TUGraz OPEN Library

Lapkova, Kostadinka

Monatsh Math (2018) 186:663–673
https://doi.org/10.1007/s00605-017-1061-y
Explicit upper bound for the average number
of divisors of irreducible quadratic polynomials
Kostadinka Lapkova1
Received: 6 December 2016 / Accepted: 19 May 2017 / Published online: 27 May 2017
© The Author(s) 2017. This article is an open access publication
Abstract Consider the divisor sum
∑
n≤N τ(n2 + 2bn + c) for integers b and c.
We extract an asymptotic formula for the average divisor sum in a convenient form,
and provide an explicit upper bound for this sum with the correct main term. As
an application we give an improvement of the maximal possible number of D(−1)-
quadruples.
Keywords Number of divisors · Quadratic polynomial · Dirichlet convolution
Mathematics Subject Classification Primary 11N56; Secondary 11D09
1 Introduction
Let τ(n) denote the number of positive divisors of the integer n and P(x) ∈ Z[x] be
a polynomial. Due to their numerous applications average sums of divisors
N∑
n=1
τ (P(n)) (1.1)
have obtained a lot of attention, e.g. in [4,7–9,20,21]. The current paper aims to
improve the author’s results from [11] concerning explicit upper bound for divisor
Communicated by A. Constantin.
B Kostadinka Lapkova
lapkova@math.tugraz.at
1 Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24/II,
8010 Graz, Austria
123
664 K. Lapkova
sums over certain irreducible quadratic polynomials P(x). More precisely, while the
upper bound obtained in [11] appears to be the first one of the right order of magnitude
N log N with completely explicit constants, it does not provide the expected constant
in the main term. Here we fill this gap achieving the correct main term as in the
asymptotic formula for the corresponding divisor sum. A more detailed introduction
to the subject can be found in [11,15].
Only very recently [5,12] provided asymptotic formulae for the sum (1.1) for some
reducible quadratic polynomials, whereas the asymptotic formulae for irreducible
quadratic polynomials are classical, due to Scourfield [20], Hooley [10] and McKee
[13–15].
Let P(x) = x2 + Bx +C for integers B and C such that P(x) is irreducible. Denote
ρ(d, P) = ρ(d) := # {0 ≤ m < d : P(m) ≡ 0 (mod d)} . (1.2)
Then we have
∑
n≤N
τ(P(n)) ∼ λN log N ,
as N → ∞, for some λ depending on B and C . Hooley [10] showed that for irreducible
P(x) = x2 + C we have
λ = 8
π2
∞∑
α=0
ρ(2α)
2α
∑
d2|C
d odd
1
d
∞∑
l=1
l odd
1
l
(−C/d2
l
)
,
while for the more general P(x) = x2 + Bx + C with  = B2 − 4C not a square,
McKee [15] obtained
λ =
{
12H∗()/(π
√||) if  < 0,
12H∗() log /(π2
√
) if  > 0, (1.3)
where H∗() is a weighted class number, and  for  > 0 is a fundamental unit.
Nevertheless we would prove independently the asymptotic formula for the specialized
irreducible quadratic polynomials in which we are interested. We achieve a compact
main term with a constant λ which is easier to formulate and easy to compare with the
main term in our explicit upper bound.
Our first result is the following theorem.
Theorem 1 Let b and c be integers, such that the discriminant δ := b2 − c is non-
zero and square-free, and δ 
≡ 1 (mod 4). Then for N → ∞ we have the asymptotic
formula
N∑
n=1
τ(n2 + 2bn + c) = 2
ζ(2)
L(1, χ)N log N + O(N ),
123
Explicit upper bound for the average number of divisors… 665
where χ(n) = ( 4δ
n
) for the Kronecker symbol ( .
.
)
.
In particular, when we consider the polynomial f (n) = n2 + 1 we have δ = −1
and χ(n) = (−4
n
)
is the non-principal primitive Dirichlet character modulo 4. Then
L(1, χ) = ∑∞n=0(−1)n/(2n + 1) = π/4 by the formula of Leibniz. We substitute
ζ(2) = π2/6 in Theorem 1 and recover the well-known asymptotic
N∑
n=1
τ(n2 + 1) = 3
π
N log N + O(N ). (1.4)
We note that 4δ =  is a fundamental discriminant and the weighted class number
H∗() in McKee’s formula (1.3) is the usual class number for  > 0 and  < −4.
