SIEVE FUNCTIONS IN ARITHMETIC BANDS, II Giovanni Coppola∗and Maurizio Laporta∗∗ ∗Via Partenio 12, 83100 Avellino (AV), Italy
∗∗Dipartimento di Matematica e Applicazioni ”R. Caccioppoli”,
Universit`a degli Studi di Napoli ”Federico II”,
Compl. di Monte S. Angelo, Via Cinthia, 80126 Napoli (NA), Italy
e-mails: [email protected]; [email protected]
(Received 8 April 2017; after final revision 29 May 2017; accepted 25 July 2017)
An arithmetic functionf is called a sieve function of rangeQif its Eratosthenes transformg=
f ∗µis supported in[1, Q]∩N, whereg(q) ¿ε qε(∀ε > 0). We continue our study of the
distribution off(n)over short arithmetic bands,n≡ ar+b(modq), withn ∈(N,2N]∩N,
1≤a≤H =o(N)andr, b∈Zsuch thatg.c.d.(r, q) = 1. In particular, the optimality of some results is discussed.
Key words : Eratosthenes transform; arithmetic progressions; short intervals.
1. INTRODUCTION
Given an arithmetic functionf :N→C, for everyN ∈Nandα∈Rlet us set
c
fN(α) def
= X
n∼N
f(n)e(nα),
wheren∼Nmeans thatn∈ N def= (N,2N]∩Nande(α)def= e2πiα, as usual. Iff is the convolution product ofgand the constantly1function, i.e.
f(n) = (g∗1)(n) = X
d|n
g(d),
we say, with Wintner [11], thatg = f ∗µis the Eratosthenes transform off (where µis the well-known M¨obius function). We callf a sieve function of rangeQ if its Eratosthenes transformg is
Indian J. Pure Appl. Math., 49(2): 301-311, June 2018
c
essentially bounded, namelyg(d) ¿ε dε,∀ε >0, and vanishes outside[1, Q]for someQ∈ N, that is
f(n) = X
d|n d≤Q
g(d).
Note thatf =g∗1is essentially bounded if and only ifgis.
As usual, ¿ is Vinogradov’s notation, synonymous to Landau’sO-notation. In particular, ¿ε means that the implicit constant might depend on an arbitrarily small and positive real numberε, which might change at each occurrence.
In this paper we continue our study of the distribution of a real sieve functionf over short arith-metic bands, i.e. over the set
[
1≤a≤H
{n∈(N,2N] :n≡a(q)}, withH =o(N).
Henceforth we writen≡a(q)to abbreviaten≡a( mod q)and omita≥1in sums likePa≤H. Let us recall that in [2] we have proved that the inequality
Tf(q, N, H)def= X
a≤H ³ X
n∼N n≡a(q)
f(n)−fcN(0)
q
´
¿εNε ³N
q +q+Q
´
(1.1)
holds for every real sieve functionf of rangeQ¿N, after assuming thatH =o(q), asq → ∞, and
q=o(N), asN → ∞. Such conditions are required in order to avoid overlapping and sporadicity of the arithmetic bands, respectively.
Here we generalize the previous inequality for
Tf(q, r, b, N, H) def= X
a≤H
³ X
n∼N n≡ar+b(q)
f(n)−fcN(0)
q
´
, (1.2)
where r, b ∈ Z are such that(r, q) = 1 (hereafter (r, q) = g.c.d.(r, q), as usual). In particular, note thatTf(q,1,0, N, H) = Tf(q, N, H). Beyond the fact that we give a more accurate proof, in the present paper we also take the chance to provide with two further corollaries and discuss the optimality of the following inequality.
Theorem 1.1 — Letq, N, H, Q∈Nbe such thatQ¿ N,H = o(q)andq =o(N). For every
sieve functionf :N→Rof rangeQand allr, b∈Zsuch that(r, q) = 1, one has
Remark 1.1 : BeingTf(q, r, b, N, H)¿N1+εH/qa trivial bound, both (1.1) and (1.3) are non-trivial once the widthθ def= loglogHN is positive, q ¿ √N1−δH andqQ ¿ N1−δH, for a suitable
δ > 0. Since it is assumed thatQ ¿ q ¿ Qhereafter, we get a boundQ ¿ √
N1−δH and thus we can go beyond the square-root ofN (forθ > 0). Consistently with the terminology introduced in [2], we stop at the barrier loglogNQ < 1+2θ. In the final section of [2], our study of the distribution of sieve functions in arithmetic bands is compared to the classical results on the distribution in arithmetic progressions, which deal mostly with the overcoming of the so-called level1/2. Our present results, insofar they generalize our previous ones, already go beyond level1/2. In this paper we discuss the possibility of going beyond1/2+θ/2in the present contest of the sieve functions in arithmetic bands, namely by takingQ >
√
N1+δH for a certain small and fixedδ > 0, so thatN1+εH/Q = o(Q), provided δ > ε. In §3 we exhibit a particular sieve function whose behavior in arithmetic bands confirms the optimality of such level.
