Nevertheless, we are able to overcome this difficulty with a very simple optimal construction which, in the majorant case, requires some basic results in the theory of de Branges spaces. We recall that, without assuming RH, explicit bounds for ζ 1 ζ psq are given by Trudgian [13] in a zero-free region for ζpsq.
As always, the crucial tool to work with sums as in Lemma 4 is the Guinand-Weil explicit formula (see [4,Lemma 8]), which for even functions reads as follows.
Proposition 5 (Guinand-Weil explicit formula). Let hpsq be analytic in the strip |Im s| ď 1 2`ε , for some ε ą 0, such that |hpsq| ! p1`|s|q´p 1`δq , for some δ ą 0. Assume further that h is even. Then ÿ ρ hˆρ´1 2 i˙" 1 2π where ρ " β`iγ are the non-trivial zeros of ζpsq and and λ " πa∆. Then: holds for all real x, L a,∆ P L 1 pRq and its Fourier transform is supported in r´∆, ∆s (i.e. L a,∆ is of exponential type at most 2π∆); and any other function F ‰ L a,∆ having the same properties as L a,∆ in item p1q has integral strictly less than the integral of L a,∆ .
Proof. Note first that the constants A, B were chosen so the numerator of L a,∆ vanishes doubly at z "˘ia. We then see that L a,∆ is entire, of exponential type at most 2π∆ and belongs to L 1 pRq. Therefore, the Paley-Wiener Theorem guarantees its Fourier transform is supported in r´∆, ∆s. Since B ą A ą 0 we have L a,∆ pxq ď f a pxq for all real x. This proves item p1q. We now prove item p2q. Suppose F is an L 1 pRq-function, F pxq ď f a pxq for all real x and p F is supported in r´∆, ∆s. Poisson summation implies where the last identity is due to the fact that L a,∆ interpolates (in second order) f a in 1 ∆ Z. Equality is attained if and only if F pxq " L a,∆ pxq in second order for all x P 1 ∆ Z. However, this completely characterizes F " L a,∆ (see [14,Theorem 9]). Finally, using that p f a pyq "´2π 2 |y|e´2 πa|y| , identity (2.2) can easily be derived using Poisson summation over 1 ∆ Z.
It turns out that because f a pxq has a local maximum at x " ? 3 a, the bandlimited majorant of f a with minimal total mass will have to be adjusted when πa∆ is small. This adjustment will require some de Branges spaces theory.
λ " πa∆ and λ 0 " 0.771 . . . is such that 2λ 0 tanhpλ 0 q " 1. Then: (1) The inequality f a pxq ď U a,∆ pxq holds for all real x, U a,∆ P L 1 pRq and its Fourier transform is supported in r´∆, ∆s (i.e. U a,∆ is of exponential type at most 2π∆); Moreover, any other function F ‰ U a,∆ having the same properties as U a,∆ in item p1q has integral strictly greater than the integral of U a,∆ .
Proof. Note that the constants pC, D, Eq are chosen so that U a,∆ is entire, that is, its numerator vanishes doubly at z "˘ia. Since U a,∆ is visibly of exponential type at most 2π∆ and belongs to L 1 pRq, the Paley-Wiener Theorem guarantees its Fourier transform is supported in r´∆, ∆s. Noting that C, D ě 0 we have f a pxq ď U a,∆ pxq for all real x, and this proves item p1q. We now show item p2q. Suppose F is an L 1 pRq-function, F pxq ě f a pxq for all real x and p F is supported in r´∆, ∆s. We now apply the generalized Poisson summation formula of Littmann for bandlimited functions [11, Theorem 2.1] for γ " pπEq´1 with E ą 0. It translates to where Bpzq " cospπzq´Eπz sinpπzq. Note when E " 0, that is, λ ě λ 0 , this is the classical Poisson summation over 1 ∆ p 1 2`Z q. Equality is attained if and only if F pt{∆q " U a,∆ pt{∆q in second order for all real t with Bptq " 0. We claim this completely characterizes F " U a,∆ . The trick is to use the theory of de Branges spaces and the interpolation formula [9, Theorem A] (the introduction of [9] gives a solid short background on the necessary de Branges spaces theory which we will use here without much explanation). First we note that the function Epzq " pi`πEzqe´π iz is of Hermite-Biehler class (i.e. |Epzq| ă |Epzq| for all z with Im z ą 0) and therefore the de Branges space HpE 2 q exists, and it consists of all entire functions of exponential type at most 2π belonging to L 2 pR, dx{p1`E 2 π 2 x 2 qq. Note also that Bpzq " ipEpzq´Epzqq{2. Moreover, it is not hard to show that all conditions of [9, Theorem A] are satisfied by Epzq, and thus we conclude that any function G P HpE 2 q is completely characterized by its values Gptq and G 1 ptq for all real t with Bptq " 0. Now it is simply a matter to note that pi`πEzq 2 F pz{∆q and pi`πEzq 2 U a,∆ pz{∆q both belong to HpE 2 q, and so they must be equal 2 .
Finally, in the case λ ě λ 0 one can use Poisson summation over 1 ∆ p 1 2`Z q to evaluate the integral of U a,∆ and obtain p U a,∆ p0q " π 2 ∆ cosh 2 pπa∆q .
Proof. First we deal with the minorant. Using that p f a pyq "´2π 2 |y|e´2 πa|y| and the Fourier transforms of 1 x 2`a2 and a 2 px 2`a2 q 2 are π a e´2 πa|y| and π 2ˆ| y|`1 2πa˙e´2 πa|y| , respectively, we obtain where T h is the operator of translation by h and Id is the identity operator. These operators come from the (distributional) Fourier transform of sin 2 pπ∆xq. We claim that the function e 2πay p L a,∆ pyq is convex in the range 0 ă y ă ∆, which would show that p L a,∆ pyq is negative in the same range since it is negative at y " 0 and vanishes at y " ∆. For 0 ă y ă ∆ we have The majorant case is simpler, since if λ " πa∆ ě λ 0 a similar computation leads to and so the desired inequality follows because D ą C ě 0.

