1. Introduction and Statement of the Results
In the milestone paper [
1], Selberg introduced an important tool in the study of the distribution of prime numbers in
short intervals [
x,
x +
H] with
H =
o(
x) as
x →
∞, that is, the integral
()
where Λ is the von Mangoldt function:
if
n =
pr for some prime
p and
; otherwise,
. Thus, Λ(
n) is a weighted characteristic function of prime numbers generated by −
ζ′(
s)/
ζ(
s) (hereafter,
ζ is the Riemann zeta function). The Selberg integral, being a quadratic mean, pertains precisely to the study of the distribution of primes in
almost all short intervals [
x,
x +
H], that is, with at most
o(
N) exceptional integers
x ∈ [
N, 2
N] as
N →
∞. Here, we define the
Selberg integral of any
as
()
where
x ~
N means
N <
x ≤ 2
N and
Mf(
x,
H) is the expected
mean value of
f in short intervals (abbreviated as s.i. mean value). In order to avoid trivialities, we assume that
H goes to infinity with
N. In view of nontrivial bounds, the discrete version
JΛ(
N,
H) may be considered close enough to the original continuous integral; so, we are legitimate to use the same symbol for both. Similar considerations hold for the
f we work with, mostly the
k-divisor function
dk for
k ≥ 3, where
dk(
n) is the number of ways to write
n as a product of
k positive integers (see [
2] and compare Section
4). Let us denote the Selberg integral of
dk as
()
with the s.i. mean value of
dk given by
()
where
Pk−1 is the
residual polynomial of degree
k − 1 such that
.
The first author [
2] has proved the lower bound
NHLk+1 ≪
k Jk(
N,
H) for
Nε ≪
H ≪
N1/k−ε, where
hereafter. We formulate, for the so-called modified Selberg integral of
d3,
()
where
M3(
x,
H) is the same s.i. mean value as in
J3(
N,
H), the following conjecture. In Section
7, Propositions
16 and
18 justify the admissibility of such a choice for the s.i. mean value in modified Selberg integrals, even in the generalization for any divisor function
dk (everything comes from Proposition
14; see Section
7).
Hereafter, we abbreviate A(N, H)⋘B(N, H) whenever A(N, H) ≪ε NεB(N, H), for all ε > 0.
Conjecture CL. If H ≪ N1/3, then .
A first consequence of our conjecture is the following result.
Theorem 1. If Conjecture CL holds, then J3(N, H)⋘NH3/2.
In Section
5, we deduce this from a general link (Lemma
10), between the Selberg integral
Jf(
N,
H) and the corresponding
modified Selberg integral,
()
in the special case of an essentially bounded (see Section
4) real function
f such that
Mf(
x,
H) vanishes identically.
Moreover, for the width of H, that is, (more generally, θ is defined by xθ ≪ H ≪ xθ for x ~ N), we will implicitly assume that any inequality θ > θ0 (resp., θ < θ0) means that ∃δ > 0 fixed and absolute constant such that θ ≥ θ0 + δ (resp., θ ≤ θ0 − δ). In particular, 0 < θ < 1 works as δ ≤ θ ≤ 1 − δ.
Noteworthy, Theorem 1 improves Ivić’s results [3] for d3, both in the bound and in the low range, while Ivić’s bound is nontrivial for N1/6+δ ≪ H ≪ N1−δ, that is, for width 1/6 < θ < 1; ours is nontrivial for 0 < θ ≤ 1/3, that is, for Nδ ≪ H ≪ N1/3 (we think to be able to get a wider range). Remarkably, under Conjecture CL, the first author has recently derived the even better J3(N, H)⋘NH6/5 (see http://arxiv.org/).
As one may expect, our study applies to every divisor function
dk, though the conjectured estimate for
()
becomes less and less meaningful as
k grows: this is due to known bounds for
αk in (compare Section
4)
We generalize Conjecture CL for the modified Selberg integral of dk with k ∈ {4,5, 6,7} as follows.
General Conjecture CL.
Fix k ∈ {4,5, 6,7}
and assume that αk−1 ∈ [0,1)
in (*). If H ≪
N1/2, then
()
Consequently, setting , for every width , there exists an ε1 =
ε1(
θ,
k) > 0
such that
()
From the last inequality, our second main result follows, proved together with Theorem 1 in Section 5.
Theorem 2. If General Conjecture CL holds, then for k ∈ {4,5, 6,7}.
