On the number variance of zeta zeros and a conjecture of Berry

Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta-function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed nonzero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non-universal regime. In this range, GUE statistics do not describe the distribution of the zeros. We also calculate lower-order terms in the second moment of the logarithm of the modulus of the Riemann zeta-function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).


Introduction
Understanding the distribution of the zeros of the Riemann zeta-function, ζ(s), is an important problem in number theory. Let N (t) be the number of zeros ρ = β + iγ of ζ(s) such that 0 < γ ≤ t and 0 < β < 1 (counted with multiplicity, where the zeros with γ = t are counted with weight 1 2 ). It is known that where, for t = γ, the function is defined by S(t) = 1 π arg ζ 1 2 + it with the argument obtained by a continuous variation along the straight line segments joining the points 2, 2 + it, and 1 2 + it starting with the value arg ζ(2) = 0. If t = γ for a zero of ζ(s), we define By equation (1.1) and the well-known estimates S(t) log t and T 0 S(t) dt log T , we can think of S(t) as the difference between the actual and average number of zeros around height t.
From (1.1), we expect that there are about δ zeros of ζ(s) with ordinates in the interval [t, t + 2πδ log T ] when 0 < t ≤ T and T is large. We define the number variance of the zeros of ζ(s) by This quantity has been studied by a number of authors, for instance [1,2,11,12,13,14]. By (1.1), up to a small error, the integral in (1.2) is equal to As described in Section 1.4, Berry [1] (see also [2]) has given a precise conjecture for the asymptotic behavior for this integral. In the universal regime of his model, when δ = o(log T ), Berry conjectured an asymptotic formula that matches exactly the variance of eigenvalues of GUE random matrices. However, when δ log T , in the so-called non-universal regime of his model, his conjecture is no longer described by the predictions from GUE and incorporates additional input from the primes.
Building upon ideas of Selberg [25] and Goldston [15], Gallagher and Mueller [14] and Fujii [12] have given a conditional proof of Berry's conjecture in the universal regime assuming both the Riemann Hypothesis (RH) and versions of Montgomery's pair correlation conjecture. In this paper, we introduce new ideas to prove novel results on the number variance of zeta zeros in the non-universal regime when δ log T . In particular, we show that new input from both the zeros and primes is needed in this regime, requiring information on the zeros beyond pair correlation (since we no longer expect GUE behavior in this range).
In Section 1.3, we give three different formulations of these results, stated as Theorems 1.3.1 -1.3.3. In Section 1.4, we show how our results give a conditional proof of Berry's conjecture in the non-universal regime assuming RH and a conjecture of Chan [5] for the pair correlation of zeta zeros in longer ranges (which examines how often normalized gaps between zeros can be close to a fixed nonzero value). Roughly, pair correlation studies the distribution of gap sizes localized near zero with respect to the average spacing, whereas our new results require information about the distribution of gap sizes localized near other points.
Before stating our new results on the number variance of zeta zeros, we first describe the work of Selberg [25,26] and Goldston [15] on the moments of S(t) and log |ζ( 1 2 +it)| and the connection to the pair correlation of zeta zeros. Analogous to a result of Goldston for S(t), our Theorem 1.2.1 gives lower-order terms for the second moment of log |ζ( 1 2 + it)| assuming RH, in terms of the pair correlation of zeta zeros. Assuming Montgomery's pair correlation conjecture, Theorem 1.2.1 establishes a special case of a conjecture of Keating and Snaith [18]. He proved this by estimating moments of S(t), first assuming RH and then later without any conditions with the same main term and a slightly weaker error term [25,26]. Assuming RH, Selberg showed that, for k ∈ N and T ≥ 3, In other words, the moments of S(t) are Gaussian. In this way, Selberg [27] deduces a central limit theorem for S(t): e −x 2 /2 dx. (1.4) This tells us πS(t) is normally distributed for t ∈ [T, 2T ] with mean 0 and variance 1 2 log log T , when T is large. Selberg (unpublished) also considered the moments of log |ζ( 1 2 + it)|. Using techniques outlined by Tsang [29], assuming RH it can be shown that These moments can be calculated unconditionally with a slightly weaker error term. A corresponding central limit theorem for log |ζ( 1 2 + it)|, analogous to (1.4), follows from the work Selberg and Tsang. See Radziwi l l and Soundararajan [24] for a recent and simplified proof of Selberg's central limit theorem for log |ζ( 1 2 + it)|.
