On the Selberg integral of the three-divisor function d3
Giovanni CoppoladefP

1. Introduction and statement of the results.

arXiv:1207.0902v3 [math.NT] 29 Sep 2012

Recall that d3 (n) =abc=n 1 is the number of ways to write n as a product of three positive integers.
Namely, the function d3 is generated by the Dirichlet series ζ(s)3 , where ζ is the Riemann zeta function.
Given positive integers N and H = o(N ) as N → ∞, the Selberg integral of d3 is the mean-squareXdefJ3 (N, H) =XN <x≤2N2d3 (n) − M3 (x, H) ,x<n≤x+Hdefwhere M3 (x, H) = H Res s=1 ζ(s)3 xs−1 is the short intervals mean-value, i.e. the expected value ofthe d3 −short sum.
Improving on results in [CL], we give here a new non-trivial bound for J3 (N, H) by comparing it to therelated modified Selberg integral,def
Je3 (N, H) =X

N <x≤2NX0≤|n−x|≤H1−2|n − x| d3 (n) − M3 (x, H) ,H

where the same choice for the s.i. mean-value M3 (x, H) is due to elementary reasons (see [CL] introduction).
In what follows, the symbols O and ≪ are respectively the usual Landau and Vinogradov notations,while we adopt the “modified Vinogradov notation” defined as
def

A(N, H)≪ B(N, H) ⇐⇒ ∀ε > 0, A(N, H) ≪ε N ε B(N, H).
As any divisor function, d3 is bounded asymptotically by every arbitrarily small power of the variable, i.e.
it satisfies the definition:deff essentially bounded ⇐⇒ ∀ε > 0 f (n) ≪ε nε ,and we shortly write f ≪ 1. According to the “modified Vinogradov notation”, we use to write d3 ≪ 1 whend3 is restricted to ]N − H, 2N + H] as before.
The author [C] has proved the lower bound N H log4 N ≪ J3 (N, H) for H ≪ N 1/3 , while, in an attempt
to establish a non trivial upper bound, Laporta and the author [CL] have conjectured the following estimate
for the modified Selberg integral.
Conjecture CL. If H ≪ N 1/3 , then Je3 (N, H)≪ N H.

As a consequence one has the main result of the present paper.
Theorem. If Conjecture CL holds, then J3 (N, H)≪ N H 6/5 .
The proof of the Theorem is given in §3, where Conjecture CL is combined with the following general
Proposition on the Selberg integral and the modified one of an essentially bounded arithmetic function f ,respectively
X
2
def X
Jf (N, H) =f (n) − Mf (x, H) ,x∼N

def X
Jef (N, H) =x∼Nx<n≤x+HX0≤|n−x|≤H2|n − x| 1−
f (n) − Mf (x, H) .
HHere x ∼ N means that N < x ≤ 2N and the s.i. mean-value is defined asdef

Mf (x, H) = H Res F (s)xs−1 ,s=11

whenever f is generated P
by a meromorphic Dirichlet series F with (at most) a pole in s = 1. It is not difficult
to see that Mf (x, H) ∼ x<n≤x+H pc (log n) with good remainders, where pc is the so called logarithmic
polynomial of f of degree c = ords=1 F (s) − 1 (see [CL]). If Mf (x, H) vanishes identically (that is the casewhen F is regular at s = 1), then we say that f is balanced.
Proposition. If f : N → C is an essentially bounded and balanced function such that
Jef (N, H)≪ N H 1+A ,∀H ∈ [H1 , H2 ] with N δ ≪ H1 ≪ H2 ≪ N 1/2−δ ,for an absolute constant A ∈ [0, 1) and for a fixed δ ∈ (0, 1/2), then
1+3A

Jf (N, H)≪ N H 1+ 5−A ,∀H ≤ H2 .

