On the Selberg integral of the k−divisor function
and the 2k−th moment of the Riemann zeta function
Giovanni Coppola
Diima - University of Salerno (Italy)
1. Introduction.
We will link the two problems, estimating the 2k−th momentof the Riemann ζ−function (on the critical line σ = 21 )def

Ik (T ) =Z T
0|ζ( 12 + it)|2k dt

and finding non-trivial bounds for the, say, Selberg integralof the k−divisor function, dk (n) (that has Dirichlet series ζ k )def

Jk (x, h) =Z xhxεXt<n≤t+h2dk (n) − Mk (t, h) dt

where, say, Mk (t, h) is the “expected value” of the (inner) sum.
This gives dk over the (say) “short interval” [t, t + h] (ash = o(t) ∀t ∈ [hxε , x]); here and in the sequel ε > 0 will be
arbitrarily small, not the same at each occurrence.
Actually, Ivić already gave (in a paper to appear on JTNB
for JAXXV Proc.) a non-trivial bound for Jk (x, h) when thehwidth of the s.i. (abbrev. short interval), namely θ := loglog x
defis greater than θk = 2σk − 1 (with σk Carlson’s abscissa):(Ivić, JTNB) θ > θkxh2⇒ ∃δ = δ(k) > 0 : Jk (x, h)  δx1

(see the trivial bound : Jk (x, h)  xh2 (log x)c , c = c(k), here)
For example, θ3 = 61 , θ4 = 14 , θ5 = 11
30 , . . . (from values of σk ).
This result clearly gives non-trivial bounds for Jk , using
ζ−moments information (off the critical line, into the σk ).
So, knowledge of the ζ implies knowledge of the dk in a.a.
(abbrev. almost all) the s.i. (short intervals).
X(See: Jk non trivial ⇒dk (n) ∼ Mk (t, h), a.a.s.i.)t<n≤t+h

However, we can also go in the opposite direction: if we havesome kind of non-trivial information about the dk , we can improve our knowledge (at least, on the 2k−th moments) of theRiemann ζ−function. Actually, this idea is due to Ivić, whogave a link between the “(auto-)correlation” of dk with“shift-parameter” a, namelydef X
Ck (a) =dk (n)dk (n + a)a ∈ N (here x ∈ N, x → ∞)n≤x(the shift is a positive integer, here; however, Ck (−a) is closeenough to Ck (a); also, Ck (0) is relatively easy to compute).
The problem of estimating dk (auto-)correlation, Ck (a),at least for a fixed a 6= 0 (a = 0 is “trivial”) is called
the k−ary additive divisor problem (k−ary add. div.
pbm); the trivial case k = 1 (with Ck (a) = x ∀a ∈ Z) and the“binary additive divisor problem”, k = 2, are the only
two solved pbms, here.
Case k = 3 is the ternary additive divisor problem(sometimes called “Linnik problem”, but Yu.V.Linnik has
left many unsolved pbms !): some time ago, Vinogradov &Takhtadjan (right spelling ?) announced its solution but with,as yet, unfilled holes in their (extremely technical !) “proof”.
2This approach, actually, still suffers from our lack of information about SL(Z, 3); while our (enough good) state of the art
about, instead, SL(Z, 2) allowed Ivić, Motohashi and Jutila tosolve satisfactorily the binary additive divisor problem (but,different approaches work; however, with weaker remainders).
Thus, so far, no one has proved (for k > 2), given a ∈ N,(∗)k

Ck (a) = xP2k−2 (log x) + ∆k (x, a), ∆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“a−uniformity” issues: following).
(main term has P2k−2 (log x), a 2k − 2 deg. poly. in log x)
Here it comes into play the idea of Ivić (see his paperin Palanga 1996 Conference Proc.) of linking the estimateof the 2k−th moment, Ik (T ), to a sum of correlations
Ck (a) performed over a (the shift), up to (roughly, we avoidtechnicalities), say, h := Tx (the s.i. comes in !)In order to be more precise, we need also to abbreviate:
defA≪ B ⇐⇒ ∀ε > 0 A ε xε B(with x, X or even T our “main variables”, indep.& → ∞)i.e., the modified Vinogradov notation ≪ allows us toignore all the arbitrarily small powers; also, we’llsay that the arithmetic function (a.f.) f : N → R isessentially bounded, write f ≪ 1, when ∀ε > 0 f (n) ε nε(as n → ∞). For example, all the dk (∀k ∈ N) are ess.bd :∀k ∈ N dk ≪ 1
3