The distribution in arithmetic bands of truncated functions of the type
f(n) =X
d|n d≤Q
g(d)
is somehow involved in the study of convolution sums like X
n≤N
f1(n)f2(n−a).
Indeed, by assuming thatf1 =g1∗1andf2 =g2∗1, the sum X
n≤N
f1(n)f2(n−a) = X
n≤N X
d|n
g1(d) X
q|n−a
g2(q)
= X
d≤N
g1(d) X
q≤N−a
g2(q) X
n≤N n≡0 (d) n≡a(q)
1
will not be altered if one sets
f1(n) = X
d|n d≤N
g1(d) and f2(n) = X
d|n−a d≤N−a
g2(d).
Further, if the shiftais confined toa¿ H, then the conditionsn≤ N andH =o(N)clearly yieldmax(n, n−a)≤N +|a| ≤2N. Moreover, iff1andf2are essentially bounded, then trivially
X
n≤N
f1(n)f2(n)¿N1+ε
and for anya¿Hone has X
n≤N
f1(n)f2(n−a) =
X X
n1,n2≤N n2−n1=a
f1(n1)f2(n2) +Oε(NεH).
However, only under the more severe conditions gi(d) ¿ d−ε for some ε > 0, (i = 1,2), Murty, Saha and the first author [4] have recently established more explicit asymptotic formulae for such convolution sums in terms of finite Ramanujan expansions (see also [5] and [7] for a deeper understanding of such a connection). Regarding the case of sieve functionsf1, f2we have to content ourself with the asymptotic formula given in [2], Corollary 1.7, for the first generation average sum
X
a≤H X
n≤N
f1(n)f2(n−a).
By applying Theorem 1.1, here we give a slight generalization of such an asymptotic formula for the correlation of real arithmetic functionsf1 andf2 inN, i.e.
f1,f2(a) def
= X
n∼N
f1(n)f2(n−a).
Corollary 1.1 — Letq, N, H, Q1, Q2 be positive integers such thatH = o(q),q = o(N) and
Q1 ≤ Q2 ¿ N, asq, N → ∞. For any real and essentially bounded arithmetic functionsg1 and
g2 supported in [1, Q1]and[1, Q2], respectively, and for allr ∈ N,b ∈ Zsuch that(r, b) = 1and
H−b≥r, one has X
a≤H a≡b(r)
f1,f2(a) =R1(f1)R1(f2)N(H−b)/r+Oε ¡
Nε(N +Q22+Q1H) ¢
,
where, forj = 1,2, we setfj =gj∗1andR1(fj)def= P
d≤Qj gj(d)
d is the so-called first Ramanujan coefficient offj.
Finally, in [1] we studied the so-called length inertia property for weighted Selberg integrals (see [2] for the link between such integrals and the distribution of a sieve function in arithmetic bands). Such a property permits transfer of non-trivial bounds in short intervals of lengthh, say, to similar bounds in longer intervals of lengthH. Now we can show that a length inertia property holds also for the distribution of a sieve function in arithmetic bands. To this aim, forH=∞(h)(that ish=o(H), asH → ∞) let us consider (recall (1.2))
Tf(q,1,0, N,[H/h]h) = X
a≤[H/h]h ³ X
n∼N n≡a(q)
f(n)−fcN(0)
q
´
,
C
where[x]is the integer part ofx∈R.
Corollary 1.2 — Letq, N, H, Q ∈Nbe such thatQ ¿ N,h =o(q), for someh = o(H), and
q=o(N). For every sieve functionf :N→Rof rangeQ, one has
Tf(q,1,0, N,[H/h]h) = X
j≤[H/h]
Tf(q,1,(j−1)h, N, h)¿εNε(N/q+q+Q)H/h.
2. PROOFS OFTHEOREM1.1ANDITSCOROLLARIES
In [2] we have established some formulæ that relate the`th Ramanujan coefficient of a real sieve functionf =g∗1of rangeQ, i.e.