As a consequence, one would get an improvement in the low range of H with respect to the results of Ivić for the mean-square of dk in short intervals [3]. In fact, Theorem 1 in [3] holds for θ ∈ (θk, 1) with θk = θ(k) defined in terms of Carlson’s abscissa σk (see Section 6). In particular, it holds for θ4 = 1/4, θ5 = 11/30, and θ6 = 3/7, whereas from αk bounds [4], we get instead with , , .
More generally,
is a kind of
smoothing (see Section
2) for the Selberg integral
Jf(
N,
H), both in the arithmetic and in the harmonic analysis aspects. The arithmetic one goes back to Ernesto Cesaro:
()
This arithmetic mean of the inner sum in
Jf(
N,
H) justifies the choice of the same
Mf(
x,
H) in
. The analytic aspects of such a smoothing process will be clear after the introduction of the
correlation in Section
3, where it is shown that Selberg integrals
Jw,f(
N,
H) are averages of
in short intervals |
h | ≪
H. In fact (see Section
3 title), a kind of elementary dispersion method proves this link.
The next corollaries, applying such an intimate link, improve recent results [
5,
6] on an additive divisor problem for
dk (first two conditionally, third unconditionally). They concern the
deviation of
dk; that is,
()
where
is the
correlation (see Section
3) of
dk and
is the so-called
logarithmic polynomial of
dk(
n). The following consequence of Theorem
1 is proved in Section
6.
Corollary 3. Let be such that Nθ ≪ H ≪ Nθ for a fixed θ ∈ (0,1/3]. If Conjecture CL is true, then
()
In a completely analogous way, from Theorem 2, we deduce the following result on for k ∈ {4,5, 6,7}.
Corollary 4. Under the same hypotheses of Theorem 2, one has, in the same ranges and for the same ε1,
()
The aforementioned link to [
5,
6] results relies on the identity (see Section
4)
()
where one has to be acquainted that one’s notations
Pk−1 and
pk−1 are not consistent with those in [
5,
6]. In particular, from (3.8) in [
6], the main term in the [
5,
6] formulas for sums of
dk correlations is
()
Since it is easily seen that, for every
H ≪
N, one has
()
then our
is comparable with the Δ
k(
N;
h) average for
dk correlations, estimated in [
5] when
k = 3 and in [
6] for every
k ≥ 3. Thus, Corollary
3 and the bound
α3 ≤ 43/96 (see [
4]) imply
for
θ ≤ 1/3, improving [
5] in the low range of short intervals (while [
6] bounds are better for
k = 3 if
θ > 1/2); in fact, their remainders ⋘
N13/12H1/2 (worse than ours when
θ < 1/3) are nontrivial only for
θ > 1/6. Similarly, since
αk < 1, Corollary
4 improves [
6] bounds for the low range (but not for 1/2 <
θ < 1).
Our Lemma 12 (see Remark 13, Section 5) and Ivić’s Theorem 1 [3] prove the following unconditional result.
Corollary 5. For a fixed integer k ≥ 3, let Δk(N; h) be defined as in [6] and for a fixed θ > 2σk − 1, where σk is Carlson’s abscissa, let Nθ ≪ H ≪ Nθ as N → ∞. Then
()
where
ε =
ε(
k,
θ) might depend also on
ε1(
k) > 0 given in [
3].
The novelty of our approach is that, though conditionally, it improves the analogous achievements obtained via the classical moments of the Riemann zeta function on the critical line (a major example being the known upper bounds for Carlson’s abscissa in Ivić [3] results). We think that our conjectures might be approachable by elementary arguments, starting with the so-called k-folding method (Section 7, Proposition 14).
On the other hand, estimates for
Jk(
N,
H) have nontrivial consequences on the 2
kth moments of the Riemann
ζ function on the critical line (see [
7]):
()
At present, we content ourselves with having found an alternative way to pursue possible improvements on the 2
kth moments of
ζ, for at least relatively low values of
k. Indeed, in Section
8, we take a glance at the effect of hypothetical estimates for Selberg integrals on the 2
kth moments, through Theorem 1.1 of [
7], whereas the first author (see
http://arxiv.org/) provides the best known estimate for the 6th moment (under CL).
In Section 8, we prove next link, between Ik(T) bounds and Jk(N, H) bounds (compare [8] conditional bounds).