whence they contribute individually small powers (ignored);
Shiu estimates (see Jk triv.est.quoted above), a kind of BrunTitchmarsh for (suitable multiplicative a.f., like) dk , let thesegive, on average over (all) s.i., powers of log. By the way,L := log x (or L := log X, even L := log N )is the abbreviation for the logarithm of our main variable.
For example, the (conjectured, ∀k > 2) m.t. of (∗)k isxP2k−2 (log x) k xL2k−2 ≪ x.
Here, it seems that the first to propose explicitly this form for(∗)k is Ivić, who also gave explicitly the polynomial P2k−2 ,that is essentially bounded (w.r.t. x). However, as we’ll see ina moment, it depends, also, on the shift a > 0.
We’ll give now, avoiding technicalities, Ivić’ s argument.
After some work (expand the square & mollify, take relevant
ranges, . . .) he gets that Ik (T ) is
Ik (T ) = Ik00 (T ) + Oε (T ε T )withdef
Ik00 =1MXX

dk (n)dk (n + a)a≤M 1+ε T −1 M <n≤M 0Z 2T

φ(t)eita/n dt,

T2

where M < M 0 ≤ 2M , with M ≪ T k/2 , say h≪ M/T , thesmooth (i.e., C ∞ ) test-function φ has support into ]T /2, 2T [,φ([3T /4, 4T /3]) ≡ 1, and has good decay
φ(R) (t) R T −R , ∀R ∈ N.
4

We should keep in mind, here, that for fixed k ∈ N we seekZ T
Ik (T ) =|ζ( 12 + it)|2k dt≪ T (“2k−th moment pbm”)
0that (for k > 2) is our aim; in fact, English school gave first 2
cases: Ingham in 1926 asymptotics for k = 1 (not too difficult!)
and k = 2 (for both, he gave only P2k−2 leading term, hence
error log x better than main term); then, Heath-Brown in 1979
gave, for k = 2 (solved 2−add.div.pbm. using Weil’s bound for
Kloosterman sums), P6 (not explicitly!) plus error E2 ≪ T 7/8 .
Starting from ’94 & ’95, a series of Ivić & Motohashi papers(applying SL(Z, 2) considerations for the binary add.div.pbm.)√gave E2 ≪ T 2/3 and, in mean-square, even E2 ≪ T . (Herelog-pows, not small pows!). Ivić explicited P2k−2 (when k = 2).
Like the binary add.div.pbm., this is not the whole story !
The case k = 3, again (recall C3 (a) pbm), is unsolved.
The boundI3 (T )≪ T
is called the “sixth moment” pbm (actually, this is the weakversion) & has a link (in Ivić, Proc. Cardiff 1996 Symposium)with the ternary additive divisor pbm.
Another interesting moment (Heath-Brown ’79): I6  T 2 Lc .
One glimpse, 2k−th pbm. (k ≤ 2, see Ivić & Motohashi,2quoted): predicted asymptotics Ik ∼ C(k)T Lk , ∀k ≥ 1(English school,again!) applying Random Matrix Theory(Quantum Physics concepts inspiration!) in (2000s) seminal
works of J. Keating & N. Snaith (at Bristol); many others(Conrey, Ghosh to name two). The RMT−ζ link originated(Dyson-Montgomery coffee-break) in 1972. (Sorry for brevity.)5

Now on (that’s why we mentioned weak 2k−th pbm !) wecan ignore (in bounds for Ik ) all terms which are ≪ T .
We give an idea of the polynomial, P2k−2 , given by Ivić, before
to proceed. It’s (see Palanga Conference Proc. for details)
1
P2k−2 (log x) =x
def

Z xX
∞

cq (a) 2
Rk (log t)dt,2q
q=10

def

with, say, Rk (log t) =C−k (q)C1−k (q)C−2 (q)logk−1 t+
logk−2 t+. . .+log t+C−1 (q)(k − 1)!
(k − 2)!
1!
depending on q, but not on a (this is vital); also, w.r.t. x,Rk (log t)≪ 1 and this is very useful ! We’ll see in a momentthat the shape of these Cj (q) is important only in case q = 1.
By
P∗the way, here cq (a) is the Ramanujan sum, defined as(is over q−coprime js)
X q
def X∗
cq (a) =eq (ja) =dµdd|qj(mod q)

d|a

def

Hence, say, S(a) = max(0, h − |a|) (Ŝ is Fejér’s kernel) givesj Xdef XbS=S(a)eq (ja) ≥ 0 ⇒S(a)cq (a) ≥ 0qaa

(see the link with Jk soon) and from the elementary, ∀d ∈ N,  XXh2hhS(a) = h + 2(h − db) =+d1−,dddaa≡0(d)b≤h/d6

we get (apply cq (a), above), writing 1℘ = 1 if ℘ holds, = 0 else:
XS(a)cq (a) = 1q=1 h2 +aXd|q q  h hd2 µ1−.
ddd

(It is here evident q = 1 greater importance.) Thus, (see Ivić):(1)