1.2. The variance in Selberg's central limit theorem. Selberg modeled log ζ(s) near the critical line using information from the primes and the zeros of ζ(s). He arrives at the main term in (1.3) using information from the primes. The information about the zeros is cleverly contained in his error term.
Recall that the variance of a distribution is given by its second moment, which corresponds to taking k = 1 in (1.3). Goldston [15] gave a refined estimate for the variance of S(t) in Selberg's central limit theorem utilizing finer information from both the primes and the zeros of ζ(s) in his representation of log ζ(s). He does so through methods relying, in part, on Montgomery's work [20] on the pair correlation of the zeros of ζ(s). Assuming RH, Goldston shows that as T → ∞, where the constant a is given by γ 0 is Euler's constant, and the sum over p runs over the primes. Here, the term with F (α) captures the information from the zeros of ζ(s). The function F (α) was introduced by Montgomery [20] to study the pair correlation of the zeros of ζ(s), and is defined by for α ∈ R and T ≥ 2, where w(u) = 4/(4 + u 2 ), and the double sum runs over the ordinates, γ and γ , of non-trivial zeros of ζ(s). In this way, we see that Goldston's result contains information from both the primes and the zeros in the definition of the constant a. As initially defined, the constant a actually depends on T . In Lemma 3.1.3 we show that this dependence is mild (see also [15,Theorem 2]).
Our first theorem is an analogue of Goldston's more precise result for the second moment of log |ζ( 1 2 + it)|, refining the case k = 1 in (1.5). Theorem 1.2.1. Assume RH and let F (α) be defined by (1.7). Then, as T → ∞, where the constant a is given by (1.6).
Though the statement of our first result is very similar to Goldston's theorem, the proofs are considerably different. One reason for this is easy to explain. From the formula for N (t) in (1.1), we see that the function S(t) is bounded near the zeros of ζ( 1 2 + it), with a jump discontinuity at each zero. On the other hand, log |ζ( 1 2 + it)| is not bounded near the zeros, and can be arbitrarily large in the negative direction. These logarithmic singularities do not substantially change the end result, but they do cause technical difficulties within the proof. Another major difference from Goldston's work is that our proof relies on a delicate cancellation of main terms, which we accomplish by introducing the function g(x) in Section 2. Though an analogous cancelation of main terms was not present in Goldston's work, it is present in the work of Chirre and the third author who introduced related functions to obtain similar cancellations in the study of the second moment of the iterated antiderivatives of S(t) (see [8,Lemma 4]). However, as we shall see, when considering log |ζ( 1 2 + it)| there are important new technical differences in the properties of our functions due to the unbounded discontinuities.
Remark. The error term in Theorem 1.2.1 could be improved using additional assumptions such as a quantitative form of the twin prime conjecture (see [7]), or a more precise conjectural formula for pair correlation by Bogomolny and Keating [3] or Conrey and Snaith [9]. For details, see the work of Chan [6].
1.3. Number variance of zeta zeros. In a series of papers, Fujii [11,12,13] considered the 2kth moments of the difference S(t + ∆) − S(t). Using Selberg's methods, for T sufficiently large, Fujii [11] showed when 0 < ∆ 1 and, assuming RH, Fujii [13] showed that (1.8) gives an asymptotic formula when ∆ log T goes to infinity with T (sufficiently slowly). If ∆ log T 1, then the main term and error term in this result are the same order of magnitude and this result does not give an asymptotic formula. Fujii's result in (1.9) has an error term of the same order of magnitude as Selberg's conditional result in (1.3). In particular, when k = 1, the error term in (1.9) is O(T ). In this case, similarly to Goldston's result for the second moment of S(t), the contribution from the zeros of ζ(s) gives a term of size T , and Selberg's method cannot be used be used to identify this main term. Realizing this, Fujii [12] applies Goldston's methods [15] to his own work and, assuming RH, he as T → ∞. Gallagher and Mueller [14] had previously given a similar estimate in the limited range ∆ 1 log T assuming both RH and Montgomery's pair correlation conjecture. A calculation related to (1.10) can also be found in recent work of Heap [17,Proposition 9]. Notice that the expression of the main term in (1.10) is stated using information from the zeros, in the form of F (α). As we shall see, more information about the distribution of the zeros of ζ(s) is required in order to accurately describe the situation when ∆ 1.