Theorem 6. Let k ≥ 3 be fixed. If Jk(N, H)⋘N1+AH1+B holds for H ≪ N1−2/k and for some constants A, B ≥ 0, then Ik(T)⋘T1+k(A+B)/2−B (beside the ε dependence, implicit constant may depend on k).
This result encourages us to seek nontrivial bounds for the dk Selberg integrals, in the future.
As already said, we will give some Propositions, besides above results; we will not prove them here. We state Proposition 9 (about proximity of two different forms of d3 expected value in short intervals) in Section 4, while Propositions 14, 16, and 18 (about, resp., the k-folding, the nontrivial arithmetic bound of dk weighted Selberg integrals, and the mean-square proximity of dk expected value in s.i., for all k ≥ 3) are stated in Section 7.
We will provide the necessary Lemmas for the proof of our results in due course, in next sections. In particular, we prove Lemma 7 (our elementary version of Linnik’s method) in Section 3 and Lemma 10 (linking 2nd and 3rd generation) and Lemma 12 (linking 1st and 2nd generation) in Section 5.
1.1. Some Notation and Conventions
If the implicit O and ≪ constants depend on parameters like ε > 0, mostly, we write Oε and ≪ε, but we omit subscripts for the ⋘ symbol. As usual, ε > 0 is arbitrarily small, changing from statement to statement. The relation f ~ g means that f = g + o(g) as the main variable tends to infinity typically. There is no confusion with the dyadic notation, x ~ N, that means . The Möbius function is , if n is the product of r distinct primes, and otherwise. Symbol 1 denotes constant 1 function and 1U is the U characteristic function. The Dirichlet convolution product of the arithmetic functions f1 and f2 is . The k-fold Dirichlet product of f is , so , for all k ≥ 2. For any , we call g the Eratosthenes transform of f if g = f*μ or equivalently f = g*1 (Möbius’ inversion formula): 1 is the Eratosthenes transform of , the divisor function. We omit a ≥ 1 in sums like ∑a≤X . The distance of from is and {α} is the fractional part of α. As usual, , for all , and , for all , for all .
2. Introducing Weighted Selberg Integrals
Given positive integers
N and
H =
o(
N), the
w-
Selberg integral of an arithmetic function
is the weighted quadratic mean
()
where the complex valued
weight w has support in [−
cH,
cH] for some fixed real number
c > 0, so that the inner sum is genuinely finite. The term
Mf(
x,
w) is the expected mean value of
f weighted with
w in the short interval of length ≪
H and depends on
w, when
f has
logarithmic polynomial pf(log
n); see Section
4; set
()
Here, c = deg(pf), with the convention that constant pf ≠ 0 has degree c = 0, while pf = 0 has c = −1.
Clearly, these
Jw,f include the most celebrated Selberg integral
JΛ(
N,
H) =
Ju,Λ(
N,
H), with
being the characteristic function of [1,
H]; more generally,
Jf(
N,
H) is the
u-Selberg integral of
f, while the modified Selberg integral
(introduced in [
9]) is recognizable by taking the
Cesaro weight; say
()
Since
()
where
is the correlation of the weight
u (see next Section
3), it follows that the Cesaro weight is the
normalized correlation of
u. We say that we
smooth Jw,f(
N,
H) when we get the
modified w-Selberg integral of
f:
()
where the new weight
is the
normalized correlation of
w; that is,
()
As a general strategy, from a nonpositive definite weight
w (with support of length ≪
H), we
smooth it; the new weight
has a nonnegative exponential sum
(see next Section
3).
Another important instance, studied by the first author (see [
10–
13]), is the
symmetry integral
()
where
,
for
t ≠ 0, and
Mf(
x, sgn) vanishes identically for every
f. Such a study has been motivated by the link found by Kaczorowski and Perelli [
14], between the classical Selberg integral and the symmetry properties of the prime numbers.
We will exploit the links between the Selberg integral Jf(N, H), the symmetry integral Jsgn,f(N, H), and the modified Selberg integral (also, within other weighted integrals) in the future (see http://arxiv.org/).
Finally, the considerations in Section 4 suggest that a satisfactory general theory on the weighted Selberg integrals may be built within the Selberg Class (see [15] and http://arxiv.org/abs/1205.1706 in which v3 refers to the arxiv paper with same title of present paper).