When δ = −1 we have an adjustment by the corresponding root number. Indeed, for
negative discriminants H∗() is the Hurwitz class number for which the precise rela-
tion with the usual class number is described for example in ([3], Lemma 5.3.7). Then
the Dirichlet class number formula yields the equality of the constant 2L(1, χ)/ζ(2)
with λ from (1.3) for the polynomials we consider in Theorem 1. Our proof, however,
is different than McKee’s [13–15], and is rather similar to Hooley’s argument from
[10].
The novelty in this paper is the following explicit upper bound, which achieves the
correct main term as in the asymptotic formula from Theorem 1.
Theorem 2 Let f (n) = n2 + 2bn + c for integers b and c, such that the discriminant
δ := b2 − c is non-zero and square-free, and δ 
≡ 1 (mod 4). Assume also that for
n ≥ 1 the function f (n) is non-negative. Then for any N ≥ 1 satisfying f (N ) ≥ f (1),
and X := √ f (N ), we have the inequality
N∑
n=1
τ(n2 + 2bn + c) ≤ 2
ζ(2)
L(1, χ)N log X
+
(
2.332L(1, χ) + 4Mδ
ζ(2)
)
N + 2Mδ
ζ(2)
X
+ 8Mδ N√
X
+ 4Mδ
√
X
+ 8Mδ NX + 4Mδ,
where χ(n) = ( 4δ
n
) for the Kronecker symbol ( .
.
)
and
Mδ =
{ 4
π2
δ1/2 log 4δ + 8
π2
δ1/2 log log 4δ + 3δ1/2 , if δ > 0;
1
π
|δ|1/2 log 4|δ| + 2
π
|δ|1/2 log log 4|δ| + 2|δ|1/2 , if δ < 0.
123
666 K. Lapkova
Note that we can formulate the theorem by extracting positive constants C1, C2, C3
such that
N∑
n=1
τ(n2 + 2bn + c) ≤ 2
ζ(2)
L(1, χ)N log N + C1 N + C2
√
N + C3,
because X = √ f (N ) = N + O(1). However we restrained ourselves from doing
so in order to keep our result more precise for eventual numerical applications. Still,
improvements of the lower order terms are very likely, for example by applying the
theory developed by Akhilesh and Ramaré [1].
When we know the exact form of the quadratic polynomial and the correspond-
ing character, we might achieve even better upper bounds. This is the case of the
polynomial f (n) = n2 + 1 when the following corollary holds.
Corollary 3 For any integer N ≥ 1 we have
∑
n≤N
τ(n2 + 1) < 3
π
N log N + 4.264N + 8√N + 8.
Just as in [11] we give an application of the latter inequality. Define a D(n) − m-
tuple for a nonzero integer n and a positive integer m to be a set of m integers such that
the product of any two of them increased by n is a perfect square. It is conjectured that
there are no D(−1)-quadruples but currently it is only known that there are at most
4.7 · 1058 D(−1)-quadruples [11]. The latter number upgraded the previous maximal
possible bound 3.01×1060 due to Trudgian [19], which in turn improved results from
[2,6].
Plugging the upper bound of Corollary 3 in the proof of ([6], Theorem 1.3) from
the paper of Elsholtz, Filipin and Fujita we obtain another slight improvement.
Corollary 4 There are at most 3.713 · 1058 D(−1)-quadruples.
2 Proof of Theorem 2
We start with proving the explicit upper bound in Theorem 2, as the claims required
for the proof of Theorem 1 can be easily adapted by the lemmae from this section.
Let us consider the polynomial f (n) = n2 + 2bn + c with integer coefficients b
and c, and the function ρ(d) defined in (1.2) counts the roots of f (n) in a full residue
system modulo d. Let δ = b2 − c and χ(n) = ( 4δ
n
)
for the Kronecker symbol
(
.
.
)
.
The core of the proofs of both Theorems 1 and 2 is the following convolution lemma
which we proved in [11].
Lemma 1 ([11], Lemma 2.1) Let δ = b2 − c be square-free and δ 
≡ 1 (mod 4).
Then we have the identity
ρ(d) =
∑
lm=d
μ2(l)χ(m).
