PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 88(102) (2010), 99–110 DOI: 10.2298/PIM1002099C

ON THE SELBERG INTEGRAL OF THE k-DIVISOR FUNCTION AND THE 2k-TH MOMENT OF THE RIEMANN ZETA-FUNCTION

Giovanni Coppola

Communicated by Aleksandar Ivić

Abstract. In the literature one can find links between the 2k-th moment of the Riemann zeta-function and averages involving d_{k}(n), the divisor function generated by ζ^{k}(s). There are, in fact, two bounds: one for the 2k-th moment of ζ(s) coming from a simple average of correlations of the d_{k}; and the other, which is a more recent approach, for the Selberg integral involving d_{k}(n), applying known bounds for the 2k-th moment of the zeta-function. Building on the former work, we apply an elementary approach (based on arithmetic averages) in order to get the reverse link to the second work; i.e., we obtain (conditional) bounds for the 2k-th moment of the zeta-function from the Selberg integral bounds involving d_{k}(n).

1. Introduction and statement of the result

We shall link the 2k-th moment of the Riemann ζ-function ζ(s) on the (critical) line (σ = Re s = \frac{1}{2}), see [10]:

I_{k}(T) =^{def} ∫_{0}^{T} |ζ(\frac{1}{2} + it)|^{2k}dt

(which we shall abbreviate with I_{k}, not to be confused with the similar 2k-th moment off the line, i.e.,

I_{k}(σ, T) =^{def} ∫_{0}^{T} |ζ(σ + it)|^{2k}dt (\frac{1}{2} < σ < 1),

2010 Mathematics Subject Classification: Primary 11M06; Secondary 11N37.

compare with [10]), with the Selberg integral of the k-divisor function, d_{k}(n) (having the generating Dirichlet series ζ^{k}(s))

J_{k}(x, h) =^{def} ∫_{hx^{ε}}^{x} | ∑_{t<n⩽t+h} d_{k}(n) − M_{k}(t, h)|^{2}dt,

(compare with [1]; henceforth we abbreviate J_{k}(x, h) by J_{k}) where, say, M_{k}(t, h) is the “expected value” of the (inner) sum.

This gives d_{k} over the (say) “short interval” [t, t+h] (as h = o(t) ∀t ∈ [hx^{ε}, x]); here and in the sequel ε > 0 may be arbitrarily small, but not necessarily the same at each occurrence.

Actually, Ivić gave in [12] a nontrivial bound for J_{k}(x, h) when the width of the s.i. (abbreviation for short interval), namely θ := \frac{log h}{log xn} is greater than θ_{k} =^{def} 2σ_{k} −1 (with σ_{k} the Carlson’s abscissa, i.e., inf{σ ∈ ]1/2, 1[ : I_{k}(σ, T) ≪ T }, here):

θ > θ_{k} ⇒ ∃δ = δ(k) > 0 : J_{k}(x, h) ≪ \frac{xh^{2}}{x^{δ}}

(with the trivial bound: J_{k}(x, h) ≪ xh^{2}(log x)^{c}, where c = c(k) > 0, see the following).

This result clearly gives nontrivial bounds for J_{k}, using the information for zeta-moments off the critical line. For example, θ_{3} = \frac{1}{6}, θ_{4} = \frac{1}{4}, θ_{5} = \frac{11}{30}, … (from the known values of σ_{k}).

Thus the knowledge of the moments of ζ(s) provides information on d_{k}(n) in almost all short intervals (a.a.s.i.).

(See: J_{k} nontrivial ⇒ ∑_{t<n⩽t+h} d_{k}(n) ∼ M_{k}(t, h), a.a.s.i.)

However, we can also go in the opposite direction: if we have some kind of nontrivial information about the distribution of d_{k}(n), we can improve our knowledge (at least, on the 2k-th moments) of the Riemann ζ-function. Actually, this idea is due to Ivić, who linked I_{k} to the “(auto-)correlation” of d_{k} with “shift-parameter” a, i.e.,

𝒞_{k}(a) =^{def} ∑_{n⩽x} d_{k}(n)d_{k}(n + a), a ∈ ℕ (here x ∈ ℕ, x → ∞)

(the shift is a positive integer: 𝒞_{k}(−a) is close enough to 𝒞_{k}(a) and 𝒞_{k}(0) is relatively easy to compute).

Here it comes into play the idea of Ivić (see [11]) of linking the estimate of the 2k-th moment, I_{k}(T), to a sum of correlations 𝒞_{k}(a) performed over a (the shift), up to (roughly, we avoid technicalities), say, h := \frac{x}{T} (the s.i. comes in!), where x ≪ T^{k/2}.