Our next results refine Fujii's calculation in (1.10) by giving an asymptotic formula of similar precision but with a much larger range of ∆. This requires expressing the main term in a different manner, giving a better understanding of the behavior of the number variance for zeta zeros for different sizes of ∆. To achieve this, we must overcome significant technical challenges, as new main terms arise and a more careful consideration of the error terms is required. Our result relies on finer information from both the primes and the zeros of ζ(s). In particular, we require a variation of Montgomery's function F (α) introduced by Chan [5] in his study of the pair correlation of zeros in longer ranges. We define 1 and we prove the following theorem.    Then, as T → ∞, (1.14) Using Theorem 1.3.1, the function c(v) can also be written in terms of the Taylor series expansion of about t = 0. Note that, for 1 ≤ y < 2, we have E(y) = −y log y + y, and the prime number theorem Our second reformulation of Theorem 1.3.1 illustrates the input from the primes and the zeros in a simpler way. As we shall see in the next section, this has the advantage of allowing for a simple comparison with a conjecture of Berry [1].  2 dt = T π 2 n≤T Λ 2 (n) n log 2 n (1 − cos(∆ log n)) The proofs of all of our theorems rely on knowledge of F (α) and F ∆ (α) for |α| ≤ 1. It is known that F (α) is real-valued, positive, and even. Moreover, refining Montgomery's original work [20], Goldston and Montgomery [16] showed that In contrast, the function F ∆ (α) is no longer positive, nor real, nor even; however, it satisfies the symmetry relations Combining the methods of Chan [5, Theorem 1.1] with Goldston-Montgomery [16], it can be shown that uniformly for 0 ≤ α ≤ 1 and small ε > 0.

1.4.
A conjecture of Berry. The Hilbert-Pólya conjecture states that the imaginary parts of the zeros of ζ(s) correspond to the eigenvalues of some self-adjoint operator, and this would imply RH. In 1973, as a consequence of his work on the pair correlation of zeros, Montgomery [20] was led to conjecture that the zeros of ζ(s) are distributed as the eigenvalues of a random matrix from the Gaussian unitary ensemble (GUE), giving support to a spectral interpretation of the zeta zeros. Montgomery's conjecture is supported by numerical evidence of Odlyzko [23], which suggests that the GUE model holds for short-range statistics between zeros, such as the distribution of the gap between consecutive zeros. However, Odlyzko's evidence shows that the GUE model fails for long-range statistics, such as the correlation between zeros that are very far apart. In this case, Berry [1] suggested that these long-range statistics are better described in terms of primes, instead of GUE statistics.
Berry [1] proposed a conjectural model for the zeros of ζ(s) as the eigenvalues of a quantum Hamiltonian operator. His model is expected to conform to the behavior of both short-range and long-range statistics of zeros, as described above. In 1988, Berry used his model to conjecture an asymptotic formula (in terms of our notation) for (1.18) As described above, the universal regime of his model is when δ = o(log T ), while the non-universal regime corresponds to δ log T .
