Abstract
Assuming the Riemann hypothesis, we prove that
where \(N_k(T)\) is the number of zeros of \(\zeta ^{(k)}(s)\) in the region \(0<\mathfrak {I}s\le T\). We further apply our method and obtain a zero counting formula for the derivative of Selberg zeta functions, improving earlier work of Luo (Am J Math 127(5):1141–1151, 2005).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \(\zeta (s)\) be the Riemann zeta function, and let
be the zero counting function for \(\zeta (s)\). Here and throughout, \(\rho =\beta +i\gamma \) is a generic zero of \(\zeta (s)\). It is known that
where
The unconditional bound is known as the Riemann–von Mangoldt formula (see [17, Theorem 9.4]), and the conditional bound is due to Littlewood [9].
There has also been considerable interest in zeros of derivatives of \(\zeta (s)\). Let \(\zeta ^{(k)}(s)\) be the k-th derivative of the Riemann zeta function, and let
be the zero counting function for \(\zeta ^{(k)}(s)\). Here and throughout, \(\rho _k=\beta _k+i\gamma _k\) is a generic zero of \(\zeta ^{(k)}(s)\). In [2] B. C. Berndt proved that
where
This should be compared to the Riemann–von Mangoldt formula. In view of (1) one may expect to prove that, assuming RH,
for all positive integers k. The first result in this direction is due to Akatsuka [1], who showed that if RH is true then
Yet this bound is weaker than (2). The second author [15] extended this estimate to higher derivatives and showed that on RH
for all positive integers k.
Recently, the first author [5] was able to prove (2) for \(k=1\), namely,
A key ingredient in his proof is an upper bound for the number of zeros of \(\zeta '(s)\) close to the critical line, and the idea there has its origin in Zhang’s work [18]. However, the method for \(k=1\) is not readily applicable for larger k. The purpose of this note is to modify the method in [5] and show that the estimate (2) holds for all positive integers k.
Theorem 1
Assume RH. Then we have
as \(T\rightarrow \infty \).
We remark that Littlewood’s conditional bound
was first proved in 1924. Later in 1944 Selberg [11] gave a different proof for this result. In 2007, Goldston and Gonek [7] showed that we can take the implied constant to be 1/2. The current best known constant is 1/4, and this is due to Carneiro et al. [3] who proved it using two different methods in 2013. It seems difficult to reduce the size of the bound (3), and this suggests that the bounds in Theorem 1 might be best possible within current knowledge.
On the other hand, using interesting heuristic arguments Farmer et al. [4] have conjectured that \(E_0(T) = O(\sqrt{\log T \log \log T})\). This raises the question of what bounds one should expect for \(E_k(T)\). We have the following
Theorem 2
Assume RH and suppose that \(E_0(T)=O(\Phi (T))\) for some increasing function \(\log \log T \ll \Phi (T)\ll \log T\). Then we have
Clearly Theorem 1 is a consequence of Theorem 2, so we shall only prove the latter. We also remark that our method works well for some other zeta and L-functions in the T-aspect. In Sect. 4 we give a brief discussion on this. In particular, we prove an analogue of Weyl’s law for the derivative of Selberg zeta functions.
2 Lemmas
Throughout, let \(\Phi (T)\) be an increasing function satisfying \(\log \log {T}\ll \Phi (T)\ll \log {T}\) and assume that \(E_0(T)\ll \Phi (T)\). Further, we use the variables k and \(\ell \) to denote orders of differentiation, where they are always positive integers.
We first express the error term of \(N_k(T)\) in terms of arguments of certain functions.
Lemma 3
Assume RH. Let \(k\ge 2\) be an integer. For \(T\ge 2\) satisfying \(\zeta (\sigma +iT)\ne 0\) and \(G_k(\sigma +iT)\ne 0\) for all \(\sigma \in \mathbb {R}\), we have
where
and the argument is defined by continuous variation from \(+\infty \), with the argument at \(+\infty \) being 0.