In order to be more precise, we need to abbreviate (with x, X or even T our main variables, all independent and → ∞):

A ≪ B ⟺^{def} ∀ε > 0 A ≪_{ε} x^{ε}B

SELBERG INTEGRAL OF d_{k} AND 2k-TH MOMENT OF ζ 101

i.e., the modified Vinogradov notation ≪ allows us to ignore all the arbitrarily small powers. We shall also say that the arithmetic function (a.f.) f : ℕ → ℝ is essentially bounded, write f ≪ 1, when ∀ε > 0 f(n) ≪_{ε} n^{ε} (as n → ∞). For example, all the d_{k} (∀k ∈ ℕ) are essentially bounded:

∀k ∈ ℕ d_{k} ≪ 1,

whence they contribute individually small powers, which for our purposes may be ignored. Shiu [19] obtains (see J_{k} for trivial estimate above) a kind of Brun– Titchmarsh estimate for (suitable multiplicative a.f., like) d_{k}, and these give, on average over (all) s.i., powers of log. By the way,

L := log T (or L := log x)

is the abbreviation for the logarithm of our main variable.

We quote the formula (proved ∀k ⩽ 2, see §3) for the d_{k} correlations:

(1.1)  𝒞_{k}(a) = xP_{2k−2}(log x) + Δ_{k}(x, a), Δ_{k}(x, a) = o(x);

here, the (conjectured, ∀k > 2) main term is xP_{2k−2}(log x) ≪_{k} xL^{2k−2} ≪ x (since P_{2k−2}(z) is a polynomial of degree 2k − 2 in z, see the following).

Here, it seems that the first to propose explicitly this form for (1.1) is Ivić, who also gave explicitly the polynomial P_{2k−2}, that is essentially bounded (w.r.t. x). However, as we shall see in a moment, it depends, also, on the shift a > 0.

We shall sketch now, avoiding technicalities, Ivić’ s argument. After some work (expand the square & mollify, take relevant ranges, …) he gets that I_{k}(T) is

I_{k}(T) = I′′_{k}(T) + O_{ε}(T^{ε}T)

with

I′′_{k}(T) =^{def} \frac{1}{M} ∑_{a⩽h} ∑_{M<n⩽M′} d_{k}(n)d_{k}(n + a) ∫_{T/2}^{2T} ϕ(t)e^{ita/n}dt,

where M < M′ ⩽ 2M, with M ≪ T^{k/2}, say h ≪ M/T, the smooth (i.e., ∈ C^{∞}) test-function ϕ has support in ]T/2, 2T [, ϕ([3T/4, 4T/3]) ≡ 1, and has good decay

ϕ^{(R)}(t) ≪_{R} T^{−R}, ∀R ∈ ℕ.

From now on (see the reason in next section) we can ignore (in bounds for I_{k}) all terms which are ≪ T.

We give an idea of the shape of P_{2k−2}, given by Ivić, before we proceed. It is (see [11] for details)

P_{2k−2}(log x) =^{def} \frac{1}{x} ∫_{0}^{x} ∑_{q=1}^{∞} \frac{c_{q}(a)}{q^{2}}R_{k}^{2}(log t)dt,

with, say,

R_{k}(log t) =^{def} \frac{C_{−k}(q)}{(k − 1)!} log^{k−1} t + \frac{C_{1−k}(q)}{(k − 2)!} log^{k−2} t + ⋯ + \frac{C_{−2}(q)}{1!} log t + C_{−1}(q)

depending on q, but not on a (this is vital); also, w.r.t. x, R_{k}(log t) ≪ 1 and this is very useful. We shall see in a moment that the shape of these C_{j}(q) is important only in the case q = 1. By the way, here c_{q}(a) is the Ramanujan sum, defined as (^{∑∗} denotes summation over j (< q) coprime with q)

c_{q}(a) =^{def} \begin{matrix}∑ & ∗ \\ j (mod q) \end{matrix} e_{q}(ja) = ∑_{\begin{matrix}d|q \\ d|a \end{matrix}} dμ(\frac{q}{d}).

Hence, say, S(a) =^{def} max(0, h − |a|) (Ŝ is Fejér’s kernel) gives (J_{k} is coming soon)

Ŝ(\frac{j}{q} ) =^{def} ∑_{a} S(a)e_{q}(ja) ⩾ 0 ⇒ ∑_{a} S(a)c_{q}(a) ⩾ 0

and from the elementary identity, ∀d ∈ ℕ, (hereon n ≡ r(q) is n ≡ r (mod q) abbreviation)

∑_{\begin{matrix}a \\ a≡0 (d) \end{matrix}} S(a) = h + 2 ∑_{b⩽h/d}(h − db) = \frac{h^{2}}{d} + d\{\frac{h}{d}\}(1 − \{\frac{h}{d}\}),

we get (apply c_{q}(a), above), writing 1_{℘} = 1 if ℘ holds, = 0 else:

∑_{a} S(a)c_{q}(a) = 1_{q=1}h^{2} + ∑_{d|q} d^{2}μ(\frac{q}{d})\{\frac{h}{d}\}(1 − \{\frac{h}{d}\}).

