-NOTATIONWe let N ∈ N, N → ∞ and (x is an integer) write x ∼ N for N < x ≤ 2NX
Sf± (x, h) :=f(n)sgn(n − x)
SYMMETRY SUM of f|n−x|≤h(f is a generic arithmetical function, usually real-valued)

If± (x, h) :=X

Sf± (x, h)

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SYMMETRY INTEGRAL of fx∼N(actually, we use this “discrete” version instead of the integral; like for :)Jf± (x, h) :=X2X

f(n) − Mf (x, h)

SELBERG INTEGRAL of fx∼N 0<|n−x|≤h(actually, Selberg’s integral is

R 2N P| x<n≤x+h Λ(n)−h|2 dx; here Λ(pk ) = log p)
Nwhere Mf (x, h) is the MEAN VALUE of f (esp. f = Λ ⇒ Mf (x, h) = 2h)-MOTIVATION
Our work starts from a Kaczorowski-Perelli paper RELATING If AND Jf IN
THE CASE OF f(n) = Λ(n).
The link is (in suitable ranges of h)
If “small” ⇔ Jf “small”(in both cases, “small” means that “has a non-trivial bound”).
Bounding IΛ is hopeless at the moment.
Thus we studied “easier” arithmetical functions f (often REAL ones).

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-SOMEESTIMATES• f(n) = d(n)FOR If (N, h)divisor function

C.-Salerno, AA(2004).

divisor sums (σ > 0)

C., INTEGERS(2004).

• f(n) = µ2 (n)square-free numbers

C., R.M.UN.PARMA(2004).

• f(n) = µ2 (n)g(n)sq.-f. supported fC., JIPAM(2004).

• f(n) = λ(n)Hecke eigenvalues

C.-Iwaniec, unpublished.

• f(n) =X

d−sd|n-TECHNIQUES
Two kinds of methods :

“divisor sums”(σ ≥ 0)
square-free numbers
Heckeeigenvalues
=LARGE SIEVE=DISPERSION & ASYMPTOTICS of:Pn∼N λ(n)λ(n + k) (available)(actually, here we need estimates for “short intervals” shifts k, i.e. k  h)This last approach works also for the other cases, asX
d(n)d(n + k)
(BINARY ADDITIVE DIVISOR)n∼NXµ2 (n)µ2 (n + k)(“BINARY ADDITIVE SQUAREFREE”)n∼N

HAVE KNOWN ASYMPOTIC FORMULAE.


NOT always KNOWN ASYMPTOTICS forPn∼N f(n)f(n + k).

THEN, if possible, USE LARGE SIEVE.

2-RECENT DEVELOPMENT
Since the Large-Sieve “BREAKS AT LEVEL 1/2”, I.E.
√THE MAGNITUDE OF d (DIVISORS) CAN’T GO BEYOND

Nwe can introduce a new ingredient to circumvent the problem.
For another (“NEW”) CLASS of functions we CAN USE L-S INEQUALITY.
This isf(n) =Xg(d),with g(n) d|n1, ε > 0.
nεIn this case we use RAMANUJAN EXPANSION (cq (n)=Ramanujan sum)f(n) =∞X

aq (f)cq (n),q=1which (thanks to WINTNER’S LEMMA & the bound for g) is ABSOLUTELYCONVERGENT ∀n ∈ N, WITH aq (f) (RAMANUJAN COEFFICIENT)
aq (f) =∞Xg(m)11 X g(jq)=ε 1+ε .
mqjq
m=1j∈Nm≡0( mod q)

This fact (splitting “LOW” and “HIGH” divisors) gives
L-S inequality ⇒ If (N, h) NON-TRIVIAL.
PThus we recover some previous cases, esp. d|n d−s (BUT WITH σ > 0)
Also, we get new ones, like the Jordan TOTIENT functions.
(The Large Sieve ALON E requires g(n)  n1ε WITH ε > 1/2).

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- D E T A I L S and I D E A S
In the sequel we will assume the following.
We will implicitly assume N, h to be natural numbers, with h = h(N ) → ∞ as
N → ∞ and h = o(N ).
(In this way we’re studying our f in ALMOST ALL SHORT INTERVALS of
the type [x − h, x + h]; here N < x ≤ 2N .)In the following we’ll write L := log N .

• f(n) = d(n)DIVISOR FUNCTION

For the divisor function f(n) = d(n) we get both results for the symmetry
integral Id & for the Selberg integral Jd .
We quote here our main result about Id in [C-Salerno,AA]:h ≤ N 1/2−η for η > 0 ⇒ Id (N, h) =√pN
163
Nhlog+ O(N hL5/2 log L)
2πh(See Corollary 1 and compare Theorem 1)

MAIN INGREDIENTSDIRICHLET’S HYPERBOLA (“FLIPPING”)“ASYMPTOTIC” LARGE SIEVE4

• f(n) =P−s:= σ−s(n),d|n dsay (σ = <(s) > 0)

DIVISOR SUMS

Here the result is a little bit more involved. First of all, assume σ > 0.