123
Explicit upper bound for the average number of divisors… 667
The proof of Lemma 1 is based on elementary facts about the function ρ(d) at prime
powers and manipulations of the Dirichlet series it generates.
Further we need the following explicit estimates.
Lemma 2 For any integer N ≥ 1 we have
∑
n≤N
μ2(n) = N
ζ(2)
+ E1(N ),
where |E1(N )| ≤ 2
√
N + 2.
Proof Let us denote Q(N ) := ∑n≤N μ2(n) for N ≥ 1. Then we can write
Q(N ) =
∑
n≤N
∑
d2|n
μ(d) =
∑
1≤d≤√N
μ(d)
∑
1≤n≤N
d2|n
1 =
∑
1≤d≤√N
μ(d)
[
N
d2
]
Using
∑∞
n=1 μ(d)/d2 = 1/ζ(2) we see that
E1(N ) = Q(N ) − N
ζ(2)
= −
∑
1≤d≤√N
μ(d)
{
N
d2
}
− N
∑
d>
√
N
μ(d)
d2
,
hence
|E1(N )| ≤
√
N + N
∣
∣
∣
∣
∣
∣
∑
d>
√
N
μ(d)
d2
∣
∣
∣
∣
∣
∣
. (2.1)
On its turn for every real x > 1
∣
∣
∣
∣
∣
∑
d>x
μ(d)
d2
∣
∣
∣
∣
∣
≤
∑
d>x
1
d2
<
∫ ∞
x−1
dt
t2
= 1
x − 1
and N/(
√
N − 1) ≤ √N + 2 for N ≥ 4. Now from (2.1) it follows that Q(N ) =
N/ζ(2) + E1(N ) and |E1(N )| ≤ 2(
√
N + 1) for N ≥ 4. It is trivial to check that the
inequality
∣
∣
∣
∣Q(N ) −
N
ζ(2)
∣
∣
∣
∣ ≤ 2
√
N + 2
holds also for N ∈ {1, 2, 3} and this proves the statement of the lemma. unionsq
The following effective Pólya–Vinogradov inequality is due to Pomerance.
123
668 K. Lapkova
Lemma 3 ([16], Theorem 1) Let
Mχ := max
L ,P
∣
∣
∣
∣
∣
P∑
n=L
χ(n)
∣
∣
∣
∣
∣
for a primitive character χ to the modulus q > 1. Then
Mχ ≤
{ 2
π2
q1/2 log q + 4
π2
q1/2 log log q + 32 q1/2 , χ even;
1
2π q
1/2 log q + 1
π
q1/2 log log q + q1/2, χ odd.
The next lemma is critical for obtaining the right main term in the explicit upper
bound of Theorem 2.
Lemma 4 For any integer N ≥ 1 we have
∑
n≤N
χ(n)
n
= L(1, χ) + E2(N ),
where |E2(N )| ≤ 2Mδ/N and Mδ is the constant defined in Theorem 2.
Proof This statement, especially the estimate of the error term, is much less trivial
and follows from the effective Pólya–Vinogradov inequality of Pomerance. First we
notice that
∑
n≤N
χ(n)
n
=
∞∑
n=1
χ(n)
n
−
∑
n>N
χ(n)
n
= L(1, χ) + E2(N ).
Let us denote X (N ) := ∑1≤n≤N χ(n) for any positive integer N . Then by Abel’s
summation it follows that for any positive integer Z > N we have
∣
∣
∣
∣
∣
∣
∑
N<n≤Z
χ(n)
n
∣
∣
∣
∣
∣
∣
=
∣
∣
∣
∣
X (Z)
Z
− X (N )
N
+
∫ Z
N
X (t)
t2
dt
∣
∣
∣
∣
≤ |X (Z)|
Z
+ |X (N )|
N
+
∫ Z
N
|X (t)|
t2
dt.