(It is here evident that q = 1 has a greater importance.) Thus, (see Ivić [11] and compare [1]):

(1.2)  ∑_{a} S(a)xP_{2k−2}(log x) = h^{2} ∫_{hx^{ε}}^{x} R_{k}^{2}(1, log t)dt + tails,

where we mean, by “tails”, remainders which are ≪ h^{3}. Here, the part of R_{k}(log t) term with q = 1 is, say,

R_{k}(1, log t) =^{def} \frac{C_{−k}(1)}{(k − 1)!} log^{k−1} t + \frac{C_{1−k}(1)}{(k − 2)!} log^{k−2} t + ⋯ + \frac{C_{−2}(1)}{1!} log t + C_{−1}(1)

and gives (see the above) the term M_{k}(log t) into the Selberg integral; as it should be, since (from an elementary version of Linnik’s Dispersion method, compare [1, Lemmas]), assuming (1.1) with this P_{2k−2}, we get

(1.3)  J_{k}(x, h) ∼ ∑_{a} S(a)𝒞_{k}(a) − h^{2} ∫_{hx^{ε}}^{x} M_{k}^{2}(log t) dt ∼ ∑_{a} S(a)Δ_{k}(x, a),

where ∼ means ignoring “tails” (see above) and “diagonals” i.e., remainders ≪ xh. We remark that both these errors are negligible (at least, for k = 3, 4, see Section 5), since they both contribute ≪ T to I_{k}(T).

Then, due to the expression for I′′_{k} , Ivić [11] made a hypothesis about (avoiding technicalities) the sums of Δ_{k}(x, a) (remainders into (1.1) above), performed over the shift a, say G_{k}, which implies the bound I_{k}(T) ≪ T (for the same k > 2). Henceforth we assume that k > 2.

Of course, he does not need (1.1) to hold individually ∀a (⩽ h, here), but he observes that he is summing up, into G_{k}, without the modulus over the remainder, Δ_{k}(x, a), so some a-cancellation can take place.

So far, he passes from an asymptotic formula (1.1) to an a-averaged form of it, which is easier to prove (however, nobody has done it yet!).

Here, with applications in mind, we pass from a single average to a double average.

Building on his expression for I′′_{k}, it is possible to make a less stringent hypothesis, to have a more flexible procedure for the remainders Δ_{k}(x, a).

We use, also, our previous work on the Selberg integral of the a.f. f (essentially bounded and real), compare [1], in order to let the Selberg integral of d_{k}, i.e., J_{k}(x, h), come into play. (It is a kind of “double average” of Δ_{k}(x, a).)

Unfortunately, due to an exponential factor multiplying d_{k}(n)d_{k}(n + a) into 𝒞_{k}(a) we cannot get a link with I_{k}(T) using only J_{k}(x, h) (with h ≪ \frac{x}{T}, x ≪ T^{k/2}), but we also need to make a hypothesis on another double average of remainders Δ_{k}(x, a). Furthermore, our bounds are also affected by the limitation k ⩽ 4, due to some error terms arising from the Linnik method (compare (5.1) proof in Section 5).

We shall formulate our original Theorem in Section 4, and then present its proof.

There is a way to improve the result (with a different proof, see Section 5) as follows.

Theorem 1.1. Let T → ∞ and, ∀ε > 0, T^{1+ε} ≪ M ≪ T^{k/2}. Then ∀k > 2 we have

I_{k}(T) ≪ T (1 + max_{T ≪M≪T^{k/2}} \frac{T}{M^{2}} max_{0<h≪M/T} J_{k}(M, h)) .

In the next two sections we shall briefly mention some history of I_{k} and the (related) additive divisor problems. Then, we shall prove our result in the subsequent section, before some remarks, and in the concluding section we shall prove Theorem 1.1.

Acknowledgement. The author wishes to express his gratitude to the referee for pointing out how to obtain the improved version of Theorem 1.1.