Beyond Lemma 1 of [CL] (whose formulæ are quoted in §2), for the proof of the Proposition (see §3) it isapplied an enhanced version of Gallagher’s Lemma for the exponential sums (see [Ga], Lemma 1) establishedby the author in the joint paper [CL1] with Laporta. Unfortunately at the moment Conjecture CL remains
unproved after a serious gap occurred in the proof of the “Fundamental Lemma” given in a former versionof [CL]. Nevertheless, we think it is worthwhile to make further attempts, first for the strongest conjecture
J3 (N, H)≪ N H, ∀H ≪ N 1/3 seems to be out of reach by any method, then because the bound of theTheorem has an impressive consequence on the 6-th moment of ζ on the critical line (compare [CL] andversion 2 of the present paper on arXiv). Actually, the author has already explored in his paper [C2] a link
between the Selberg integral Jk (N, H) of the k-divisor function dk and the 2k−th moment of Riemann zetafunction on the critical line, i.e.
Z 2T  2k1
defIk (T ) =
ζ+ it
dt
2T
(under suitably conditions on N, H, T ). Somehow the same link holds also between Jek (N, H) and Ik (T ),since in Gallagher’s Lemma the right hand side is a Selberg integral de facto and, as showed in [CL1], it
can be replaced by the corresponding modified Selberg integral (see Lemma in §2). This is still true forthe Dirichlet polynomials case, namely Theorem 1 of [Ga] can be modified in the same fashion. Thus, byapplying such a Dirichlet polynomial version of Gallagher’s Lemma within the method of [C2] instead ofTheorem 1 of [Ga] one gets the following outstanding consequence of Conjecture CL.
Corollary. If Conjecture CL holds, then I3 (T )≪ T .
In the literature this result is known as the “weak 6−th moment”, since it gives no asymptotic equality forI3 (T ), but just an upper bound with additional sufficiently small powers. The proof of the Corollary will begiven in a forthcoming paper.
2. Notation and preliminary formulæ.
The correlation of f with shift h is defined as
def

Cf (h) =Xn∼Nf (n)f (n − h) =XXf (n)n∼Nf (m) + O

m∼N
n−m=h



max

[N −|h|,2N +|h|]|f |2defn∈[N −|h|,N ]∪[2N,2N +|h|]Recall that from the orthogonality of the exponentials e(β) = e2πiβ (β ∈ R) one hasXn∼N

wheref (n)f (m) =Xf (n)e(nα) =0m∼Nn−m=hdeffb(α) =Z 1

X

n|fb(α)|2 e(−hα)dα,Xn∼N

2f (n)e(nα)

X

1 .

is truncated in the range ]N, 2N ] in order to avoid convergence problems. If f ≪ 1, then (compare [CL])Cf (h) =

Z 1
0



|fb(α)|2 e(−hα)dα + Oε N ε |h| ,∀h 6= 0.

The correlation of an uniformly bounded weight w : R → C, vanishing outside [−H, H], is conveniently defined asX
def X
Cw (h) =w(a)w(b).
aba−b=hIn particular, the main weight involved here is the characteristic function of the integers in [1, H], saydefu(n) = 1[1,H] (n), whose correlation isdef X

Cu (h) =u(a)aXu(b) =X Xa≤H

ba−b=h

1 = max(H − |h|, 0).

b≤H
a−b=h

If f is essentially bounded and balanced, then Lemma 1 of [CL] provides the formulæXXJf (N, H) =Cu (h)Cf (h) + Oε (N ε H 3 ),Jef (N, H) =CCu /H (h)Cf (h) + Oε (N ε H 3 ),hh
Cu (h)|h| 
where we find the Cesaro weight= max 1 −, 0 , that is the “normalized” correlation of u.
HHIn what follows such formulæ will be applied in the integral form
Jef (N, H) =Z 1/2

−1/2fb(α)

2 |bu(α)|4H2

dα + Oε (N ε H 3 ),Jf (N, H) =

Z 1/2

−1/2

def P

obtained by using the exponential sum ub(α) =(∗)