Recall the definition of the quantity Mχ in Lemma 3. Then
∣
∣
∣
∣
∣
∣
∑
N<n≤Z
χ(n)
n
∣
∣
∣
∣
∣
∣
≤ Mχ
(
1
Z
+ 1
N
+
∫ Z
N
dt
t2
)
= 2Mχ
N
,
which is uniform in Z . Therefore |E2(N )| =
∣
∣∑
n>N χ(n)/n
∣
∣ ≤ 2Mχ/N . Now by
the conditions on δ to be square-free and not congruent to 1 modulo 4 it follows that
123
Explicit upper bound for the average number of divisors… 669
the discriminant 4δ is fundamental, thus the character χ(n) = ( 4δ
n
)
is primitive with a
conductor 4|δ|. Then according to Lemma 3 Mχ ≤ Mδ and this proves the statement.
unionsq
The following lemma is due to Ramaré. One could easily extract the right main term
from Lemma 2 through Abel summation but the estimate of the minor term requires
computer calculations.
Lemma 5 ([17], Lemma 3.4) Let x ≥ 1 be a real number. We have
∑
n≤x
μ2(n)
n
≤ log x
ζ(2)
+ 1.166.
Let us start with the proof of Theorem 2. Note that from the condition f (N ) ≥ f (1)
and the convexity of the function f (x) we have f (N ) = max1≤n≤N f (n). Using the
Dirichlet hyperbola method we have
∑
1≤n≤N
τ( f (n)) =
∑
1≤n≤N
∑
d| f (n)
1 ≤ 2
∑
1≤n≤N
∑
d≤√ f (n)
d| f (n)
1
= 2
∑
1≤d≤√ f (N )
∑
1≤n≤N
d+O(1)≤n
∑
d| f (n)
1 ≤ 2
∑
1≤d≤√ f (N )
∑
1≤n≤N
d| f (n)
1.
Recall the definition (1.2) of the function ρ(d). Then the innermost sum equals [N/d]
copies of ρ(d) plus a remaining part smaller than ρ(d). Recall that X = √ f (N ).
Then
∑
1≤n≤N
τ( f (n)) ≤ 2
∑
1≤d≤X
(
N
d
ρ(d) + ρ(d)
)
= 2N
∑
1≤d≤X
ρ(d)
d
+ 2
∑
1≤d≤X
ρ(d).
(2.2)
From the convolution identity stated in Lemma 1 it immediately follows that
∑
d≤X
ρ(d) =
∑
lm≤X
μ2(l)χ(m) =
∑
l≤X
μ2(l)
∑
m≤X/ l
χ(m).
Recall the definition of Mχ and the Pomerance bounds Mχ ≤ Mδ . Then using also
Lemma 2 we get
∑
d≤X
ρ(d) ≤ Mδ
∑
l≤X
μ2(l) ≤ Mδ
(
X
ζ(2)
+ 2√X + 2
)
. (2.3)
Similarly by the convolution property of the function ρ(d) and Lemma 4 we can write
∑
d≤X
ρ(d)
d
=
∑
l≤X
μ2(l)
l
∑
m≤X/ l
χ(m)
m
≤
∑
l≤X
μ2(l)
l
(
L(1, χ) + 2Mδ lX
)
. (2.4)
123
670 K. Lapkova
One could go further using Lemma 5 and again Lemma 2. We obtain
∑
d≤X
ρ(d)
d
≤ L(1, χ)
∑
l≤X
μ2(l)
l
+ 2Mδ
X
∑
l≤X
μ2(l)
≤ L(1, χ)
(
log X
ζ(2)
+ 1.166
)
+ 2Mδ
X
(
X
ζ(2)
+ 2√X + 2
)
. (2.5)
Plugging the estimates (2.3) and (2.5) in (2.2) gives the statement of Theorem 2.
Let us prove Corollary 3. For the polynomial f (n) = n2 + 1 we can be more
precise as the condition d ≤ √ f (n) is equivalent to d ≤ n. Then after applying more
carefully the hyperbola method and changing the order of summation we obtain
∑
n≤N
τ(n2 + 1) = 2
∑
n≤N
∑
d≤n
d|n2+1
1 = 2
∑
1≤d≤N
∑
d≤n≤N
d|n2+1
1
= 2
∑
1≤d≤N
⎛
⎜
⎜
⎝
∑
1≤n≤N
d|n2+1
1 −
∑
1≤n<d
d|n2+1
1
⎞
⎟
⎟
⎠ ≤ 2N
∑
1≤d≤N
ρ(d)
d
.