2. A concise history of moments of the Riemann zeta-function

We should keep in mind, here, that for fixed k ∈ ℕ we seek the bound

I_{k}(T) = ∫_{0}^{T} |ζ(\frac{1}{2} + it)|^{2k}dt ≪ T (2k-th moment problem)

(it is “on the line”, since σ = \frac{1}{2} is the critical line: “off the line” means with \frac{1}{2} < σ ⩽ 1) that (for k > 2) is our aim; in fact, the English school gave first 2 cases: first Hardy and Littlewood [5] in 1916 gave asymptotics for k = 1 (not too difficult!) and then Ingham [8] in 1927 for k = 2 (actually, for both k = 1, 2 he gave only P_{2k−2} leading term, hence error log x better than main term); then, Heath-Brown in 1979 [7] gave, for k = 2 (solved 2-add.div.pbm., i.e., the binary additive divisor problem, see Section 3, using Weil’s bound for the Kloosterman sums), P_{6} (not explicitly!) plus error E_{2} ≪ T^{7/8}.

In a series of papers from the early ’90’s, Ivić and Motohashi (applying considerations from SL(ℤ, 2) for the binary add.div.pbm.) obtained E_{2} ≪ T^{2/3}L^{c} and, in the mean-square, even E_{2}(T) ≪ \sqrt{T} log^{C} T. Ivić explicited P_{2k−2} (when k = 2). Like the binary additive divisor problem, this is not the whole story.

The case k = 3 (recall 𝒞_{3}(a) problem), is again unsolved.

The bound I_{3}(T) ≪ T is called the “sixth moment” problem (actually, this is the weak version and has a link (see Ivić [13]) with the ternary additive divisor problem.

Another interesting result on higher moments is (Heath-Brown [6]): I_{6}(T) ≪ T^{2}L^{c}.

One glimpse at the higher moments (for k ⩽ 2, see [14]): the predicted asymptotics is I_{k} ∼ C(k)T L^{k^{2}}, ∀k ⩾ 1 by applying the methods from Random Matrix Theory in the 2000’s seminal works (early 2000’s) of Keating and Snaith [15] (at Bristol); many others (Conrey, Ghosh [2] to name two).

3. A history of some additive divisor problems

The problem of proving (1.1) (at least fixed a > 0) is called the k-ary additive divisor problem: trivial case k = 1 (𝒞_{1}(a) = x ∀a ∈ ℤ) and the binary additive divisor problem, k = 2, are the only solved problems.

The case k = 3 is the ternary additive divisor problem (sometimes called Linnik problem): some time ago, Vinogradov and Takhtadzhjan (see below k = 2) announced its solution but with, as yet, unfilled holes in their (very technical) “proof”. Their approach still suffers from our lack of information about SL(ℤ, 3); while our (enough good) state of the art about, instead, SL(ℤ, 2) (actually, through the application of the Kuznetsov trace formula, see [20]) allowed (starting from [18] approach) Ivić, Motohashi and Jutila to solve satisfactorily, see especially [14] (and recent Meurman’s [16]), the binary additive divisor problem (different approaches work, with weaker remainders). We mention (still k = 2), in passing, Kloosterman sums bounds (like Weil’s) in the δ-method of Duke–Friedlander–Iwaniec for “determinantal equations” (especially [4]). An even more general problem than this last one has been solved by Ismoilov [9].

Thus, so far, no one has proved (for k > 2), given a ∈ ℕ,

𝒞_{k}(a) = xP_{2k−2}(log x) + Δ_{k}(x, a), Δ_{k}(x, a) = o(x),

as x → ∞ (the k-ary additive divisor problem), not even for a single shift a > 0 (already k = 2 has delicate “uniformity” issues: see [14]).

4. Statement and proof of the original theorem

We state here our original theorem, together with its proof.

Theorem 4.1. Let M < M′ ⩽ 2M, T^{1+ε} ⩽ M ≪ T^{k/2} and H = M^{1+ε}/T, with double average G̃_{k} = G̃_{k}(M, T) defined as

G̃_{k} =^{def} sup_{M⩽x⩽M′, t⩽H}(\frac{1}{t}J_{k}(x, t) + \frac{1}{t}|∑_{h⩽t} ∑_{h<a⩽t} Δ_{k}(x, a)|).

Then for k = 3, 4 we have

I_{k}(T) ≪ T (1 + sup_{T ≪M≪T^{k/2}} G̃_{k}(M, T)/M).