3. Weighted Selberg Integrals Are Correlation Averages
The
correlation of an arithmetic function
is a shifted convolution sum of the form
()
where
is the
shift. Observe that one may restrict
f to
. Further, a correlation of shift
h is strictly related to a weighted count of
such that
n −
m =
h; namely,
()
We use the last formula to define the correlation of a weight
w by neglecting the
O-term conveniently in the present context; namely,
()
The reason of such a different definition will be clarified after the next lemma, where we prove a strict connection between correlations and weighted Selberg integrals by applying an elementary
dispersion method.
Lemma 7. Let N, H be positive integers such that H → ∞ and H = o(N) as N → ∞. For every uniformly bounded weight w with support in [−cH, cH] and every arithmetic function f, one has
()
where
.
Proof. It is readily seen that, after expanding the square and exchanging sums, it suffices to show, say,
()
Since we may clearly assume that
and
, we write
()
By noting that the condition
n −
b =
m −
a ∈ (
N, 2
N] is implied by
n,
m ∈ (
N +
cH, 2
N −
cH], one has
()
Remark 8. Essentially the remainder term comes from the estimate for short segments of length ≪H within long sums of length ≫N. We refer to these short segments as the tails in the summations. To simplify our exposition, the symbol (T) within some of the next formulas will warn the reader of some tails discarded to abbreviate the formulas themselves.
Thus, by using the
exponential sum (hereafter, we will not specify that
is a finite sum)
()
we write
()
An appealing aspect is that the exponential sums, whose coefficients are correlations of a weight
w, are nonnegative. More precisely,
()
In particular, for the correlations of
u = 1
[1,H], one gets the well-known
Fejér kernel
()
More generally, the Fejér-Riesz theorem [
16] states that every nonnegative exponential sum is the square modulus of another exponential sum. A particularly easy instance of this theorem follows by recalling that the Cesaro weight is the normalized correlation of
u; that is,
(see Section
2). Hence, again Fejér’s kernel makes its appearance in
()
We also say that the Cesaro weights are
positive definite and think that this is an advantage, whereas the Selberg integral and the symmetry integral have weights
u and sgn that are far from being positive definite.
We expect to be able to exploit such a positivity condition, in the future.
4. Essentially Bounded, Balanced, Quasi-Constant, and Stable Arithmetic Functions
The wide class of arithmetic functions under our consideration consists of functions bounded asymptotically by every arbitrarily small power of the variable, in agreement with the definition (i.e.,
f satisfies one of the
Selberg Class axioms, the so-called
Ramanujan hypothesis (see especially [
15])),
()
that we denote briefly by
f⋘1. A well-known prototype of an essentially bounded function is the divisor function
dk, whose Dirichlet series is
ζ(
s)
k. Similarly, the Dirichlet series
()
is defined in at least the right half-plane
, whenever the generating function
f is essentially bounded. Assuming that
F is meromorphic, recall that the expansion of
F at
s = 1 leads to an asymptotic formula,
()
defining
Pf as the
residual polynomial of
f, which either has degree (the
polar order ord
s=1F is the order of the pole of
F at
s = 1. In particular,
k is the polar order of
ζk) ord
s=1F − 1 or vanishes identically when ord
s=1F < 1 (compare [
15] for
F hypotheses). For the remainder term
Rf(
x), a good estimate would be
()
with a suitable 0 ≤
α(
f) < 1 (negative values of
α(
f) are discarded as meaningless); see [
15] formulas.
This is the case for any divisor function
dk. Indeed, from
(*) of Section
1, one has
()
where the degree of the residual polynomial
Pk−1 (see Section
1) is
k − 1, because the polar order of
ζk is
k. Here,
αk ≤ 1 − 1/
k comes, inductively, from the elementary Dirichlet hyperbola method applied to
k = 2 (precisely, Δ
k(
x) ≪
x1−1/klog
k−2x). We find, by partial summation, the logarithmic polynomial
pk−1 such that
()
Thus, the
balanced part of
dk(
n) is obtained by subtracting the very slowly increasing function
pk−1(log
n):
()
Note that the product of
xs/
s and the Dirichlet series of
has zero residue at
s = 1.
Moreover,
(*) is equivalent to
This invites us to formulate the following definitions (although with a different meaning, such a terminology has been coined by Ben Green and Terence Tao. Mainly, Green [
17] calls a function
f −
δ balanced when
f is a characteristic function of a set with density
δ):
()
(i.e.,
F has an analytic continuation in
s = 1),
()
Of course, well-balanced implies balanced (because, from previous bound,
F(
s) is regular at
s = 1). However, the converse needs not to be true; particularly, Λ(
n) − 1 is a balanced function (by the prime number Theorem), but the existence of an exponent
α < 1 is far from being proved (compare [
18]).
