Introducing weighted Selberg integrals

Giovanni Coppola

University of Salerno

1Introducing weighted Selberg integrals
Giovanni Coppola
Atle Selberg, in 1946, studied prime numbers in almost all
short intervals [S] estimating
Z 2NX
2Λ(n) − H dx,
Nx<n≤x+H

where Λ is the well-known von Mangoldt function, namelydef

Λ(pr ) = log p, ∀p primes and ∀r ∈ N, and 0 outside primepowers. This is the “classical”, so to speak, Selberg integral.
In the 1990s, Kaczorowski & Perelli [KP] discovered aremarkable link between classic Selberg integral above andthe “symmetry integral of the primes”,
Z 2NXX2
Λ(n) −Λ(n) dx.
Nx<n≤x+Hx−H≤n<x

Discrete versions of these integrals are, for any arithmetic
function f , the Selberg integral of f ,XX2
def
Jf (N, H) =f (n) − Mf (x, H) ,N <x≤2Nx<n≤x+H

Mf (x, H) is the “short interval mean-value”(expected valueof short sum) and the symmetry integral of f ,XX2
def
Jsgn,f (N, H) =sgn(n − x)f (n) .
N <x≤2Nx−H≤n≤x+H2

I introduced(2010) the modified Selberg integral of fXdefJef (N, H) =N <x≤2N1 X
HXf (n)−Mf (x, H)

2h≤H x−h<n<x+hwhere Mf (x, H) is the same mean-value above in Jf (N, H).
Defining the Cesaro weight,


|a|
def
C(a) = max 1 −,0 ,H
we see that the modified Selberg integral of f isJef (N, H) =X

N <x≤2NX

2C(n − x)f (n) − Mf (x, H) .

nIn fact, these three integrals are particular instances ofso-called “weighted Selberg integral”of f ,def

Jw,f (N, H) =XN <x≤2NX2w(n − x)f (n) − Mf (x, w) ,nthat we (with Laporta) [CL] introduced in Generations.
(We’ll refer to “Generations”, here, arxiv’s 1205.1706v3.)
Here w : [−H, H] → C is absolutely bounded and Mf (x, w)
depends on w. Selberg integral and symmetry integral havew equal to “u”, characteristic function of (0, H] and to signfunction “sgn”, resp.; modified Selberg integral comes with
the Cesaro weight C.
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For these weights w : [−H, H] → C the mass isdef Xw(0)b=w(a)
aP
and we define, for f (n) = d|n g(d), the arithmetic form
(see Generations’ §2) of s.i. mean-value as
X g(q)def
Mf (x, w) = w(0)
b,
q
q

def P
compare Generations’ §3. We call g(q) =d|q µ(d)f (q/d)
the Eratosthenes transform of f .
Actually, in case f = dk things are more complicatedand s.i. mean-value is defined in terms of residues, say, byanalytic form; also the arithmetic form depends strongly
on x so that our Generations’ “k−folding method”, say,proves proximity of these two forms.

However, now on, for clarity we will confine to functions
f = g ∗ 1 (above), where g nor its support depend on x.
I started study of these, say, “sieve functions”, in the
2000s. For them, the s.i. mean-value doesn’t depend on x
so may be defined as above.
Methods used for these sieve functions in the case of
Selberg integral and symmetry integral successfully adapt
to a class of more general weights. We’ll see this in theTheorems 2 & 3, following.
I start with a very recent result : it roughly says that a
non-trivial symmetry integral is “inherited”, from Eratosthenes transform g, to f = g ∗ 1. Proof uses Large Sieve.
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We will abbreviate H  N Θ for N Θ  H  N Θ , say, Θ is
the width of the short interval [x − H, x + H]. Also, writeh  N θ for N θ  h  N θ , say, θ is the width of shorterinterval [x − h, x + h].
We abbreviate g≪ 1 and say g is “essentially bounded”,
when ∀ε > 0 g(n) ε nε , n → ∞.
We link the symmetry integral of our f = g ∗ 1, to thesymmetry integral of its Eratosthenes transform, g.
Theorem 1. Fix Θ ∈ (1/3, 1), let H  N Θ as N → ∞.
Let g : N → R be essentially bounded. Then ∃G ∈ (0, 1):XX
2
sgn(q − x)g(q)  N h2−G , ∀θ ∈ (0, Θ]x∼Nx−h≤q≤x+h⇓
Xx∼NXsgn(n − x)(g ∗ 1)(n)

2

 NH

2−G0,x−H≤n≤x+Hwhere G0 > 0 depends on G > 0 and Θ.
Thus, we have inheritance of non-trivial symmetry integral,from g to g ∗ 1 (from f Eratosthenes transform to f ).
Unfortunately, the width here has to be Θ > 1/3, due totechnique (Large Sieve inequality) used in the Proof. Thisis rather elementary, since it applies only Cauchy inequality.
Our second result is even more elementary (not even aCauchy inequality!), while our third applies state-of-the-art
very sophisticated techniques (due to Duke, Friedlander &Iwaniec, [DFI], for bilinear forms of Kloosterman fractions).
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We define the correlation Cw of w : [−H, H] → C asdef