Proof. First of all, the main terms in (1.2), with P_{2k−2}, are treated like Ivić does [11]; actually, since his G_{k} is the supremum (see [11, Theorem 1]) of (the absolute value of)

∑_{a⩽t} Δ_{k}(x, a),

we split it as

\frac{1}{t} ∑_{h⩽t} ∑_{a⩽h} Δ_{k}(x, a) + \frac{1}{t} ∑_{h⩽t} ∑_{h<a⩽t} Δ_{k}(x, a),

whence

|∑_{a⩽t} Δ_{k}(x, a)| ⩽ |\frac{1}{t} ∑_{h⩽t} ∑_{a⩽h} Δ_{k}(x, a)| + |\frac{1}{t} ∑_{h⩽t} ∑_{h<a⩽t} Δ_{k}(x, a)|,

where the second (double) sum is in our G̃_{k}; in the sequel, we shall let the Selberg integral appear from the other term; that is, in fact, the arithmetic mean

\frac{1}{t} ∑_{h⩽t} ∑_{a⩽h} Δ_{k}(x, a)

(a kind of average, something like the C^{1} process in Fourier series) and can be expressed as (on exchanging summations)

\frac{1}{t} ∑_{a⩽t}(t − a + 1)Δ_{k}(x, a) = \frac{1}{t} ∑_{a⩽t}(t − a)Δ_{k}(x, a) + \frac{1}{t} ∑_{a⩽t} Δ_{k}(x, a).

Before we proceed further, we need to express how the diagonal remainders into the estimate of J_{k}(x, t), i.e., the terms ≪ xt, and the tails, i.e., ≪ t^{3}, appear in our final estimate for I_{k}: compare the calculations soon after this proof. This last equation especially has a term

\frac{1}{t} ∑_{a⩽t} Δ_{k}(x, a) ≪ x,

from the trivial estimate Δ_{k}(x, a) ≪ x, giving diagonal terms; and the easily proved relation Δ_{k}(x, −a) = Δ_{k}(x, a) + O_{ε}(x^{ε}a) gives tails (another diagonal: Δ_{k}(x, 0) ≪ x):

\frac{1}{t} ∑_{a⩽t}(t − a)Δ_{k}(x, a) ∼ \frac{1}{2t} ∑_{0⩽|a|⩽t}(t − |a|)Δ_{k}(x, a)

(the ∼ means “+ diagonals and tails”); in all, the term in G̃_{k} with J_{k} is

\frac{1}{t} ∑_{h⩽t} ∑_{a⩽h} Δ_{k}(x, a) ∼ \frac{1}{2t} ∑_{0⩽|a|⩽t}(t − |a|)Δ_{k}(x, a)

and, since S(a) = max(t − |a|, 0), ∀0 ⩽ |a| ⩽ t, applying (1.3) (compare [1]),

∑_{0⩽|a|⩽t}(t − |a|)Δ_{k}(x, a) ∼ J_{k}(x, t),

we obtain the desired bound with the Selberg integral (and the double average). □

We remark that, in spite of the fact that our “new” theorem holds ∀k > 2 (integer), we have here some trouble in handling Selberg’s integral tails, since they contribute to G̃_{k} as (in the sup above)

≪ \frac{1}{t}t^{3} ≪ H^{2} ≪ \frac{M^{2}}{T^{2}}

which gives to I_{k}(T) a contribution (the other sup above)

≪ T (\frac{M^{2}}{T^{2}} M^{−1}) ≪ \frac{M}{T} ≪ T^{k/2−1}.

This is ≪ T only when k/2 ⩽ 2, i.e., k ⩽ 4 (diagonals give ≪ T, good ∀k.)

The tails arise naturally when applying the Linnik method and even a more careful analysis will almost surely not eliminate them. While they are negligible for the Selberg integral, they are not negligible for our present approach.

We remark, in passing, that the “additional” double average, in this approach, cannot be dispensed with.

5. Proof of Theorem 1.1

First of all, we may restrict ourselves to the following definitions of I_{k}, J_{k} :

I_{k}(T) := ∫_{T/2}^{T} |ζ(\frac{1}{2} + it)|^{2k}dt, J_{k}(M, h) := ∫_{M/2}^{3M} | ∑_{t<n⩽t+h} d_{k}(n) − M_{k}(t, h)|^{2}dt.

In fact, we can combine a dissection argument (for I_{k}) and (for J_{k}) a positivity and monotonicity argument to reduce these integrals to the ones defined in the introduction. The logarithmic factor(s) will be in the ≪ (let us fix ε > 0). In order to keep the exposition clearer, we write ≪ even when it is only ≪, in the sequel.

We start by choosing a Dirichlet series (C_{j} is not to be confused with C_{j}(q) above)

f_{k}(s) = ∑_{n=1}^{∞} a_{k}(n)n^{−s} =^{def} ∑_{j=0}^{k−1} C_{j}ζ^{(j)}(s), with ζ^{k}(s) − f_{k}(s) holomorphic at s = 1.