R`(f)def= X
d≤Q d≡0 (`)
g(d)
d ,
to the values offcN attained at rational numbers, whereas it is easily seen that
R`(f) = 1` X
m≤Q/`
g(`m)
m ¿ε Qε
` . (2.1)
In order to prove Theorem 1.1 and Corollary 1.1, we need to quote such properties that here are included in the following lemma.
Lemma 2.1 — Letf =g∗1be a sieve function of rangeQ¿N, withQ, N ∈N. Then
c
fN(0) = R1(f)N +Oε ¡
Q1+ε¢; (2.2)
c
fN ³j
`
´
= R`(f)N +Oε((`Q)ε(Q+`)), (2.3)
∀` >1,∀j ∈Z∗` ={1≤j≤`: (j, `) = 1}.
PROOF: From X
n∼N n≡a(q)
f(n) = X
n∼N n≡a(q)
X
d≤Q d|n
g(d) =X
d≤Q
g(d) X
n∼N n≡a(q) n≡0 (d)
1
= X
d≤Q (d,q)|a
g(d) X
n∼N/d nd≡a(q)
1 = X
d≤Q (d,q)|a
g(d)
µ
N
dq(d, q) +O(1)
¶
= N
q
X
d≤Q (d,q)|a
g(d)
d (d, q) +Oε
¡
Q1+ε¢,
we get (2.2) as long asq= 1. Now, let us assume that1≤j < `with(j, `) = 1, and write
c
fN(j/`) = X
d≤Q
g(d) X
v∼Nd
e`(jdv)
= X
d≤Q d≡0 (`)
g(d)
³N
d +O(1)
´
+O³ X
d≤Q d6≡0 (`)
|g(d)| kjd/`k
´
,
wherekrkdef= minn∈Z|r−n|,∀r∈R. Thus, (2.3) follows from X
d≤Q d≡0 (`)
g(d)
³N
d +O(1)
´
=R`(f)N+Oε ³
Qε
³Q
` + 1
´´ , and X d≤Q d6≡0 (`) 1
kjd/`k ≤ X
0<|r|≤`/2 X
d≤Q jd≡r(`)
1
|r/`| ¿` X
r≤`/2
1
r
³Q
` + 1
´
¿ε`ε(Q+`).
The proof is completed. 2
PROOF OF THEOREM 1.1 : By the orthogonality of additive characterseq(t)def= e(t/q), (q ∈N,
t∈Z), we get
Tf(q, r, b, N, H) = 1
q
X
a≤H ³ X
n∼N
f(n)X
k≤q
eq ¡
k(ar+b−n)¢−fcN(0)
´ = 1 q X k<q X
a≤H
eq(k(ar+b))fcN(−k/q)
= 1 q X `|q `>1 X
j∈Z∗ ` c fN µ −j ` ¶ X
a≤H
e`(j(ar+b)),
where we have set`=q/(k, q)andj =k/(k, q). Since(r, q) = 1, for any`|qthere exists an integer
rsuch thatrr≡1 (mod`). Therefore we write
Tf(q, r, b, N, H) = 1
q
X
`|q `>1
X
j∈Z∗ ` c fN µ −jr ` ¶
e`(jrb)X
a≤H
e`(ja).
By applying (2.1), (2.3) and the well-known inequality (see [6], Ch. 25) X
V1<v≤V2
e(vα)¿min
³
V2−V1,kα1k ´
we conclude that
Tf(q, r, b, N, H) ¿ε 1q X
`|q `>1
³
|R`(f)|N+ (`Q)ε(Q+`) ´ X
j∈Z∗ `
1
kj/`k
¿ε N ε q X `|q `>1 ³N
` +Q+`
´
`¿εNε ³N
q +Q+q
´
,
that is (1.3). 2
Remark 2.1 : Besides (1.3), from the previous proof it transpires that also the upper bound of
Tf(q, r, b, N, H) = 1
q
X
`|q `>1
X
j∈Z∗ ` c fN µ −jr ` ¶
e`(jrb) X
a≤H
e`(ja)
¿ 1 q X `|q `>1 ` X j≤`/2 (j,`)=1 1
j maxj∈Z∗ `
|fcN(j/`)|
is independent of bothbandrsuch that(r, q) = 1(that is to say, it is the same upper bound obtained forr= 1andb= 0).