An essentially bounded arithmetic function is said to be quasi-constant if there exists such that (the condition on the derivative A′ implies that A is essentially bounded, provided that a depends only on n. However, we do not exclude the possibility that a and A might depend on auxiliary parameters) A′(t)⋘1/t and A = a on . Clearly, the logarithmic polynomial pk−1(logn) is quasi-constant with respect to n. This, together with the fact that is a well-balanced arithmetic function of exponent αk, suggests the following further definition.
An arithmetic function
f is stable of exponent
α if there exist a quasi-constant function
a and a well-balanced function
b of exponent
α such that
f =
a +
b, whereas the amplitude of
f is defined as
()
In what follows, we will assume that stable
f have a logarithmic polynomial (above is the general definition).
Recall that the Dirichlet divisor problem requires to prove the conjectured amplitude α2 = α(d) = 1/4, while one infers α2 ≤ 1/2 by the Dirichlet hyperbola method and the best known bound at the present moment is α2 ≤ 141/416 (Huxley, 2003). In the sequel, αk = α(dk) is the best possible exponent in (*) and (*~).
According to Ivić [
3], the mean value in the Selberg integral of an arithmetic function
f, with Dirichlet series
F(
s) converging absolutely (at least) in
and meromorphic in
, has
analytic form given by
()
where
is the derivative of the residual polynomial
Pf. We remark that
Mf(
x,
H) is linear in
f and is
separable; that is, the variables
H,
x are separated. The
logarithmic polynomial has the property
()
For
f =
dk, this identity permits us to compare [
5,
6] results with ours (see Section
1).
In particular, the analytic form of the mean value in the Selberg integral of
d3 is explicitly given by
()
with
, where
γ is the Euler-Mascheroni constant and
()
Regarding the (short) sum of
d3 weighted with Cesaro weights, that is,
()
Proposition
16 in Section
7 (see Remark
17 too) suggests (from Proposition
14 in Section
7) that the expected mean value is
()
where
N3 = [
N1/3]. Next, Proposition
9 justifies the interchange of
analytic form M3(
x,
H) and
arithmetic form of the mean value
, inside
; its proof is designed for the specific case
k = 3.
Proposition 9. Uniformly, for every x ~ N, one has
()
The idea of the proof (details on arxiv: v3 of this paper) is to apply Amitsur’s formula [
19] with Tull’s error term [
20] (Amitsur derived a symbolic method to calculate main terms of asymptotic formulas. Tull gave a refined partial summation that allows us to transfer the error terms from the formula for the sum ∑
q≤Q d(
q), like Dirichlet’s classical
, to this formula); that is,
()
Finally, by Lemma
7, the Selberg integrals and the modified one for a real and essentially bounded function
f are related to the correlation averages (see Remark
8), respectively, as
()
In particular, when
f is also balanced, that is,
Mf(
x,
H) vanishes identically, from these formulas, we get
()
5. Averages of Correlations: Smoothing Correlations’ Formulas by Arithmetic Means
Recalling that
()
with
, one easily infers the formulas
In particular, for every balanced real function
f⋘1, from the last formulas of the previous section, we get
()
Formulas
(I),
(II), and
(III) correspond, respectively, to the following iterations of correlations’ averages.
-
1st generation:
()
-
2nd generation:
()
-
3rd generation:
()
Such an obstinate process of averaging is motivated by the fact that it is rarely possible to prove an asymptotic formula for the single correlation
. This counts the number of
h-twins (see Section
3) not only when
f is a pure characteristic function (the von Mangoldt function is a typical case). In general, the underlying Diophantine equation is a binary problem still out of reach; see Section 3 of [
7] (and also
http://arxiv.org/: v3 of present paper). On the other hand, for higher generations of correlations’ averages, we get smoother averages; for which, consequently, there is more hope of proving nontrivial estimates. However, even for the 2nd generation, this hope is quite frustrated by the lack of efficient elementary methods to bound the Selberg integral directly. Indeed, Ivić [
3] applies the 2
kth moments of
ζ, since
Jk has a strong connection with them (see Section
8).
Further, it is interesting to analyze the cost of the loss when nontrivial information on the correlations’ averages at some nth generation level is transferred to the (n − 1)th generation (see Lemmas 10 and 12).