Proof
This is standard. Apply the argument principle to \(\frac{G_k}{G_{k-1}}(s)\) on the rectangular region with vertices \(1/4+i,\sigma _k+i,\sigma _k+iT,1/4+iT\), where \(\sigma _k\) is large so that \(G_k\) is dominated by 1 to the right of \(\sigma _k\). See also [15, Proposition 3.1] for an alternative proof. \(\square \)
Lemma 4
Assume RH and let \(\ell \ge 1\) be an integer. Then for \(1/2+\frac{(\log \log T)^2}{\log T}<\sigma <1\), we have
Proof
This is follows from [15, Lemma 2.3] by taking \(\epsilon _0=(4\log T)^{-1}\) there. \(\square \)
Lemma 5
Let \(\ell \ge 1\) be an integer. For all t sufficiently large we have
uniformly for \(1/2\le \sigma \le 1\).
Proof
This can be proved in a standard way. See Theorem 9.6 (A) in [17] for example. \(\square \)
Lemma 6
Assume RH and let \(\ell \ge 1\) be an integer. Then
holds for \(0<\sigma \le 1/2\) and sufficiently large t whenever \(\zeta ^{(\ell -1)}(\sigma +it)\ne 0\).
Proof
Put
Then we have
Using Hadamard factorization we easily see that for large t,
and
where \(\rho _{\ell -1}\) runs over all zeros of \(\zeta ^{(\ell -1)}(s)\). We can rewrite the latter as
By [8, Corollary of Theorem 7], RH implies that \(\zeta ^{(\ell )}(s)\) has at most finitely many non-real zeros in \(\mathfrak {R}(s)<1/2\). This implies that the second sum is O(1). Meanwhile [14] shows that
Thus
when t is large.
Taking the real part, we have
Hence using Stirling’s formula for the Gamma function, we have
which is negative for \(\sigma \le 1/2\) and t large. \(\square \)
Lemma 7
Assume RH. Let \(\mathcal {Z}_\ell =\{z_i\}_i\) be the collection of distinct ordinates of zeros of \(\zeta , \zeta ',...,\zeta ^{(\ell )}\) on \(\mathfrak {R}(s)=1/2\). For large T and \(Y\le T\), we have
Proof
It follows from Lemma 6 that for all \(j\in \mathbb {N}\), zeros of \(\zeta ^{(j)}\) on the critical line at large heights can only occur at zeros of \(\zeta ^{(j-1)}\). Therefore, for sufficiently large T,
\(\square \)
Write \(\mathcal {D}=\mathcal {D}(T)\) for the region \(\{w: \mathfrak {R}w\ge 1/2, |\mathfrak {I}w-T| \le 1\}\). Divide \(\mathcal {D}\) into N parts, as follows. Let \(B_j=\{w: 1/2\le \mathfrak {R}w\le 1/2+Y_j, |\mathfrak {I}w-T|\le Y_j\}\) where \(Y_j=2^jX\) and \(X=(\log {T})^{-1/2}\). We can write \(\mathcal {D}=\cup _{j=1}^N R_j\) where \(R_1=B_1\) and \(R_j=(B_j-B_{j-1})\cap {\mathcal {D}}\) for \(j\ge 2\). Note that \(2^NX \approx 1\).
A key ingredient in [5] is an upper bound for the number of zeros of \(\zeta '(s)\) in regions like \(R_j\)’s. To prove our Theorem 2 we need such bounds for higher derivatives, and the following result provides us the desired estimates.
Lemma 8
Let \(N_k(R_j)\) be the number of zeros of \(\zeta ^{(k)}\) in \(R_j\). Then \(N_k(R_j) \ll _k Y_j\log T+\Phi (2T)\).
