Micah B. Milinovich
A BSTRACT

1. Introduction
Let ((s) denote the Riemann zeta-function. This article is concerned with estimating discrete moments of the form where k is a natural number and γ < γ + are successive ordinates of non-trivial zeros of ζ(s). As usual, the function
denotes the number of zeros ρ = β + iγ of ζ(s) up to height T , counted with multiplicity. There have been a number of results concerning the behavior of the moments M k ( T ), and other closely related sums, for small values of k . See, for instance, [1–4, 6, 8, 13, 14, 15, 17]. In this article, we prove the following theorem, which provides upper and lower bounds for M k ( T ), for every positive integer k , which are very nearly the same order of magnitude. T HEOREM 1.1.  Assume the Riemann hypothesis (RH). Let k be a natural number and let ε > 0 be arbitrary. Then, letting γ and γ + denote consecutive ordinates of non-trivial zeros of ((s), we have for sufficiently large T where the implied constants depend on k and ε . With considerably more work, it might be possible to remove the ε appearing in the exponent of log T in the lower bound for M k ( T ) using a method of Rudnick and Soundararajan [11, 12]. This would be similar to the situation in [9], where lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of ζ(s) are given. Assuming the RH, Conrey and Ghosh [2] have shown that
Also, assuming the RH, Conrey [1] established the inequalities
Here, we have used the notation A ≤ B to indicate that A ≤ (1 + o(1)) B . Slightly stronger upper bounds for M 2 ( T ) have been given by Hall [4, 6]. It seems reasonable to conjecture that, for each natural number k,
Theorem 1.1 certainly lends strong support for this conjecture. In fact, the estimates in ( 1.3 ) could be established if one knew enough about the behavior of the continuous moment of (( s ) and its derivative on the critical line (see the remark at t he end of § 3). It may actually be the case that M k (T) ~ A k (log T) k 2 , when k ∈ N . However, there does not seem to be a reasonable conjecture for the values of A k when k ≥ 2.
2. Initial manipulations
The functional equation for the Riemann zeta-function, in asymmetric form, is
and that Z changes sign when t is an ordinate of a zero of ζ(s) on the critical line of odd multiplicity. If γ and γ + denote consecutive ordinates of non-trivial zeros of ζ(s) on the critical line, then Z(γ) = Z(γ + ) = 0 and by Rolle’s Theorem there is a zero λ of Z ′ ( t ) satisfying γ ≪ λ ≪ γ + . In fact, on RH, it is known that there is exactly one zero λ of Z ′ ( t ) satisfying γ ≪ λ ≪ γ + when γ > 0; see Hall [5, Theorem 2]. Therefore, assuming the RH, we see that
for each γ > 0 and each natural number k . Since there are exactly two zeros of Z ′ ( t ) between
for each k ∈ N. The sum over the zeros of Z ′ ( t ) appearing on the right-hand side of the above equation can be replaced by an integral involving Z(t) and Z'(t) plus a small error term. Assuming the RH, let λ denote the unique zero of Z ′ ( t ) between two consecutive zeros γ and γ + of Z ( t ). Then
for every natural number k . Thus, it follows from (2.3), (2.4), and the Lindelöf hypothesis (which is a consequence of the RH) that
for each k ∈ N and every ε > 0. 3. Auxiliary theorems and the proof of Theorem 1.1 In order to prove Theorem 1.1, we require an upper and a lower bound for the integral appearing on the right-hand side of (2.5). These bounds will be deduced as a consequence of the following two theorems. Our proof only requires the estimates in Theorem 3.2 in the cases where ℓ is either 0 or 1. Assume the RH. Let T be large and let k be a natural number. Then T HEOREM 3.1.
where the implied constant depends on k . T HEOREM 3.2.  Assume the RH. Let T  be large, k  be a natural number, ℓ be a non-negative integer, and ε > 0 be arbitrary. Then
and
where the implied constants in (3.2) and (3.3) depend on k , l , and ε . Theorem 3.1 is proved using standard Dirichlet polynomial methods. The proof of Theorem 3.2 requires upper bounds for moments of ζ(s) off of the critical line. We deduce these from a recent result of Soundararajan [14] and a convexity estimate for moments of the Riemann zeta-function due to Heath-Brown [7]. Proof of Theorem 1.1.   When k = 1, the theorem follows from the estimate in (1.2). Hence, we may assume that k ≥ 2 so that 2k − 4 is a non-negative integer. By Hölder’s inequality and two applications of Theorem 3.2 (with ℓ = 0 and ℓ = 1), it follows that
Combining this estimate with (1.1) and (2.5) establishes the upper bound in Theorem 1.1. To establish the lower bound in Theorem 1.1, we observe that two more applications of
Thus, by (2.5) and Theorem 3.1, we find that
Dividing both sides of the above inequality by N ( T ) and using ( 1.1 ), we establish the lower bound in Theorem 1.1. ● R EMARKS . (1) In order to establish the estimate for the lower bound for M k ( T ) in the above proof, one can use other exponents in Hölder’s inequality in place of 2/3k, (k – 2)/3k, and  ç . For instance, we could have instead used 3/4k,(2k – 3)/4k, and  ç . Other choices of exponents are possible, as well. (2) If it could be shown that
assuming the RH,
This would establish the correct order of magnitude for the moments M k ( T ).