PROOF OFCOROLLARY1.1 : Let us write
f1,f2(a) = X
n∼N
f1(n)f2(n−a) = X
n∼N
f1(n) X
q|n−a q≤Q2
g2(q)
= X
q≤Q2
g2(q) X
n∼N n≡a(q)
f1(n).
Now, note that, since without loss of generality we can assume|b| ≤r, fromH =o(q)one has that(H−b)/r=o(q), asq → ∞. Therefore, from (2.2) and Theorem 1.1 we obtain
X
a≤H a≡b(r)
f1,f2(a) = X
q≤Q2
g2(q) X
s≤(H−b)/r X
n∼N n≡sr+b(q)
f1(n)
= X
q≤Q2
g2(q) Ã
H−b qr
X
n∼N
f1(n) +Tf1 ³
q, r, b, N,(H−b)/(qr)
´!
= H−b
r
³ X
q≤Q2
g2(q)
q
´ ¡
R1(f1)N +Oε¡Q1+1 ε¢¢
+Oε
³
Nε X
q≤Q2 ³N
q +q+Q1
´´
= R1(f1)R1(f2)N(H−b)/r+Oε ¡
Nε(N+Q22+Q1H) ¢
,
C
C
that is the formula of the Corollary. 2
PROOF OFCOROLLARY1.2 : Since
Tf(q,1,0, N,[H/h]h) = X
j≤[H/h]
³ X
(j−1)h<a≤jh X
n∼N n≡a(q)
f(n)−h
qfcN(0) ´
= X
j≤[H/h] ³ X
a≤h
X
n∼N n≡a+(j−1)h(q)
f(n)−h
qfcN(0) ´
= X
j≤[H/h]
Tf(q,1,(j−1)h, N, h),
the Corollary follows immediately from Theorem 1.1. 2
Remark 2.2 : Note that the above equation holds for anyf, not necessarily a sieve function.
3. SUPPORTING THEOPTIMALITY OF THELEVEL
Let us setsgn(x)def= x/|x|for allx∈R\ {0}andsgn(0)def= 0. Then, for any fixedq∈N∩(Q,2Q], we define the arithmetic functiongas
g(d) =g(d, q, N, H)def= sgn³ X
a≤H
³ X
m∼N/d md≡a(q)
1−1
q
X
m∼N/d
1
´´
,
ifd∈ N∩(Q,2Q], andg(d) = 0otherwise. It is plain that gis the Eratosthenes transform of the sieve functionf(n) =f(n, q, N, H)def= Pd|ng(d)of range2Q. After noting thatsgn(x)x=|x|for allx∈R, we write
|Tf(q, N, H)| = ¯ ¯ ¯X
a≤H ³ X
n∼N n≡a(q)
f(n)−1
q
X
n∼N
f(n)
´¯¯ ¯ = ¯ ¯ ¯X
d∼Q
g(d)X
a≤H
³ X
m∼N/d md≡a(q)
1−1
q
X
m∼N/d
1
´¯¯ ¯
= X
d∼Q ¯ ¯ ¯X
a≤H ³ X
m∼N/d md≡a(q)
1−1
q
X
m∼N/d
1
´¯¯ ¯.
We want to show that|Tf(q, N, H)| ÀN H/q, so we set def
=
n
d∈N∩(Q,2Q] : X
a≤H X
m∼N/d md≡a(q)
1≥1
o def
S,
|Tf(q, N, H)| = X¯¯¯ X a≤H
³ X
m∼N/d md≡a(q)
1−1
q
X
m∼N/d
1
´¯¯
¯+XH
q
X
m∼N/d
1
qQ À N H
q .
def
d|n
X
a≤H X
m∼N/d md≡a(q)
1≤ X a≤H
X
n∼N n≡a(q)
d(n)¿ε
N1+εH
Q .
If Q > √
N1+δH for a certainδ > 0, then N1+ε
Summarizing,Q > √
N1+δH gives a trivial bound for T
f(q, N, H), if Q < q ≤2Q. In this sense, the above functionf is a counterexample (depending onq, N, H) to the possibility for the variableQof going beyond the barrier√N H.
4. FURTHERREMARKS
Sieve functions are ubiquitous in analytic number theory. Besides the truncated divisor sumΛR(see [8]), that is a linear combination of sieve functions of rangeR (see [2] for our remarks onΛR), we quote the so-called restricted divisor function
τQ(n)def= (1Q∗1)(n) = X
d|n,d≤Q
1,
whose Eratosthenes transform is the indicator1Qof[1, Q]∩N. We refer the reader to [10] for results on the distribution ofτQ in short arithmetic progressions. Here we wish to stress that in [10] one finds also conjectures and average results concerning the distribution in arithmetic bands of the more complicated function
τQ,R(n)def= (1Q∗1R)(n) = X
(d,t) dt=n
1Q(d)1R(t).