We first briefly describe Berry's conjecture following his notation. For E > 0, define be the renormalized zeros of ζ(s), so that the sequence x m has average spacing 1. In what follows, we let E, x, and ∆x be three large parameters, which we think of as going to infinity, satisfying the relations ∆x = o(x) and In particular, note that We define the variance as Finally, we let τ * be another parameter, such that τ * = Φ(E) log(E/2π) for some function Φ(E) that goes to infinity as E → ∞. Berry's conjectural formula [1,Eq. (19)] states that The right-hand side does not depend on the choice of τ * , as E → ∞. The universal regime is when where the term in the first set of brackets is the dominant (leading order) term. See also [1, Eqs. (20) and (21)] for simplifications in different ranges of L. Translating to our normalization and our notation, Berry made the following conjecture.
In 1990, Fujii [12] proved an asymptotic formula for (1. uniformly for α in compact intervals as T → ∞. 2), and it again arises from estimating a sum over primes in Lemma 4.2.1 below. In the following sections, we attempt to state each lemma in the largest possible range of ∆ to clarify where these restrictions appear.

2.
A representation formula for log |ζ(1/2 + it)| 2.1. Some auxiliary functions. Following the ideas developed by Goldston [15], we must obtain a representation formula for log |ζ(1/2 + it)| in terms of a Dirichlet polynomial supported over prime powers and a sum over the zeros of ζ(s). This is based on an explicit formula of Montgomery [20] and, in our case, requires introducing three auxiliary, real-valued functions, whose technical properties play important roles in our proof. For u ∈ (0, 2), define (a) we have g ∈ C ∞ (−2, 2) and g is even; Remark. We highlight that h has an unbounded but integrable singularity at the origin, which is different from the situation in both [15] and [8]. We also note that f, g, and their derivatives are uniformly bounded on the interval [0, 1].
Proof. First we consider g(u) as defined in (2.2). By the Dominated Convergence Theorem, we have that g ∈ C ∞ (−2, 2). The fact that g is even follows from the fact that cosh(y) is even. Now let u ∈ (0, 2). Then for all u > 0, we know Using this representation for 1 u , it follows that as claimed. Next we consider h(u) defined in (2.3). First, note that h(u) is even by construction. Next, we which implies h ∈ L 1 (R). Next, we calculate the Fourier transform of h(v) using the well known Fourier pair ϕ(y) = e −2π|y| and ϕ(ξ) = 1 π Let a ∈ R. Since h is even, we may assume a ≥ 0. Using the variable change w = v u and (2.4), it follows that Clearly, h ∈ L 2 (R), and therefore h ∈ L 2 (R). This completes the proof.
Proof. We begin with the fact that, for t = γ, we have Now we use a slightly modified version of an explicit formula of Montgomery [20] (see [8, Eq. (2.6)]). For This immediately follows from [8, Eq. (2.6)] by letting n = 1 therein and taking real parts. Dividing by and integrating (2.5) from 1 2 to infinity, for x ≥ 4, t ≥ 1, and t = γ we have By using the substitution, u = (σ − 1 2 ) log x, the integral in the second main term of (2.6) yields where f is defined in (2.1). Again, by the same substitution, for the first main term of (2.6), we have where h is defined in (2.3). Finally, the fourth term of (2.6) equals The other terms are error terms and can be treated similarly to the proof of [15, Lemma 1]. Combining all the terms of (2.6) completes the proof.
Note that we have the extra main term log 2 log t/2π log x when compared to Goldston's formula for S(t) in [15, Lemma 1]. This comes from Stirling's formula when analyzing the real part of ζ ζ (s), and it does not appear when taking the imaginary part. We use Lemma 2.2.1 to obtain an expression for the quantities we want to compute in Theorems 1.2.1 and 1.3.2. We now adopt some notation for the expressions we will consider.
Henceforth, let T ≥ 4 and ∆ = ∆(T ) be a function of T such that 0 < ∆ T b , for some fixed 0 < b < 1.
For t ≥ 1, denote so that A(t) contains the information on primes and B(t) contains the information on zeros in our expression In the next result, we use Lemma 2.2.1 to write the objects in Theorems 1.2.1 and 1.3.2 in terms of the above expressions G i , H i , and R i .