Similarly to (2.4) we have
∑
1≤d≤N
ρ(d)
d
=
∑
l≤N
μ2(l)
l
∑
m≤N/ l
χ(m)
m
≤
∑
l≤N
μ2(l)
l
(
L(1, χ) + 2Mχ lN
)
.
In this case Mχ ≤ 1 and L(1, χ) = π/4, and application of Lemmas 5 and 2 gives
∑
1≤d≤N
ρ(d)
d
≤ π
4
∑
l≤N
μ2(l)
l
+ 2
N
∑
l≤N
μ2(l)
≤ π
4
(
log N
ζ(2)
+ 1.166
)
+ 2
N
(
N
ζ(2)
+ 2√N + 2
)
.
After numerical approximation of the constants we conclude the validity of Corollary 3.
3 Proof of Theorem 1
Let again f (n) = n2 + 2bn + c. We can assume that f (n) is non-negative for n ≥ 1
as for N large enough f (N ) = max1≤n≤N f (n) and the contribution of the finitely
many integers n for which f (n) < 0 will be negligible compared to the main term.
123
Explicit upper bound for the average number of divisors… 671
As in the proof of Theorem 2 and using (2.2) and (2.3) it is easy to see that
∑
n≤N
τ( f (n)) = 2N
∑
d≤N
ρ(d)
d
+ O
⎛
⎝
∑
d≤N
ρ(d)
⎞
⎠ + O(N )
= 2N
∑
d≤N
ρ(d)
d
+ O(N ). (3.1)
We need the following estimates.
Lemma 6 For N → ∞ we have the asymptotic formulae
∑
n≤N
μ2(n)
n
= log N
ζ(2)
+ O(1)
and
∑
n≤N
χ(n)
n
= L(1, χ) + O(1/N ).
Proof The first asymptotic formula follows easily after Abel transformation of the
classical formula
∑
n≤N
μ2(n) = N
ζ(2)
+ O(√N ),
which in its turn obviously follows from the explicit version in Lemma 2. The second
statement can be deduced from Lemma 4. unionsq
Then the asymptotic estimate corresponding to (2.4) is
∑
d≤N
ρ(d)
d
=
∑
l≤N
μ2(l)
l
(L(1, χ) + O(l/N )) = L(1, χ)
∑
l≤N
μ2(l)
l
+ O
⎛
⎝ 1
N
∑
l≤N
μ2(l)
⎞
⎠ = L(1, χ)
(
log N
ζ(2)
+ O(1)
)
+ O(1)
= L(1, χ)
ζ(2)
log N + O(1). (3.2)
Now plugging (3.2) into (3.1) gives Theorem 1.
123
672 K. Lapkova
4 Some examples
In our preceding paper [11] we gave several examples of polynomials for which
Theorem 2 can provide effective upper bound of the average divisor sum. There we
compared the coefficient of the main term in the asymptotic formula of McKee with the
one from our upper bound. The current paper achieves equality of the two coefficients
but there is one more computation to be made, so that our result could be useful
numerically, this of the special value of the Dirichlet L-function L(s, χ) at s = 1.
Formulae for computing L(1, χ) in the two separate cases δ < 0 and δ > 0 are
given in ([22], Theorem 4.9). When δ < 0 this is implemented by the SAGE function
quadratic_L_function_exact(1, 4δ) whose values are computed for several
examples in the table below. When δ > 0 one can find the approximate value of
L(1, χ) by finding first the corresponding Dirichlet character (of modulus 4δ, order
2, even and primitive) in the Refine search platform from LMFDB (The L-functions
and modular forms database [18]).
The following table lists some examples.
f (n) δ χ L(1, χ)
n2 + 1 −1
(−1
.
)
π/4–0.7854
n2 + 10n + 27 −2
(−8
.
) √
2/4π–1.1108
n2 + 4n + 10 −6
(−24
.
) √
6/6π–1.2826
n2 + 10n − 10 35
(
140
.
)
∼0.8377
n2 + 20n + 9 91
(
364
.
)
∼1.6887
Acknowledgements Open access funding provided by Graz University of Technology. This work was
supported by a Hertha Firnberg grant of the Austrian Science Fund (FWF) [T846-N35]. The author thanks
Christian Elsholtz for drawing her attention to this problem, Olivier Ramaré for commenting her preceding
work [11] and the referee for his/her suggestions which led to improvement of the exposition.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-
tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
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