This is done, in order to give the expected main term of the short sum ∑_{n} d_{k}(n), that comes from the residue of \frac{h}{x}ζ^{k}(s)\frac{x^{s}}{s} at s = 1; like (we shall see in a moment) for the other short sum ∑_{n} a_{k}(n), which is this main therm, together with ≪ 1, remainder terms. Here we obtain the coefficients a_{k}(n) = ∑_{j} C_{j}(− log n)^{j}, since we recall

ζ^{(j)}(s) = ∑_{n=1}^{∞}(− log n)^{j}n^{−s}

and we still have to calculate the coefficients C_{j} . This can be done using the Laurent expansion at s = 1 of the Riemann zeta-function:

ζ(s) = \frac{1}{s − 1} + ∑_{n=1}^{∞} γ_{n}(s − 1)^{n}

and (taking k-th powers) we get, from this, that

C_{0}ζ(s) + C_{1}ζ′(s) + ⋯ + C_{k−1}ζ^{(k−1)}(s),

here, has to have the same principal at s = 1 as ζ^{k}(s). This is accomplished by studying a linear system in k equations and unknowns C_{0}, …, C_{k−1}, which has a unique solution.

Then we isolate the mean-square of f_{k}:

I_{k}(T) ⩽ ∫_{T/2}^{T} |ζ(\frac{1}{2} + it)^{k} − f_{k}(\frac{1}{2} + it)|^{2}dt + ∫_{T/2}^{T} |f_{k}(\frac{1}{2} + it)|^{2}dt,

since it is relatively easy to bound (analogously as the mean square of the zetafunction, compare with Section 2):

∫_{T/2}^{T} |f_{k}(\frac{1}{2} + it)|^{2}dt ≪ T

and, in the following, we shall ignore (see why in Sections 1 and 2) all such kind of remainders. We wish to express ζ^{k} − f_{k} at s = 1/2 + it, with t ≍ T (i.e., T ≪ t ≪ T now on), as ( ≪ log T) smoothed Dirichlet polynomials such that there holds (leaving ≪ T)

(5.1) ∫_{T/2}^{T} |ζ^{k}(\frac{1}{2} + it) − f_{k}(\frac{1}{2} + it)|^{2}dt ≪ ∫_{T/2}^{T} | ∑_{M⩽n⩽2M} w(n)\frac{d_{k}(n) − a_{k}(n)}{n^{1/2+it}} |^{2}dt,

where T^{1+ε} ≪ M ≪ T^{k/2} and w here is a C^{1} weight supported in [M, 2M], with

w(x) ≪ 1, w′(x) ≪ \frac{1}{M} ∀x ∈ [M, 2M].

This can be done applying the functional equation (∀k > 2) for ζ^{k}, as in [10]; plug t ≍ T, α = \frac{1}{2} + 1/ log T, Y ≍ M ≍ T^{k/2}, h = log^{2} T (do not confuse with our h at last), into [10, Theorem 4.4] to obtain

ζ^{k}(\frac{1}{2} + it) = ∑_{n⩽2Y} d_{k}(n)e^{−(n/Y) log^{2} T} n^{−1/2−it} + χ^{k}(\frac{1}{2} + it) ∑_{n⩽M} d_{k}(n)n^{−1/2+it}

−\frac{1}{2πi}∫\begin{matrix}Re(w)=1/ log T \\ | Im(w)|⩽log^{4} T \end{matrix} χ^{k}(\frac{1}{2} + it + w)∑_{n⩽M}d_{k}(n)n^{w−1/2+it}Y^{w}Γ(1+ \frac{w}{log^{2} T} )\frac{dw}{w} + o(1).

Subtracting f_{k}(1/2 + it) from the left-hand side and inserting a_{k}(n) into n−sums we get that

ζ^{k}(\frac{1}{2} + it) − f_{k}(\frac{1}{2} + it)

can be expressed (apart from the ≪ 1-terms) by sums of the following types:

∑_{n⩽2Y} a_{k}(n)e^{−(n/Y) log^{2} T} n^{−1/2−it} − f_{k}(\frac{1}{2} + it), (5.2)

∑_{n⩽M} a_{k}(n)n^{−1/2+it}, (5.3)

∑_{n⩽M} a_{k}(n)n^{\frac{1}{log T} +iu−1/2+it}, (|u| ⩽ log^{4} T), (5.4)

∑_{n⩽2Y} (d_{k}(n) − a_{k}(n))e^{−(n/Y)} log^{2} T n^{−1/2−it}, ∑_{n⩽M}(d_{k}(n) − a_{k}(n))n^{−1/2+it}, (5.5)