Proof. By rearranging the terms in Lemma 2.2.1, we have Squaring the above expression and then integrating from 1 to T yields where we used the Cauchy-Schwarz inequality to bound the first error term on the left-hand side. For the second error term on the right-hand side of (2.9), since f (v) is uniformly bounded for all v ∈ [0, 1], | cos(v)| ≤ 1 for all v ∈ R, and T 1 n it log t dt log T for n ≥ 2, we see that To control the last error term on the right-hand side of (2.9), consider the antiderivative of log |ζ( 1 2 + it)|. Assuming RH, it is known that (See [4, Lemma 2.2] for a slightly stronger estimate.) Thus, using integration by parts, we obtain By combining and rearranging all the calculations for the terms in (2.9), we complete the proof of part (a).
For the proof of part (b), since ∆ T b , we observe that for t > 1, x ≥ 4, and ε > 0 the mean-value theorem implies that so that this term is absorbed into the error bound. The rest of the proof of part (b) is analogous to the proof of part (a). Consequently, the proof is complete.
In order to conclude the proofs of Theorems where the implied constants are universal.
Proof. Consider the identity where the implied constants are universal.
Proof. By interchanging γ and γ , we may assume that ∆ ≥ 0. We have say. We use the inequality 4|u| 4 + u 2 ≤ min 1, 4 |u| and the fact that there are O(log T ) zeros in the interval [T − ∆ − 2, T − ∆] to estimate Z 1 as follows: For part (b), since 0 < H ≤ T , we use that for 0 ≤ n ≤ T + H) there are O(log T ) zeros in the interval (n, n + 1) to obtain: In the last line, since the summand is positive, we may bound the sum over n by a sum over all integers and then use the fact that function converges to a continuous periodic function of x ∈ R. In particular, it is uniformly bounded.
Proof. First, we prove a pointwise estimate for F ∆ (α, T + H) that holds in the larger range |∆| ≤ T and is useful for both parts (a) and (b). By the mean-value theorem, for θ ∈ R, we have Therefore, for ∆ ∈ R with |∆| ≤ T , we have To bound the last error term, one can see that say. By the mean-value theorem and part (a) of Lemma 3.1.2, we find that Y 1 H T |α| log 2 T . Since w(u) ≥ 0, we may extend the sum in Y 2 to apply part (b) of Lemma 3.1.2. Therefore, Y 2 1 T (H + |∆| + 1) log T. We estimate Y 3 by further dividing the sum into two parts: where we used that w(u) ≥ 0 to extend the first sum and applied parts (b) and (c) of Lemma 3.1.2, respectively. This yields Y 3 1 T (H + |∆| + 1) log 2 T . Y 4 can be treated similarly to Y 3 , since we may interchange γ and γ , use that w(u) is even, and replace ∆ with −∆. Combining the above estimates, we obtain that uniformly for α ∈ R, T ≥ 4, 0 < H ≤ T , and ∆ ∈ R with 0 ≤ |∆| ≤ T .
We now use the pointwise estimate (3.2) to prove part (a) as follows. It is known that uniformly for β ≥ 0 and T ≥ 4 (see [15,Lemma A]). Integrating by parts, for β ≥ 1 this implies that Therefore, by the case ∆ = 0 of the estimate (3.2), we obtain This proves part (a). Part (b) is similar, using that |∆| ≤ log 2 T and Lemma 3.1.1 in place of (3.3).

3.2.
Unbounded discontinuities. In this section, our goal is to express R i as a sum over pairs of zeros of ζ(s) in order to apply Montgomery's pair correlation method to estimate R i . The arguments of Montgomery and Goldston consist of localizing the sum to zeros in the interval [0, T ] and then extending the integral in the definition of R i in (2.8) to infinity, up to small errors. However, due to the unbounded discontinuity of our weight function h at the origin, their arguments do not apply directly. This leads to difficulties, and we must use a different and delicate approach to control the error terms in this case. The first part of this approach lies in the introduction of a sequence of T n 's for which the following lemmas will hold. The idea of using such a sequence is classical (for instance, see [10,Ch.17]). Since N (T + 1) − N (T ) log T , by the pigeonhole principle, for every n ∈ N we can find a sequence {T n } satisfying n ≤ T < n + 1 and |γ − T n | 1 log n . (3.4) In this way, we obtain similar results to Goldston on a sequence of points tending to infinity, despite the unbounded discontinuity of our function h. Now, we define and we prove the following lemma.