In this regard, if
f is real, essentially bounded, and balanced, then the trivial inequality
gives
()
Then, in order to obtain an inequality in the opposite direction, we prove the following lemma.
Lemma 10. For every essentially bounded, balanced, and real arithmetic function f and for every H = o(N),
()
Proof. By applying Cauchy’s inequality and Parseval’s identity, we see that
()
The Lemma is proved (using
H3 ≪
N1/2H5/2), because the formulas for
Jf(
N,
H) and
from
(II) and
(III) yield
()
Now, let us prove Theorems 1 and 2.
Proof of Theorems 1 and 2. First, we note that Mk(x, H) ~ ∑x<n≤x+H pk−1(logn), where pk−1 is the logarithmic polynomial of dk and the implicit remainders give a negligible contribution in view of the stated estimate for Jk(N, H). Then, from Lemma 10, it follows that if, for some H, a nontrivial estimate of the form
()
holds with some gain
G →
∞, then, for the same range of
H, one has
()
Thus, using these inequalities for the balanced part of
dk(
n), that is,
, Theorems
1 and
2 follow by taking, respectively,
G =
HN−ε in the Conjecture CL for
k = 3 and
in the General Conjecture CL for
k = 4,5, 6,7.
Remark 11. Lemma 10 reveals that, applying only Cauchy inequality, a third generation gain G ≪ N/H leads to the gain for the second generation. We say that the exponent gain has halved.
What about the trade of information from the second generation to the first one?
Let us define the deviation of a stable arithmetic function
f, with logarithmic polynomial
a(
n) =
pf(log
n), as
()
Lemma 12 estimates in terms of the Selberg integral of f (when f⋘1 is real and stable).
Lemma 12. Let f be an essentially bounded, stable, and real arithmetic function with amplitude α = α(f). Then, for every H = o(N), one has
()
Proof. Let f = a + b where a(n) = pf(logn) as above and let b be well-balanced. Then, we write
()
By the Cauchy inequality, one has
()
where the mean value is (see Section
2)
, since
a is quasi-constant. Hence, we get
()
and the lemma follows applying partial summation to ∑
x~N a(
x)
b(
x), since
f is stable with amplitude
α.
Remark 13. From previous Lemma, if, for a suitable G = o(H), G → ∞, one has
()
then
()
In particular, for
f =
dk, we get (the essence of Corollary
5; see above)
()
In conclusion, Lemmas
10 and
12 provide the following chain of implications of nontrivial bounds:
()
The exponent gains halves at each step. If it is a neat positive one, we say that
f is
stable through generations.
In the future, we will explore the hard case of stable f (possibly) having no logarithmic polynomial.
6. Asymptotic Formulas for d3 in Almost All Short Intervals
We turn our attention to
J3 nontrivial bounds, postponing those on
Jk to the next section. Ivić [
3] proved that
()
for the width
θ > 1/6, with a neat exponent gain
ε1 > 0. In other words, if
θ >
θ3 is the width range for such an inequality to hold, Ivić has proved
θ3 = 1/6.
This comes from
σ3 ≤ 7/12, where
σk is the
Carlson’s abscissa for the Riemann zeta 2
kth moment:
()
(see [
4] to get some of the known upper bounds for
σk).
Conjecture CL provides improvements on Ivić’s result, since, for every width θ ≤ 1/3, it yields the best possible estimate, that is, the square-root cancellation (compare with the abovementioned lower bound J3(N, H) ≫ NHlog4N holding for 0 < θ < 1/3; see [2]), which in turn implies the optimal θ3 = 0 through the arguments of Section 5 (compare Lemma 10 and Theorems 1 and 2 proofs).
Our estimate for J3 leads to improvements on [5, 6] for d3 deviation. That is what we prove now.
Proof of Corollary 3. From the decomposition introduced in Section 4, one gets
()
where we recall that
d3,
are essentially bounded and
p2(log
n) is a quasi-constant function of
n. Therefore,
()
By applying partial summation and
(*~), one has
()
Since
J3(
N,
H) ≫
NHL4 holds for at least width 0 <
θ < 1/3 (see [
2] and recall that
L = log
N), the conclusion from Theorem
1 follows, because Cauchy’s inequality implies
()
where (compare Section
2)
()
7. Main Theme: From All Long Intervals to Almost All Short Intervals
The inductive identity dk = dk−1*1 invites us to explore the possibility to infer formulas for dk in almost all short intervals, from suitable information on dk−1 in long intervals.