∑_{n⩽M}(d_{k}(n) − a_{k}(n))n^{\frac{1}{log T} +iu−1/2+it} (|u| ⩽ log^{4} T). (5.6)

For our purposes, the bounds χ^{k}(\frac{1}{2} + it) ≪ 1 and χ^{k}(\frac{1}{2} + \frac{1}{log T} + it) ≪ 1 suffice (see e.g., [10]), with uniform constants ∀k. Also, we can transform the n-sums into integrals, which (apart from these sums) give an essentially bounded contribution. We first treat the terms (5.2), (5.3), (5.4) like remainders, while (5.5) and (5.6) will be into our final (5.1). Recall that there will be ≪ log T (i.e., ≪ 1) of such sums. Analogously as in Theorem 4.11 of Titchmarsh’s book [21]

∑_{n⩽M} a_{k}(n)n^{−1/2+it′} = Res(f_{k}(s + \frac{1}{2} − it′)\frac{M^{s}}{s} , \frac{1}{2} + it′) + f_{k}(\frac{1}{2} − it′) + O_{ε}(M^{ε});

here t′ = t or t′ = t + (1/ log T) + u, with |u| ⩽ log^{4} T, an eventual perturbation to w of the kind n^{1/ log T +iu}, which gives w′ ≪ (log T)^{4}/M, into our w′ ≪ 1/M.

We perform k times an M-average over these (5.3), (5.4) terms to let the residue term O_{k}(1), while (see above) f_{k} mean-square is negligible, like O_{ε}(M^{ε}) ≪ 1. The other terms are managed like these, with a (k times) M-average (Y ≍ M), on writing

∑_{n⩽2Y} a_{k}(n)e^{−(n/Y) log^{2} T} n^{−1/2−it} − f_{k}(\frac{1}{2} + it) = ∑_{n⩽A} a_{k}(n)n^{−1/2−it} − f_{k}(\frac{1}{2} + it)

+ ∑_{n⩽A} a_{k}(n)O(\frac{n}{Y log^{2} T} )n^{−1/2−it} + ∑_{A<n⩽2Y} a_{k}(n)e^{−(n/Y) log^{2} T} n^{−1/2−it},

where the difference on the right-hand side is treated as above; also,

∑_{n⩽A} a_{k}(n)O(\frac{n}{Y log^{2} T} )n^{−1/2−it} ≪ ∑_{n⩽A} \frac{\sqrt{n}}{Y log^{2} T} ≪ \frac{A\sqrt{A}}{Y},

while partial summation allows to isolate the exponential factor, in order to average (like before) the inner sums, and, with the choice A ≍ Y^{2/3}, we get ≪ 1, so that even (5.2) can be neglected. Next, we M-average and use a dyadic dissection (to arrive to [M, 2M], there) for the remaining sums. (In the other proof, notaveraged residue terms in (5.2)–(5.4) correspond to tails, giving the limit k ⩽ 4, see Section 4.) Collecting all these sums and estimates, we get (5.1).

Now we apply Gallagher’s Lemma (see e.g., Lemma 1.10 in [17]) to the sum ∑_{n} w(n)(d_{k}(n) − a_{k}(n)) n^{−1/2−it}, to obtain

∫_{−T}^{T} |∑_{n} \frac{w(n)}{\sqrt{n}}(d_{k}(n) − a_{k}(n))n^{−it}|^{2}dt ≪ T^{2} ∫_{0}^{∞}|∑_{y}^{τy} \frac{w(n)}{\sqrt{n}}(d_{k}(n) − a_{k}(n))|^{2} \frac{dy}{y} ,

with τ := exp(1/T) (see that τ − 1 ≍ 1/T). Here, since w is supported in [M, 2M],

(5.7)  I_{k}(T) ≪ max_{T^{1+ε}≪M≪T^{k/2}} \frac{T^{2}}{M} ∫_{M/2}^{3M} | ∑_{y<n⩽τy} \frac{w(n)}{\sqrt{n}}(d_{k}(n) − a_{k}(n))|^{2}dy

(again, leaving ≪ T). Let us now denote ρ := τ − 1 ∼ 1/T.

In order to avoid the inner dependence on y, applying the partial summation we have

∑_{y<n⩽y+ρy} c(n)f(n) ≪ max_{h≪max(ρy)}| ∑_{y<n⩽y+h} c(n)| (|f(y + ρy)| + ∫_{y}^{y+ρy} |f′(v)|dv)

(where we can assume y integer, the difference giving ≪ 1 in the sum), with the (local) definitions c(n) := d_{k}(n) − a_{k}(n) and f(v) := w(v)/\sqrt{v}, where

|f(v)| ≪ \frac{1}{\sqrt{v}} and |f′(v)| ≪ \frac{1}{v\sqrt{v}} + \frac{1}{M\sqrt{v}} ∀v ∈ [y, y + ρy] (M < y ⩽ 2M)

we have

| ∑_{y<n⩽y+ρy} w(n)n^{−1/2}(d_{k}(n) − a_{k}(n))|^{2} ≪ max_{h≪M/T} \frac{1}{M} | ∑_{y<n⩽y+h}(d_{k}(n) − a_{k}(n))|^{2}.