Proof of part (a). First note that for γ = t, using an argument of Goldston [15, p. 158], we find that where I = {γ : T < γ ≤ T + 1 log x }. We now show that the terms in the sum for which γ / ∈ [0, T ] contribute an amount of size o(T ) to R 1 . Using (3.6) and (3.7), we restrict the interval of zeros within the sum in R 1 to γ, γ ∈ [0, T ]. Then by expanding the integral, we rewrite R 1 as (3.8) Integrating the third error term on the right-hand side of (3.8) gives Using the facts that h ∈ L 1 and |I| log T log x + log T log log T , the second error term on the right-hand side of (3.8) reduces to Similarly, the fourth error term on the right-hand side of (3.8) yields We introduce a parameter 1 < H < T to split the range of integration for S 1 as follows: To estimate S 11 , we note that T − t + 1 ≥ H + 1, extend the sum over γ , and use that h ∈ L 1 . We find that For S 12 , we use that T − t + 1 ≥ 1 and extend the sum slightly to obtain To balance these two error terms, we choose H = √ T . Therefore, we conclude that We estimate S 2 similarly, by splitting the range of integration from 1 to H and from H to T. We again find that By combining the estimates for S 1 and S 2 , the fourth error term on the right-hand side of (3.8) can be estimated as For the first error term of (3.8), we again split the range of integration and find that For γ ∈ I and t ∈ [1, T − 1], we know h[(t − γ) log x] 1 (t−γ) 2 log 2 x . Since h ∈ L 1 , by an argument similar to the proof of (3.7), we see where we used (3.9) in the last line. Since T ∈ {T n }, we know that |γ − T | 1 log T . Thus for t ∈ I, we have that T − 1 ≤ t ≤ T and T < γ ≤ T + 1 log x imply |t − γ| 1 log T . Since |I| < 1 and x ≥ 4, using that Hence, since γ is contained in an interval of size less than 1, it follows that for all T ∈ {T n }. Hence combining our estimates for Σ 1 and Σ 2 gives Therefore, R 1 is confined to γ, γ ∈ [0, T ] with an added error of O( √ T log 2 T ). Similarly, we may extend the integral range of [1, T ] to (−∞, ∞) with the same error. Thus, We now use the properties of h(v) expressed in Lemma 2.1.1 to simplify our expression for R 1 . Since h ∈ L 1 and it is even, we can use the substitution u = (t − γ ) log x together with convolution to find that Since h ∈ L 1 , we know that convolution is well-defined and h * h =ĥ 2 . Furthermore, from Lemma 2.1.1, we know that h ∈ L 2 , and therefore k(ξ) = 1 π 2 h(ξ) 2 ∈ L 1 . Thus by Lemma 2.1.1, (3.5), and the properties of Fourier Transform, we have as claimed.
Proof of part (b). The proof here is similar, but we highlight some important differences. Recall that where we defined B(t) in (2.7). First, by Lemma 3.1.3 and part (a) of Lemma 3.3.2, since ∆ Therefore, we find that where I ∆ = {γ : T < γ ≤ T + ∆ + 1 log x }. Note that |I ∆ | (∆ + 1) log T . By computations similar to those of part (a), we find that The next step is different from the steps in the proof of part (a). To bound the last error term in (3.11), we use the Cauchy-Schwarz inequality: (3.12) To estimate J 1 , we expand the integral, apply Cauchy-Schwarz once more, and use that h ∈ L 2 and |I ∆ | (∆ + 1) log T . This gives To estimate J 2 , we use (3.6) to extend the sum over zeros to the interval [0, T + 1], together with the bound N (T + 1) T log T and the fact that h ∈ L 1 . This yields where we used part (a) of Lemma 3.3.2. Combining (3.11), (3.12), (3.13), and (3.14), we obtain Similarly, the integral above may be extended to R up to the same error term. The rest of the proof is analogous to part (a).