We need to defineη(h) (s) :=∞Xh −sn ,nn=1s∈Cwhere kαk := minn∈Z |α − n| is the DISTANCE TO THE INTEGERS.
In this case from Corollary 1 of [Coppola,INTEGERS] we get for the symmetry
integral of f, say Is (N, h), the result

h = N θ, θ <min(σ, 1/2)
|ζ(1 + s)|2
⇒ Is (N, h) = 2N η(h) (2σ) + E(N, h),2+σζ(2 + 2σ)with the remainder E(N, h) = E(N, h, s) = o(N ) (and here the o can depend
however on s, σ, =(s), |s|).

MAIN INGREDIENTSFLIPPING
ASYMPTOTIC LARGE SIEVE(Together with partial summation & standard estimates)5• f(n) = µ2 (n)SQUARE-FREE NUMBERS

From Corollary 2 of [Coppola,RIV.MAT.UNIV.PARMA] we get√h ≤ N 4/15−η , for some η > 0 ⇒ Iµ2 (N, h)  N h.

This time we use the well-known identityµ2 (n) =X

µ(d)

d2 |n

which allows us to “lower” the level (i.e., the magnitude of the divisors).
Also, our treatment doesn’t look at the inner µ(d) (using only its boundedness),so may as well apply to any g likeg(n) =Xb(d),d2 |nwhatever is the (BOUNDED) b(d).

In fact, we use: “LOW DENSITY” OF DIVISORSLARGE SIEVE INEQUALITY
MAIN INGREDIENTSTREATMENT OF SPORADIC TERMS

6• f(n) = µ2 (n)g(n)SQUARE-FREE SUPPORTED

We quote Thm.1.1 & Thm.1.3 [Coppola, JIPAM]. Write
kgk :=max|g(n)|;N −h<n≤2N +habbreviate: compl./c. = completely, mult. = multiplicative, add. = additive.
√Assume h/L2 → ∞, as N → ∞ and J = J(N ) → ∞, J  h/L as N → ∞.
Then
XN hN h2
g compl.mult. ⇒ Ig (N, h)  L2 max+ 2 kgk2
d2Iµ2 g,22DJd dJd∼DX

g compl.mult. ⇒ Iµ2 g (N, h)  L2 max
DJd2Ig

d∼DN h,d2 d2

+N h2
kgk2
J2Also,2g c.add. ⇒ Ig (N, h)  L max
DJg c.add. ⇒ I

µ2 g

X

2d Iµ2 g

d∼D2(N, h)  L max
DJX

2d Ig

d∼DN h,d2 d2N h,d2 d2+(√N h2+ N J hL2 )kgk22J+(N h2+ N JL2)kgk2
J2

At the same time, we also have, for the symmetry integral of 2Ω(n), with Ω(n)the total number of p|n (∀ε > 0 FIXED)
√ !
N3h=o
⇒ I2Ω(n) (N, h)  N h 2 N ε .
L

MAIN INGREDIENTSLink Ig ↔ Iµ2 g : ELEMENTARY
Sym.Int.of 2Ω(n) : SYMMETRY of d(n) in AP(AP = ARITHMETIC PROGRESSIONS)

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• f(n) = λ(n)HECKE EIGENVALUES

We need, here, to introduce a “SMOOTHED SYMMETRY INTEGRAL”, i.e.
Z +∞
X2(g)
If (N, h) :=f(n)g(n)sgn(n − x) dx,−∞|n−x|≤h(here g is “smooth” and g(x) = 1, ∀x ∈ [N − h, 2N + h])
where, now, the symmetry integral is (again, for technical reasons) defined as
If (N, h) :=Z 2NNX

2f(n)sgn(n − x) dx.

|n−x|≤hUsing a POSITIVITY ARGUMENT, we get an upper bound for Iλ in terms of(g)asymptotics for Iλ , i.e. its “smoothed” version.
These asymptotics, in turn, are obtained using the DISPERSION METHOD(which applies to symmetry integrals, even better than to Selberg integrals !),combined with ASYMPTOTICS FORX
f(n)f(n + k)(k, the shift, is O(h))
n∼N(here, say, f(n) = λ(n)g(n), i.e. “SMOOTHED HECKE EIGENVALUES”)
AVAILABLE IN THE LITERATURE (due to Conrey-Iwaniec, see AA).
Summarizing (and avoiding technicalities) for the result:h  N 1/4 ⇒ Iλ (N, h) k,q N hL3
and for the proof:(k = weight, q = level) POSITIVITY
MAIN INGREDIENTSDISPERSIONASYMPTOTICS FOR SMOOTHED E.V.
(E.V. = HECKE EIGENVALUES)
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