Cw (a) =XXw(h2 )w(h1 )h2 −h1 =aand, then, we call w an arithmetic weight when its correlation satisfies(A)

Xa≡0(mod `)1X
Cw (a) =Cw (a) + O(H),` a∀` ≤ 2H

For example, our weights u, sgn, C are arithmetic.
Furthermore, we need another definition:
defw is good ⇐⇒ w is arithmetic & absolutely boundedi.e., “absolutely bounded”, here, means |w| ≤ K and K > 0
is absolute (say, 1, 2, 4), independent of H.
For example, our main weights u, sgn, C are good.
The “modified Vinogradov notation”(used by G. Kolesnik)
defA≪ B ⇐⇒ A ε N ε B,

∀ε > 0is very useful to save arbitrary small powers.
We have a general elementary result (doesn’t even use
Cauchy inequality), for weighted Selberg integrals, withgood and real w, of essentially bounded arithmetic functions f .
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Theorem 2. Let N, H ∈ N, H → ∞, H = o(N ) as
N → ∞. Assume Q = Q(N, H) → ∞ and g : N → R is
an arithmetic function with support inside [1, Q], abbreviate
f = g ∗1≪ 1. Then, for all good weights w : [−H, H] → R,Jw,f (N, H)≪ N H + Q2 H + QH 2 + H 3 .

We give a very brief sketch of the Proof.
• We apply a kind of elementary Dispersion Method (say,we open the square and exchange sums): then ourweighted Selberg integral is a correlations’ average.
• Main terms are cancelled out (+negligible remainders).
(Here we use hypothesis: w is good.)• For other terms, a positivity argument for the “discreteFourier transforms” of correlations (whose DFT are infact always non-negative: immediate).
We we will not sketch the Proof of next result, but only saythat it runs approximately as this; except, that last stepis substituted by an application (via fractional parts’ finite
Fourier expansions) of [DFI] quoted results for averages ofKloosterman sums (in turn, they, after many other tools,
apply Weil’s bound for Kloosterman sums stemming from
“Weil’s RH for curves over finite fields”).

7We state a companion result, applying a very technicalresult on bilinear forms of Kloosterman fractions [DFI].
Theorem 3. Fix 0 < θ < 1/2. Let N, H ∈ N, H  N θ , as
N → ∞. Assume Q = Q(N, H) → ∞ and g : N → R is
an arithmetic function with support inside [1, Q], abbreviate
f = g ∗ 1≪ 1. Then, for all good weights w : [−H, H] → Rand ∀δ > 0,Jw,f (N, H)≪ N H + N δ Q95/48 H 2 + N 1−2δ/3 H 2 + QH 2 .
Thus we may summarize Theorem 2 and Theorem 3 inthe following.
Corollary. Fix 0 < θ < 1/2. Let N, H ∈ N, H  N θ ,as N → ∞. Assume Q  N λ , with, say, level λ =λ(θ) in the range λ ∈ (0, max(1/2 + 1/190, (1 + θ)/2)) andg : N → R is an arithmetic function with support inside[1, Q], abbreviate f = g ∗ 1≪ 1. Then, for all good weights
w : [−H, H] → R, ∃δ = δ(θ, λ) > 0 :Jw,f (N, H)  N H 2 /N δ .
See that usual approach to correlations via the arithmetic
progressions suffers from the “level 1/2 barrier”, due to theLarge Sieve inequality. Our new approach through finite
Fourier expansions and [DFI] results enables us to bypass
this barrier: we reach level 1/2 + 1/190 in Corollary (ofcourse, not for arithmetic progressions !) .
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Remark. An application is to the well-known “truncatedvon Mangoldt functions”, with parameter R, introduced byGoldston - Pintz - Yildirim (say Q = R  N 1/2+1/190−ε ).
In fact, our f = g ∗ 1 plays the rôle of their ΛR .
As regards Theorem 1, not only symmetry is inherited, butalso proximity of Selberg integral & the modified one (weconsider weight C −u, has mass 0, then Cauchy inequality).
As we saw, there are a variety of techniques to boundweighted Selberg integrals, many to come, too...
Here is a “short bibliography”, for the talk. (By the way,see the ArXiV for the present results to come !)References[C] Coppola, G. - On the Selberg integral of the k-divisor
function and the 2k-th moment of the Riemann zetafunction - Publ. Inst. Math.(Beograd) (N.S.) 88(102)(2010), 99–110. MR 2011m : 11173 - available online[CL] Coppola, G. and Laporta, M. - Generations of correlation averages - http://arxiv.org/abs/1205.1706v3[DFI] Duke, W., Friedlander, J. and Iwaniec, H. - Bilinear
forms with Kloosterman fractions - Invent. Math.
128 (1997), no. 1, 23–43. MR 97m : 11109[KP] Kaczorowski, J. and Perelli, A. - On the distributionof primes in short intervals - J. Math. Soc. Japan 45(1993), no. 3, 447–458. MR 94e : 11100[S] Selberg, A. - On the normal density of primes in smallintervals and the difference between consecutive primes- Arch. Math. Naturvid. 47 (1943), no. 6, 87–105.
MR 7, 48e
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THANK YOU !!!

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