3.3.
A modified pair correlation approach. The next step is to introduce the weight function w(u), from (1.7), to write R 1 and R 2 in Lemma 3.2.1 in terms of Montgomery's function F (α) and Chan's function Proof. The proofs of the expressions in parts (a) and (b) are proved using similar methods, but the proof of part (b) is more involved. For this reason, we only work out part (b). Recall that k is the function defined in (3.5). We have that k(y) min(1, 1 y 2 ). From this estimate we introduce the weight function w(u), defined in (1.7), into the sum over zeros using the following argument. We consider the difference Using the facts that N (T ) T log T , there are O(log t) zeros in any given interval [t, t + 1], and that ∆ ≤ T , we have Therefore, Similarly, we may introduce the weight w(u) into the the other terms in the representations of R 1 and R 2 in Lemma 3.2.1 to complete the proof.
Using Lemma 3.3.1 and the properties of F (α) and F ∆ (α), we take x = T β and proceed to estimate R i .
where the error term on part (a) is of size O T log log T log T , and the error term on part (b) is of size Proof of part (a). Recall the definition of the function F (α) and w(u) in (1.7). Then using the definition of Fourier transform, we manipulate the sum over zeros in the representation formula for R 1 in Lemma 3.3.1 to yield Then, inputting (3.15) into part (a) of Lemma 3.3.1 gives Recall from (3.5) that k(u) is piecewise defined with a transition at u = 1/(2π). Thus, we use (1.15) and the fact that F (α) and k(u) are both even and nonnegative functions to rewrite the above integral over k For the second integral on the right-hand side in (3.17), because β is fixed, we know that To compute the first integral on the right-hand side of (3.17), we use the facts that k α 2πβ = g 2 α β for 0 ≤ α ≤ β, that 0 < β ≤ 1 is fixed, and that k is smooth near the origin and uniformly bounded. By technical yet straightforward manipulations, we find that Combining the estimates (3.18) and (3.19) yields Inputting (3.20) into the representation for R 1 in (3.16) concludes the proof of part (a).

Proof of part (b).
We consider the definition of the function F ∆ (α) in (1.11). Then using the definition of Fourier transform, we manipulate the sum over zeros in the representation formula for R 2 in Lemma 3.3.1 to yield Then, inputting (3.15) and (3.21) into part (b) of Lemma 3.3.1 gives By splitting the second integral above using the symmetry relations for F ∆ in (1.16), we have Next, we divide the integral over the intervals (0, β), (β, 1), and (1, ∞), and apply (1.17). Consequently, since k(u) is even, k(0) = g 2 (0), and T iα∆ + T −iα∆ = 2 cos(∆α log T ), we obtain By combining the above integrals, we have that From the proof of part (a) (see (3.20)), we know that By adding (3.22) and (3.23) together, our asymptotic formula for R 2 reduces to which completes the proof.

Contributions from the primes
In this section, we estimate the expressions G i + H i . First, we obtain intermediate expressions for G i and H i separately.

4.1.
Expressions for G i and H i . We begin with a useful Lemma that helps estimate the second moment of some trigonometric polynomials. Then, T 0 ∞ n=1 a n cos(t log n) The implied constants are universal.

Proof. A classical result of Montgomery and Vaughan [21, Corollary 3] states that, for complex numbers
For part (a), let z := ∞ n=1 a n n −it , and note that Re z = ∞ n=1 a n cos(t log n). Consider the identity (Re z) 2 = |z| 2 + Re (z 2 ) 2 . We write z = a 1 + ∞ n=2 a n n −it and use that, for n ≥ 2, we have n −it dt = in −it / log n. This yields Here, we used the Cauchy-Schwarz inequality to obtain Combining Then, we apply the same argument above with z = ∞ n=1 a n (n −ihn − 1) n −it , using Montgomery and Vaughan's result (4.1) with b n = a n (n −ihn − 1).
Using the previous lemma, we obtain the following expressions for G i .