4. Lemmas
In this section, we collect some estimates that we use to deduce Theorems 3.1 and 3.2.
This is Corollary A of Soundararajan [14]. Proof. ●
This is Lemma 5 of Heath-Brown [7]. Proof.
for t ≪ 0 or t ≫ 3T . Hence, using the estimate |ζ(σ + it)| ≪ c (|t| + 1)  w ich (on the RH) holds uniformly for σ ≥ ½ and | ≥ 1, we observe that
The lemma now follows by summing this estimate over the dyadic intervals [ T, T/2], [ T/2, T/4], [T/4,T/8],. . .. R EMARKS .
where d k (n) is the number of ways to express an integer n ≥ 1 as the product of exactly k positive integers. This follows, for instance, from Theorem 13.2 of Titchmarsh [16]. Therefore, the above lemma is only non-trivial if σ = σ(T) → ½ as T → ∞. lemma can be altered to show that ( 4.1 ) holds uniformly for ½ − c/logT ≪ σ ≪ ¾ for any fixed constant c > 0. In this case, the implied constant would depend on k , ε , and c .
uniformly for | t | ≪ T and T ≫ 2. Here χ ( s ) is the function defined in ( 2.1 ).
Proof. By Cauchy’s integral formula,
Applying Hölder's inequality to the integral over α on the left-hand side of the above equation,

5. Proof of Theorem 3.1
If k is 1 or 2, then the integral appearing in Theorem 3.1 can be estimated asymptotically using standard methods. Hence, we assume throughout the proof that k ≥ 2 is an integer (so that k − 1 is also a positive integer). Also, we let ε > 0 be an arbitrarily small positive constant which may not be the same at each occurrence. Throughout the proof, implied constants are allowed to depend on k and ε but are otherwise absolute. Following Conrey and Ghosh [2], we set
Using (2.1) and (2.2), one can show that | Z ′ ( t )| = | Z 1 ( z + it )| for every real number t ; this is assertion (i) of the main lemma in [2]. Hence,
Now let A(t) be a complex-valued function that is regular for real t . Then, by the Cauchy- Schwarz inequality and (5.2), it follows that
hand side of the above inequality and a lower bound for the integral appearing on the left-hand side for a specific choice of test function A(t). Let ξ = T θ for a fixed value of θ satisfying 0 < θ < 1 and set A(t) = A( ž + it) where
and d k (n) is the number of ways to express a natural number n as the product of exactly k positive integers. It t urns out that the value of θ , which we choose at t he end of the proof, is important in our argument. Since
for Rs > 1, we see that A(s) is an approximation to the kth power of the zeta-function. It is known that for each natural number k there exists a constant C k such that
and that
Using these estimates along with Montgomery and Vaughan's mean-value theorem for Dirichlet polynomials [10, Corollary 3], we see that
Assuming the RH, standard estimates for ζ(s), ζ ′ (s), χ ′ (s)/χ(s), and d k (n) imply that
uniformly for ½ < σ < ³ and 1 ≤ t ≤ T . Therefore, by Cauchy’s Theorem and (5.7), it
and
In order to estimate J 1 and J 2 , we replace ζ ′ (s) and ζ (s) by their corresponding Dirichlet series. Since these series converge absolutely, we can interchange the sums and the integrals, and then integrate term by term. We write
for Rs > 1 where
Then
Since
Here we have used the inequality | ab | ≪ | a | 2 + | b | 2 , which is valid for a, b ∈ C. Hence,
A similar calculation shows that
This completes the proof of Theorem 3.1.
6. Proof of Theorem 3.2
We may assume that ℓ > 0, as otherwise Theorem 3.2 corresponds to Soundararajan's result stated in Lemma 4.1. For α ∈ C, we claim that
uniformly for |α| < (log T )− 1 . When Rα ≥ 0, this claim is a consequence of Lemma 4.3. When Rα < 0, Stirling's asymptotic formula for the gamma function [10, Theorem C.1] can be used to show that
for some absolute constant C > 0 when | σ − ξ | ≪ (logT)− 1 and 0 ≪ t ≪ T . Therefore, by
linear combination of derivatives of ζ( ž + it ) and powers of log T , and then use ( 6.2 ). Recalling that | χ( f + it )| = | χ( f − it )| = 1 for real t , it follows from (2.2), the Leibniz rule for differentiation, and Lemma 4.4 that
for | t | ≪ T when T is sufficiently large. Thus,
as claimed. This completes the proof of Theorem 3.2. The author would like to thank Professor D. R. Heath-Brown for some Acknowledgements. helpful comments, which greatly simplified the proof of Theorem 3.2. The author would also like to thank the anonymous referee for a number of useful suggestions.