2fb(α) |bu(α)|2 dα + Oε (N ε H 3 ) ,h≤H e(hα). Note that for every 0 < ε < 1 one has|bu(α)| > [εH] =⇒ |α| <1.
2[εH]Indeed, since (∗) is trivial for α = 0, we may assume that 0 < α < 1/2, which easily implies 2α < sin(πα).
Then, (∗) follows also in this case for[εH] < |bu(α)| =1
1| sin(πHα)|≤<.
sin(πα)sin(πα)2αFinally, we quote the aforementioned modified version of Gallagher’s Lemma (see [CL1]).
Lemma. Let N, h be positive integers such that h → ∞ and h = o(N ) as N → ∞. If f : N → C isessentially bounded and balanced, thenh21Z 2h

1− 2h2fb(α) dα≪ Jef (N, h) + h3 .

3. Proofs of the Proposition and the Theorem.
Proof of the Proposition. If H were not essentially bounded, the trivial bound ≪ N H 2 would imply
Jf (N, H)≪ N immediately. Thus, let us assume that H ≫ N η for some small η > 0. Taking ε = ε(H), E =E(H) to be determined later such that 0 < ε < E and ε, E → 0 as H → ∞, we write
Jf (N, H)≪Z 1/2−1/22fb(α) |bu(α)|2 dα + H 3 ≪3≪ ε2 H 2Z

2|bu(α)|≤[εH]

fb(α) dα + E 2 H 2Z21fb(α) dα + 2E[εH]<|bu(α)|≤EH

By applying Parseval’s identity together with (∗) one has
2 22Jf (N, H)≪ N H ε + H E2Z

21fb(α) dα + 21E|α|≤ 2h≪ N H 2 ε2 + H 2 E 2

Z 1/2−1/2ZZ|bu(α)|>EHfb(α)fb(α)

2 |bu(α)|4H22 |bu(α)|4H2dα + H 3 .

dα + H 3 ≪211fb(α) dα + 2 Jef (N, H) + 2 H 3 .
1EE|α|≤ 2hdef
Now the hypothesis Jef (N, h)≪ N h1+A and the previous Lemma for h = [εH] → ∞ imply
E2H AHA .
Jf (N, H)≪ N H 2 ε2 + 1−A + 2ε
HE H2(1−A)(1−A)22

A−1

Taking ε = H − 5−A , E = H − 2(5−A) one gets ε2 = E εH
1−AA−1= HE 2 , ε = o(E) and E → 0, as desired.

Proof of the Theorem. Denoting the logarithmic polynomial of d3 with p2 , we recall that M3 (x, H) ∼Px<n≤x+H p2 (log n) where the implicit remainders give a negligible contribution to J3 (N, H). Thus, it issufficient to apply the Proposition to the balanced and essentially bounded function f (n) = d3 (n)−p2 (log n),
because Conjecture CL allows to take A = 0 and H2 ≪ N 1/3 .
Acknowledgement. The author wishes to thank Maurizio Laporta for useful comments.
References[C] Coppola, G. - On some lower bounds of some symmetry integrals - http://arxiv.org/abs/1003.4553 to appear on Afrika Mathematika (Springer)[C2] Coppola, G. - On the Selberg integral of the k-divisor function and the 2k-th moment of the Riemannzeta-function Publ. Inst. Math. (Beograd) (N.S.) 88(102) (2010), 99–110. - available online[CL] Coppola, G. and Laporta, M. - Generations of correlation averages - http://arxiv.org/abs/1205.1706[CL1] Coppola, G. and Laporta, M. - A modified Gallagher’s Lemma - http://arxiv.org/abs/[Ga] Gallagher, P. X. - A large sieve density estimate near σ = 1 - Invent. Math. 11 (1970), 329–339.
MR 43#4775Giovanni Coppola
Università degli Studi di Salerno
Home address : Via Partenio (Pal.Soldatiello) - 83100, Avellino(AV) - ITALY
e-page : www.giovannicoppola.namee-mail : gcoppola@diima.unisa.it

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