Proof
Let \(R_j^*\) be \(R_j\) without the left side boundary on the critical line. In view of Lemma 7 it suffices to prove \(N_k(R_j^*) \ll _k Y_j\log T+\Phi (2T)\). Denote by \(\Theta (\rho _k;1/2+i(T+Y_j),1/2+i(T-Y_j))\in (0,\pi )\) the argument of the angle at \(\rho _k\) with two rays through \(1/2+i(T-Y_j)\) and \(1/2+i(T+Y_j)\). Note that \(\Theta (\rho _k;1/2+i(T+Y_j),1/2+i(T-Y_j))\gg 1\) if \(\rho _k \in R_j^*\). Thus
Write
Recall (4) that
We claim that
To prove this, note that for t on the segment \((z_{i},z_{i+1})\) we can write
where \(h(s)=\pi ^{-s/2}\Gamma (s/2)\). Thus, by using the temporary notation \(\Delta \arg \) to denote the argument change along the segment \((z_{i},z_{i+1})\), we have
It follows from the well-known functional equation for \(h(s)\zeta (s)\) that
From Lemma 6, we have
for \(l=1,2,\ldots ,k\) and \(t\in (z_{i},z_{i+1})\). Moreover, by Stirling’s formula we obtain
Thus
as claimed.
It then follows from (5) and Lemma 7 that
\(\square \)
3 Proof of Theorem 2
Applying Lemma 3, we only need to show that
holds for all \(k\in \mathbb {N}\).
Let \(X=1/\sqrt{\log T}\) as defined in the paragraph preceding Lemma 8. From Lemma 4, we see that
It remains to show
From Lemma 5, we have
where \(\Theta (a;b,c)\) is the (positive) angle at a in the triangle abc. Hence, it suffices to show that
From [8, Corollary of Theorem 7], RH implies that for sufficiently large T, \(\zeta ^{(k)}\) has no zeros in the left half of the critical strip above \(T-1\). Hence we may assume that
where \(\mathcal {D}\) is the region defined in the paragraph preceding Lemma 8. Using the expression \(\mathcal {D}=\cup _{j=1}^N R_j\) and Lemma 8 we have
Recall that \(X=\frac{1}{\sqrt{\log T}}\) and \(N\ll _k \log (1/X)\ll _k \log \log T\). Thus the above bound is
as desired. \(\square \)
4 Other zeta and L-functions
Our method works well for some other zeta and L-functions in the T-aspect. Below we give two examples of the first derivative of Selberg zeta functions and Dirichlet L-functions, respectively. Dealing with higher derivatives of these functions would require information about the “trivial” zeros of those derivatives, which is not the purpose of this paper.
First, let us consider the Selberg zeta functions on cocompact hyperbolic surfaces. Precisely, let X be a compact Riemann surface of genus \(g \ge 2\), and let \(Z_X(s)\) be the associated Selberg zeta function. Denote by \({\mathcal {N}}(T)\) and \({\mathcal {N}}_1(T)\) the zero counting functions for \(Z_X(s)\) and \(Z_X'(s)\), respectively; so \({\mathcal {N}}(T)\) is the number of nontrivial zeros of \(Z_X(s)\) up to height T, and similarly for \({\mathcal {N}}_1(T)\). Weyl’s law tells us that
where \(C_X\) is a specific constant depending on X. In [10] Luo proved that
Following our method in an identical manner, we can prove that
where \(D_X\) is a specific constant depending on X. Thus (8) improves Luo’s result. (Precisely, \(C_X=g-1\) and \(D_X=-\log N(P_{00})/2\pi \) where \(N(P_{00})=\min _{P_0}N(P_{0})\); see page 1143 in [10] for more explanation of the notation.) The estimate (8) was proved by the first author (unpublished) using a different method, but our method here is simpler.
As another example, let \(L(s,\chi )\) be the Dirichlet L-function where \(\chi \) is a primitive Dirichlet character to the modulus q. Let \(N(T,\chi )\) be the number of nontrivial zeros of \(L(s,\chi )\) with heights between \(-T\) and T. Define \(N_1(T,\chi )\) similarly as the zero counting function for \(L'(s,\chi )\). It follows from Selberg’s work [12] that on the generalized Riemann hypothesis (GRH)
As for \(L'(s,\chi )\), recently the first author [6] proved that on GRH we have
where m is the smallest prime number not dividing q. This improves earlier work of the second author [16]. Our method here should give analogues of (10) for higher derivatives of \(L(s,\chi )\) once some standard information on trivial zeros of these derivatives is gathered. As a final remark, we note that in the T-aspect the error term in (10) is as good as that in (9). However, in the q-aspect \(\sqrt{m\log 2m\log qT}\) might sometimes be larger than \(\frac{\log qT}{\log \log qT}\). In fact, simple calculation shows that
when m is no greater than \(\log qT/(\log \log qT)^3\), while the largest possible value for m is about \(\log q\). It would be of interest to see if one can remove the second term in the error of (10).