Next, with the goal of studying H 1 and H 2 , we use some estimates of Goldston, together with some trigonometric identities, to obtain expressions for the real and imaginary parts of integrals involving log ζ(1/2 + it) times trigonometric functions. Some of these results appear previously in [15] (part (b)) and implicitly in [12] (part (d)). We collect them all in the following lemma, for the reader's convenience. Proof. Assuming RH, Goldston [15, p. 169], improving upon a contour argument of Titchmarsh [28], showed that This completes the proof of the lemma.
We now obtain expressions for H i from the above lemma. As mentioned in the introduction, in the next lemma we use Theorem 1.2.1 to control some of the error terms in part (b), which is the part of the lemma that is relevant to Theorem 1.3.2.
Proof. Part (a) follows from part (a) of Lemma 4.1.3 and the definition of H 1 . For part (b), we rearrange the terms and use a change of variables to find that Therefore, we see that For the second term, unconditionally, note that the prime number theorem with error term implies that P (y) := n≤y Λ 2 (n) n = log 2 y 2 + O(1).

4.2.2.
Proof of part (b): Summing by parts. Similarly, we have Using summation by parts, integration by parts, and the prime number theorem with error term, we obtain To estimate the first term on the right-hand side of (4.12), consider the quantity M (y) := n≤y Λ 2 (n) = y log y − y + E(y), so that dM (y) = log y dy + dE(y) and E(y) = O N y log N y is defined in (1.13). For this, we let 1 < < 2 be a parameter. We anticipate that we will eventually take → 1 + . Then, the sum in the first term of (4.12) is We use the change of variables u = ∆ log y in the first integral. For the second integral, we integrate by parts and use that E( ) = − log to find that [−∆ log y sin(∆ log y) + (1 − cos(∆ log y))(log y + 2)]dy (4.14) Here, we used that E(y) N y log N y (for any N > 0) to extend the last integral to infinity, up to an error term. Now, we let → 1 + . Note that Additionally, since E(y) = y − y log y for all 1 ≤ y < 2, the second integrand above satisfies, in this range, E(y) y 2 log 3 y [−∆ log y sin(∆ log y) + (1 − cos(∆ log y))(log y This shows that the second integral is absolutely convergent on (1, ∞). Therefore, recalling that x = T β and ∆ log 2 T , we may let → 1 + in (4.14) to find that uniformly for ∆ > 0. We use this fact and combine the first three terms on the right-hand side of (5.1) to where we used integration by parts in the last line. Next we consider the integral involving g 2 (v) on the right-hand side of (5.1). Using integration by parts, we similarly see that Combining these simplified expressions together gives where c(v) is defined in (1.14).
where the implied constant is universal.
Proof. This can be established using classical arguments in a similar manner to the proof of the prime number theorem (assuming RH), e.g. [22,Chapter 13].
By the mean-value theorem, cos(∆α log T + 2π∆u) = cos(∆α log T ) + O(∆|u|). We also have the identity The rest of the proof consists of controlling this last error term. This requires a technical but straightforward modification of Montgomery's arguments and definitions in [20], which we define and prove in the appendix.
In particular, we define F σ0 (α) in (6.1), which is a modification of F (α) by using a slightly different weight.
Hence the desired result now follows.  Then, integrating by parts, we find that In particular, F σ0 (α) ≥ 0, and F σ0 is even. Following Montgomery [20] (see also [16]), we have the following asymptotic formula for F σ0 (α).
We write the above as L(x, t) = R(x, T ). Note that We note that on the left-hand side of (6.4) we may use an estimate of Goldston   If we choose x = T α for 0 ≤ α ≤ 1 − ε, then following Montgomery's argument the above estimates imply that R(T α , T ) = T log T T −2ασ0 log T (1 + o(1)) + α σ 0 + o(1) .
We combine this with (6.3) to obtain the desired result for |α| ≤ 1 − ε. As remarked above, by the argument of Goldston and Montgomery [16,Lemma 7], this can be extended uniformly to |α| ≤ 1.
We also note that the following estimate holds.