XS(a)xP2k−2 (log x) = h2Z xhxεa

Rk2 (1, log t)dt + tails,

where we mean, by “tails”, remainders which are ≪ h3 . Here,def
the part of Rk (log t) term with q = 1 is, say, Rk (1, log t) =C−k (1)C1−k (1)C−2 (1)logk−1 t+
logk−2 t+. . .+log t+C−1 (1)(k − 1)!
(k − 2)!
1!
and equals the term Mk (log t) into the Selberg integral; as itshould be, since (from an elementary version of the Linnik’s
Dispersion method), assuming (∗)k with this P2k−2 , we get(2)Jk (x, h) ∼XaS(a)Ck (a) − h2

∼XZ xhxεMk2 (log t) dt

S(a)∆k (x, a),awhere ∼ means ignoring “tails” (see above) & “diagonals”,i.e. remainders ≪ xh. We remark that both these kind oferrors are negligible, since they both contribute ≪ T to
the final result for Ik (T ). (This can be easily seen.)
7

Then, due to Ik00 expression, Ivić made a hypothesis about(avoiding technicalities) sums of ∆k (x, a) (remainders into (∗)kabove), performed over the shift a, say Gk , which implies thebound Ik (T )≪ T (for the same k > 2). Now on k > 2.
Of course, he doesn’t need (∗)k to hold individually ∀a(≤ h, here), but he observes that he’s summing up, into Gk ,without the modulus over the remainder, ∆k (x, a), so somea−cancellation can take place, here.
So far, he passes from an asymptotic formula (∗)k to ana−averaged form of it, which is easier to prove (however, yetnobody has done it !).
Here, with applications in mind,
we pass from a single average to a double average
Building on his expression for Ik00 , it’s possible to make a lessstringent hypothesis, to have a more flexible procedure
for the remainders ∆k (x, a).
We use, also, our previous works on the Selberg integralof the a.f. f (essentially bounded & real) in order to let
the Selberg integral of dk , i.e. Jk (x, h), come into play. (It is akind of “double average” of ∆k (x, a).)Unfortunately, due to an exponential factor multiplyingdk (n)dk (n + a) into Ck (a) we can’t get a link with Ik (T ) using
only Jk (x, h) (with h≪ Tx , x≪ T k/2 ), but we need, also, tomake an hypothesis on another double average of remainders∆k (x, a). We will give our Theorem and a sketch of the proof.

82. Result and Remarks.
Theorem. Let M < M 0 ≤ 2M , T 1+ε ≤ M  T k/2 andek = Gek (M, T ) defined asH = M 1+ε /T , with double average G
1
1 X X
defe
Gk =supJk (x, t) +∆k (x, a)  .
ttM ≤x≤M 0 ,t≤Hh≤t h<a≤tThen for k = 3, 4 and any fixed ε > 0 we have
Ik (T ) k,ε T 1+ε1+

sup
T ≪ M T k/2

ek (M, T )/MG!

.

We remark that, in spite of the fact that Ivić ’s Theorem holds∀k > 2 (integer), we have some trouble in handling Selberg’s inek as (in the sup above)
tegral tails, since they contribute to G
1 3M22≪ t ≪H ≪ 2t
T

which gives to Ik (T ) a contribute (other sup above)
≪T

2



MM−1M≪≪ T k/2−12TTthat is ≪ T only when k/2 ≤ 2, i.e. k ≤ 4 here.
We trust the possibility to have a link as above not only forthe sixth & the eighth moment, but the tails arise naturally
when applying the Linnik method and even a more carefulanalysis will almost surely not eliminate them ! While they’renegligible for Selberg integral, not so for present application !
9In order to give an idea of the proof, we’ll avoid technics
and highlight the ideas (few & elementary) involved.
First of all, since Ivić already has a sum of remainders, sayX∆k (x, a)a≤t

we simply let the arithmetic mean appear (together with theek definition). Then, for this, see that
other double sum, see G
1 XX∆k (x, a)th≤t a≤h(a kind of average of the above, something like C 1 process inFourier series) can be expressed as (exchanging sums)
1X
1X
1X(t− a+ 1)∆k(x, a) =(t− a)∆k (x, a)+∆k (x, a)ttta≤ta≤ta≤twhich reduces to (using ∆k (x, a)≪ x)
1X
1 X∼(t − a)∆k (x, a) ∼(t − |a|)∆k (x, a),t2ta≤t0≤|a|≤t(+ diag.s & tails); and S(a) = max(t − |a|, 0), ∀0 ≤ |a| ≤ t ⇒X(t − |a|)∆k (x, a) ∼ Jk (x, t)0≤|a|≤t

we get the desired bound with the Selberg integral (& doubleaverage).
We remark, in passing, that “additional” double average
can’t be dispensed with.
10