References
Akatsuka, H.: Conditional estimates for error terms related to the distribution of zeros of \(\zeta ^{\prime }(s)\). J. Number Theory 132(10), 2242–2257 (2012)
Berndt, B.C.: The number of zeros for \(\zeta ^{(k)}(s)\). J. Lond. Math. Soc. (2) 2, 577–580 (1970)
Carneiro, E., Chandee, V., Milinovich, M.B.: Bounding \(S(t)\) and \(S_1(t)\) on the Riemann hypothesis. Math. Ann. 356, 939–968 (2013)
Farmer, D.W., Gonek, S.M., Hughes, C.P.: The maximum size of L-functions. J. Reine Angew. Math. 609, 215–236 (2007)
Ge, F.: The number of zeros of \(\zeta ^{\prime }(s)\). Int. Math. Res. Not. IMRN 5, 1578–1588 (2017)
Ge, F.: The number of zeros of \(L^{\prime }(s,\chi )\). Acta Arith. 190(2), 127–138 (2019)
Goldston, D.A., Gonek, S.M.: A note on \(S(t)\) and the zeros of the Riemann zeta-function. Bull. Lond. Math. Soc. 39, 482–486 (2007)
Levinson, N., Montgomery, H.L.: Zeros of the derivative of the Riemann zeta-function. Acta Math. 133, 49–65 (1974)
Littlewood, J.E.: On the zeros of the Riemann zeta-function. Proc. Camb. Philos. Soc. 22, 295–318 (1924)
Luo, W.: On the zeros of the derivative of the Selberg zeta function. Am. J. Math. 127(5), 1141–1151 (2005)
Selberg, A.: On the remainder in the formula for \(N(T)\), the number of zeros of \(\zeta (s)\) in the strip \(0< t< T\). Avhandlinger utgitt av Det Norske Videnskaps-Akademi i Oslo I. Mat.-Naturv. Klasse 1, 1–27 (1944)
Selberg, A.: Contributions to the theory of Dirichlet’s L-functions. Skr. Norske Vid.-Akad., Oslo I. 3, 1–62 (1946)
Speiser, A.: Geometrisches zur Riemannschen Zetafunktion. Math. Ann. 110, 514–521 (1935)
Spira, R.: Another zero-free region for \(\zeta ^{(k)}(s)\). Proc. Am. Math. Soc. 26, 246–247 (1970)
Suriajaya, A.I.: On the zeros of the \(k\)-th derivative of the Riemann zeta function under the Riemann hypothesis. Funct. Approx. Comment. Math. 53, 69–95 (2015)
Suriajaya, A.I.: Two estimates on the distribution of zeros of the first derivative of Dirichlet L-functions under the generalized Riemann hypothesis. Journal de Théorie des Nombres de Bordeaux 29(2), 471–502 (2017)
Titchmarsh, E.C.: In: D.R. Heath-Brown (ed.) The Theory of the Riemann Zeta-Function, 2nd edn. Oxford Science Publications, Oxford (1986)
Zhang, Y.: On the zeros of \(\zeta ^{\prime }(s)\) near the critical line. Duke Math. J. 110, 555–572 (2001)
Acknowledgements
This project was started when the first author was a postdoc fellow at the University of Waterloo and the second author was a member of iTHEMS under RIKEN Special Postdoctoral Researcher program.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Ade Irma Suriajaya is supported by JSPS KAKENHI Grant Number 18K13400.
Rights and permissions
About this article
Cite this article
Ge, F., Suriajaya, A.I. Note on the number of zeros of \(\zeta ^{(k)}(s)\). Ramanujan J 55, 661–672 (2021). https://doi.org/10.1007/s11139-019-00219-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-019-00219-z