Finally, inserting this in (5.7) we obtain

I_{k}(T) ≪ T max_{T^{1+ε}≪M≪T^{k/2}} max_{0<h≪M/T} \frac{T}{M^{2}} ∫_{M/2}^{3M} | ∑_{x<n⩽x+h} d_{k}(n) − M_{k}(x, h)|^{2}dx,

where it is evident that the main term M_{k}(x, h) is the a_{k}(n)−short sum in ]x, x+h].

References

1. G. Coppola, On the Correlations, Selberg integral and symmetry of sieve functions in short intervals, http://arxiv.org/abs/0709.3648v3; to appear in J. Comb. Number Theory 2(2) (2010).

2. J. B. Conrey, A. Ghosh, A conjecture for the sixth power moment of the Riemann zetafunction, Internat. Math. Res. Notices 15 (1998), 775–780.

3. H. Davenport, Multiplicative Number Theory, Third edition, Revised and with a preface by Hugh L. Montgomery, Grad. Texts Math. 74, Springer-Verlag, New York, 2000.

4. W. Duke, J. Friedlander, H. Iwaniec, A quadratic divisor problem, Invent. Math. 115 (1994), no. 2, 209–217.

5. G. H. Hardy, J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Math. 41(1) (1916), 119–196.

6. D. R. Heath-Brown, The twelfth power moment of the Riemann-function, Q. J. Math. Oxf. II. Ser. 29 (1978), 443–462.

7. D. R. Heath-Brown, The fourth power moment of the Riemann zeta function, Proc. London Math. Soc. (3) 38 (1979), 385–422.

8. A. E. Ingham, Mean-value theorems in the theory of the Riemann Zeta-function, Proc. London Math. Soc. (2) 27 (1928), 273–300.

9. D. Ismoilov, An asymptotic formula in additive number theory, Math. Notes 39 (1986), 257– 266. Translation from Mat. Zametki 39 (1986), 465–482.

10. A. Ivić, The Riemann Zeta Function, The theory of the Riemann zeta-function with applications Wiley, New York, 1985.

11. A. Ivić, The general additive divisor problem and moments of the zeta-function, in: New trends in probability and statistics, Vol. 4 (Palanga, 1996), 69–89, VSP, Utrecht, 1997.

12. A. Ivić, On the mean square of the divisor function in short intervals, J. Théor. Nombres Bordx. 21(2) (2009), 251–261.

13. A. Ivić, On the ternary additive divisor problem and the sixth moment of the zeta-function; in: Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), 205–243, London Math. Soc. Lecture Note Ser. 237, Cambridge Univ. Press, Cambridge, 1997.

14. A. Ivić, Y. Motohashi, On some estimates involving the binary additive divisor problem, Q. J. Math. Oxf. II. Ser. 46 (1995), 471–483.

15. J. P. Keating, N. Snaith, Random matrix theory and ζ(1/2 + it), Comm. Math. Phys. 214(1) (2000), 91–110.

16. T. Meurman, On the binary additive divisor problem; in: Number Theory (Turku, 1999), 223–246, de Gruyter, Berlin, 2001.

17. H. L. Montgomery, Topics in multiplicative number theory, Lect. Notes Math. 227, Springer-Verlag, Berlin–New York, 1971.

18. Y. Motohashi, The binary additive divisor problem, Ann. Sci. École Norm. Sup. (4) 27(5) (1994), 529–572.

19. P. Shiu, A Brun-Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math. 313 (1980), 161–170.

20. L. A. Takhtadzhjan, A. I. Vinogradov, The zeta-function of the additive divisor problem and the spectral expansion of the automorphic Laplacian (Russian), Automorphic functions and number theory, II. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 134 (1984), 84–116.

21. E. C. Titchmarsh, The theory of the Riemann zeta-function, Second edition. Edited and with a preface by D. R. Heath-Brown, Clarendon Press, Oxford University Press, New York, 1986.

D.I.I.M.A. (Received 25 09 2009) University of Salerno (Revised 11 02 2010 and 15 04 2010) 84084 Fisciano (SA) Italy

 gcoppola@diima.unisa.it
 giocop@interfree.it