However, the known bounds on the amplitudes
αk seem to be a first serious bottle-neck. Further, while it might be comparatively easy to attack Conjecture CL, the path climbs up drastically when
k ∈ {4,5, 6,7}. Although a general
k-folding method is available (Proposition
14, the core of our elementary approach), we want to prove the following mean-square proximity of analytic and arithmetic forms of s.i. mean value:
()
for a suitable
arithmetic mean value
. The right form is suggested by Proposition
14 that also implies (via the Large Sieve inequality) a nontrivial bound for
in short intervals of width
θ > 1 − 2/
k (for any
w satisfying a property shared by our weights
u,
CH, and sgn), indeed, Proposition
16.
On the other hand, the above mean-square proximity of Mk(x, H) and , at the present moment, is obtained with a gain weaker than N2/k; see next Proposition 18, where we apply the nontrivial estimates for Jk(N, H) given by Ivić [3] in the common range where Proposition 16 also gives nontrivial bounds.
Now, for the so-called
k-folding method, we need some ad hoc notation. For all
k ≥ 2, let us consider
()
where
is the
k-fold Dirichlet product of a fixed arithmetic function
a and the weight
is uniformly bounded with respect to
H, independent of
x. Set
and denote
()
where
()
Proposition 14. If is essentially bounded, then uniformly, for x ∈ [N + H, 2N], one has
()
The proof runs on the lines of (initial part of) the proof of Corollary 1 in [2] and we will omit it.
Remark 15. The uncovered range N < x < N + H gives in mean-square a tail contribution ⋘H3.
We keep hereafter the notation gk(q) even for a = 1 and set .
Let us state the following consequence of Proposition 14, denoting .
Proposition 16. Assume that w is uniformly bounded in [−H, H] and
()
Then, for every integer
k > 2, one has
()
The proof applies the Large Sieve inequality in the form given by Lemma 3 of [13] and will appear elsewhere.
Remark 17. The Cesaro weight CH satisfies previous assumptions: recalling Section 3 last formula and the well-known , if we set , then (for Hq > 0; otherwise, the following vanishes)
()
Similar calculations prove that w = u and w = sgn satisfy previous Proposition assumptions, too.
The replacement, in general, of analytic Mf by arithmetic lies at the heart of our arithmetic approach.
We see that the analytic form
is close to the
arithmetic form in Proposition
16:
()
Proposition 18. For every integer k ≥ 3, there exists G = G(k) > 0 such that
()
The proof follows by comparing (above quoted) Ivić’s bounds with Proposition 16 and will appear elsewhere.
Notice that, for k = 3, this Proposition yields a much weaker inequality than Proposition 9.
Remark 19. As for the case k = 3, implications in Section 5 and Propositions 16 and 18 immediately prove Corollary 4.
8. Conditional Bounds for the Moments of ζ on the Critical Line
We do not use explicitly any deep property of ζ, relying upon αk known bounds, but these, actually, follow from nontrivial Ik(T) estimates (for a clear digression on this; see the wonderful book [4] by Ivić).
On the other hand, bounds for Jk have nontrivial consequences on Ik(T), as applied in next proof.
Proof of Theorem 6. For T → ∞ and for every ε > 0, Theorem 1.1 of [7] yields (⋘ saves Oε(Tε) here)
()
Indeed, in Theorem 1.1 of [
7], one finds the same inequality, with
Jk(
N,
H) replaced by the Selberg integral
()
However, an easy dyadic argument allows one to replace [
Hxε,
x] by an interval like [
N, 2
N], while the substitution of the integral on [
N, 2
N] with
Jk(
N,
H) generates only negligible remainder terms.
Since Jk(N, H)⋘N1+AH1+B holds for H ≪ N1−2/k, by hypothesis, the previous inequality implies
()
Remark 20. Define the excess Ek to be such that ; Theorem 6 yields Ek = (k/2)(A + B) − B. Known values are E3 = 1/4, E4 = 1/2, E5 = 3/4 followed by Hölder’s inequality from E2 = 0 and E6 = 1 (see Section 2 in [7]).
According to Section 5, our Ek belong to a 2nd generation approach, while Ivić’s [8] is a 1st generation one. Under Conjecture CL, our new approach, based on a modified version of Gallagher’s Lemma [21], gives E3 = 0.