Proof of Theorem 3
Let 1 2 ă σ ă 1 and ∆ ą 0. Throughout the rest of the paper we set a " σ´1 2 and λ " πa∆. Using Lemma 4 and the evenness of the zeta-zeros we obtain as t Ñ 8, where we have used that f a pxq " Op1{x 2 q uniformly for |x| ě 1 and 0 ă a ă 1{2, hence ř γ f a pγq " Op1q. We then apply Lemmas 6 and 7 to get ÿ γ M t L a,∆ pγq`Op1q ď Reˆζ where M t " 1 2 T t`1 2 T´t`Id. Note that for each t ě 0 the functions M t L a,∆ and M t U a,∆ are even and admissible for the Guinand-Weil explicit formula (Proposition 5). We use the operator M t because its Fourier transform is the operator that multiplies by 2 cos 2 pπtxq, which is nonnegative. This will allow us to simply discard (or easily bound) the sum over primes in the explicit formula.
3.1. Proof of the lower bound. Applying Proposition 5 and Lemmas 6 and 8 we obtain ÿ γ M t L a,∆ pγq " 1 2π In this part we assume that λ ě c for some given fixed c ą 0. We now analyze the terms on the right-hand side above. The function L a,∆ depends on the parameters A and B, but both behave like (since λ ě c) Hence |L a,∆ pxq| ď Kpx 2`a2 q´1 for some K ą 0. Since ps 2`a2 qL a,∆ psq has exponential type 2π∆ and it is bounded on the real line, a routine application of the Phragmén-Lindelöf principle implies that |L a,∆ psq| ď K e 2π∆|Im s| |s 2`a2 | , s P C (3.2) (alternatively, one could derive such bound by direct computation). Using the bounds for A and B it follows that Using that M t is self-adjoint and applying Stirling's approximation to obtain we deduce that 1 2π Combining the above bounds we obtain Oˆe p1´2aqπ∆ pa 2´1 4 q 2`e π∆ t 2`1 a`1`e´2 πa∆ log t˘˙.
Choosing π∆ " log log t (which is the optimal choice) and using (3.1) we obtain or πpσ´1{2q log log t ě c. This proves the desired result.

3.2.
Proof of the upper bound. Using Proposition 5 and Lemma 7 we obtain ÿ γ M t U a,∆ pγq ď 1 2π Λpnq ? n p U a,∆ˆl og n 2π˙c os 2 p 1 2 t log nq.
When λ ě λ 0 the computations are very similar to the lower bound and we just indicate them here. We still have both C and D behaving like 8λe´2 λ`O pe´2 λ q, and a bound similar to (3.2) holds. Using Stirling's formula and Lemma 7 we get 1 2π 2π∆e´2 πa∆ log t`Oˆ1`e´2 πa∆ log t a˙.
Using the estimates for C and D it follows that Since p U a,∆ is supported in r´∆, ∆s, we estimate the sum over primes (which we cannot discard as before) using Lemma 8 and that p f a pyq "´2π 2 |y|e´2 πa|y| to get The above estimate follows from the prime number theorem (see [4, Eq. (B.2)]). Choosing π∆ " log log t and using (3.1) we obtain n the range pσ´1{2q log log t ě λ 0 and p1´σq ? log log t ě c for some fixed c ą 0; note that ∆ 3 " O c`p 1{2´aq´2plog tq 1´2a˘. This finishes the proof.

Proof of Theorem 2
To obtain the bounds for the imaginary part of the log-derivative ζpsq we will employ the interpolation technique of [4, Section 6] for functions with slow growth, which we conveniently state in the form of a lemma.
The lemma follows.