Such an essentially bounded function is linked to the pair correlation problem for fractional parts of the quadratic functionαk2, with k ∈ N andα ∈ R(compare also [9]). While, under suitable
S
N H
À | |
=
P
1is essentially bounded andq > Q, we note that
Now let us prove that| |S = o(Q). To this end, after recalling that the divisor function d(n)
X | | ≤S
d∈S
H/Q = o(Q) once ε < δ, that yields the desired conclusion.
and prove that| |=o(Q), which in turns yields|E| ÀQ. This implies
d∈S d∈E
E
conditions onH, Q, R, the search of an upper bound for X
a≤H
³ X
n≡ar(q)
τQ,R(n)− QRq ´
, with(r, q) = 1,
is pursued in [10], Conjecture 1.2, here Theorem 1.1 leads to an asymptotic formula for the inverse Eratosthenes transform ofτQ,Rin arithmetic bands, namely for
X
a≤H X
n≡ar(q)
(τQ,R∗µ)(n).
In [3] we established an asymptotic inequality for the exponential sum associated to the localized divisor functions, a family of functions including the aforementioned τQ, τQ,R, and the standard divisor function dk, k ≥ 2 (recall that dk(n) is the number of ways to write n as a product of k positive integers). The particular instance of such an inequality fordkis
X
n∼N
dk(n)e(nα)¿k,ε (N q)ε(N/q+q+N1−1/k),
for allα ∈ [a/q −1/q2, a/q + 1/q2], with(a, q) = 1, q > 1. Somehow, this can be regarded as being analogous to the inequality which follows by combining (2.3) with (2.1). Such a circumstance is remarkable even in view of the fact that anydkfalls short of being a sieve function, with a sort of Eratosthenes transform which turns out to be the sum ofdk−1 plus some restricted divisor functions (see the last section of [1]).
Finally, since the function proposed in§3 seems to be rather artificial in that it depends on a fixed modulusq ∈N∩(Q,2Q], an intriguing open question to ask is how many standard sieve functions might support the optimality of the level accomplished by Theorem 1.1 and the results of [2].
ACKNOWLEDGEMENT
The authors thank the referee for helpful comments.
REFERENCES
2. G. Coppola and M. Laporta, Sieve functions in arithmetic bands, Hardy-Ramanujan J., 39 (2016), 21-37. 1. G. Coppola and M. Laporta, Symmetry and short interval mean-squares, Analytic number theory, On the occasion of the 80th anniversary of the birth of Anatolli Alekseevich Karatsuba, Tr. Mat. Inst. Steklova,
3. G. Coppola and M. Laporta, A note on the exponential sums of the localized divisor functions, Anal. Probab. Methods Number Theory, 25-28, Proceedings of the Sixth International Conference ”Analytic
and Probabilistic Methods in Number Theory” (Palanga, Lithuania, 11-17 September 2016), A. Du-bickas et al., (Eds), 2017, Vilnius University.
4. G. Coppola, M. R. Murty and B. Saha, Finite Ramanujan expansions and shifted convolution sums of arithmetical functions, J. Number Theory, 174 (2017), 78-92.
5. G. Coppola, M. R. Murty and B. Saha, On the error term in a Parseval type formula in the theory of Ramanujan expansions II, J. Number Theory, 160 (2017), 700-715.
6. H. Davenport, Multiplicative Number Theory, 3rd edition, GTM 74, Springer, New York, 2000.
7. H. G. Gadiyar, M. R. Murty and R. Padma, Ramanujan, Fourier series and a theorem of Ingham, Indian J. Pure Appl. Math., 45(5) (2014), 691-706.
8. D. A. Goldston, On Bombieri and Davenport’s theorem concerning small gaps between primes, Mathe-matika, 39 (1992), 10-17.
9. I. E. Shparlinski, On the restricted divisor function in arithmetic progressions, Rev. Mat. Iberoam., 28 (2012), 231-238.
10. J. L. Truelsen, Divisor problems and the pair correlation for the fractional parts ofn2α, Int. Math. Res.
Not. IMRN, 2010(16) (2010), 3144-3183.
11. A. Wintner, Eratosthenian averages, Waverly Press, Baltimore, MD, 1943.
