Journal de Théorie des Nombres de Bordeaux

Shuntaro Yamagishi

An exponential sum estimate for systems  with linear polynomials

Andrew Fiori

Sub-Shimura Varieties for type $O(2,n)$

Valerio Talamanca

Heights and representations of split tori

David Simmons

Some manifolds of Khinchin type for convergence

Adriana A. Albarracín-Mantilla

Edwin León-Cardenal

Igusa’s Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials

Kevin Keating

Extensions of local fields and elementary symmetric polynomials

Boris Adamczewski

Colin Faverjon

Méthode de Mahler, transcendance et relations linéaires : aspects effectifs

Sho Yoshikawa

Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7

Markus Kirschmer

One-class genera of exceptional groups over number fields

Carlo Sanna

Carlo SANNA
A factor of integer polynomials with minimal integrals
Tome 29, no 2 (2017), p. 637-646.
<http://jtnb.cedram.org/item?id=JTNB_2017__29_2_637_0>
© Société Arithmétique de Bordeaux, 2017, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 29 (2017), 637–646
A factor of integer polynomials with
minimal integrals
par Carlo SANNA
Résumé. Pour chaque entier positif N , soit SN l’ensemble des
polynômes P (x) ∈ Z[x] de degré inférieur à N et d’intégrale po-
sitive non-nulle minimale sur [0, 1]. Ces polynômes sont liés à la
répartition des nombres premiers puisque, si l’on note ψ la fonction
de Tchebychev, on a
∫ 1
0 P (x) dx = exp(−ψ(N)). Nous démontrons
que, pour tout nombre entier positif N , il existe P (x) ∈ SN tel que
le polynôme (x(1−x))bN/3c divise P (x) dans Z[x]. Nous montrons
en fait que l’exposant bN/3c ne peut pas être amélioré. Ce résultat
est analogue à celui obtenu par Aparicio concernant les polynômes
de Z[x] de norme L∞ non-nulle minimale sur [0, 1]. En outre, il
est en quelque sorte l’amélioration d’un résultat de Bazzanella,
qui considérait xbN/2c et (1− x)bN/2c au lieu de (x(1− x))bN/3c.
Abstract. For each positive integer N , let SN be the set of
all polynomials P (x) ∈ Z[x] with degree less than N and minimal
positive integral over [0, 1]. These polynomials are related to the
distribution of prime numbers since
∫ 1
0 P (x) dx = exp(−ψ(N)),
where ψ is the second Chebyshev function. We prove that for any
positive integerN there exists P (x) ∈ SN such that (x(1−x))bN/3c
divides P (x) in Z[x]. In fact, we show that the exponent bN/3c
cannot be improved. This result is analog to a previous of Aparicio
concerning polynomials in Z[x] with minimal positive L∞ norm
on [0, 1]. Also, it is in some way a strengthening of a result of
Bazzanella, who considered xbN/2c and (1 − x)bN/2c instead of
(x(1− x))bN/3c.
1. Introduction
It is well-known that the celebrated Prime Number Theorem is equivalent
to the assertion:
ψ(x) ∼ x, as x→ +∞.
Manuscrit reçu le 4 février 2016, révisé le 18 février 2016, accepté le 27 février 2016.
Mathematics Subject Classification. 11A41, 11C08, 11A63.
Mots-clefs. Integer polynomials, Chebyshev problem, prime numbers.
638 Carlo Sanna
Here ψ(x) is the second Chebyshev function, defined for x ≥ 0 as
ψ(x) :=
∑
pm≤x
log p,
where the sum is extended over all the prime numbers p and all the positive
integers m such that pm ≤ x.
In 1936, Gelfond and Shnirelman proposed an elementary and clever
method to obtain lower bounds for ψ(x) (see Gelfond’s comments in [5,
pp. 285–288]). In 1982, the same method was rediscovered and developed
by Nair [9, 10].
The main idea of the Gelfond–Shnirelman–Nair method is the follow-
ing: Given a positive integer N , let PN (x) be a polynomial with integer
coefficients and degree less than N , say
PN (x) =
N−1∑
n= 0
anx
n,
with a0, . . . , aN−1 ∈ Z. Now consider the integral of PN (x) over [0, 1], that
is
I(PN ) :=
∫ 1
0
PN (x) dx =
N−1∑
n= 0
an
n+ 1 .
Clearly, I(PN ) is a rational number whose denominator divides
dN := lcm{1, 2, . . . , N},
hence dN |I(PN )| is an integer. In particular, if we suppose I(PN ) 6= 0, then
dN |I(PN )| ≥ 1. Now dN = exp(ψ(N)), so we get
(1.1) ψ(N) ≥ log
( 1
|I(PN )|
)
.
Finally, from the trivial upper bound
|I(PN )| =
∣∣∣∣∫ 1
0
PN (x) dx
∣∣∣∣ ≤ ∫ 1
0
|PN (x)| dx ≤ max
x∈ [0,1]
|PN (x)| =: ‖PN‖,
we obtain
(1.2) ψ(N) ≥ log
( 1
‖PN‖
)
.
At this point, if we choose PN to have a sufficiently small norm ‖PN‖, then
a lower bound for ψ(x) follows from (1.2). For example, the choice
PN (x) = (x(1− x))2b(N−1)/2c
gives the lower bound
ψ(N) ≥ log 2 · (N − 2) > 0.694 · (N − 2).
A factor of integer polynomials with minimal integrals 639
This motivates the study of the quantities
`N := min{‖P‖ : P (x) ∈ Z[x], deg(P ) < N, ‖P‖ > 0},
CN :=
1
N
log
( 1
`N
)
,
and the set of polynomials
TN := {P (x) ∈ Z[x] : deg(P ) < N, ‖P‖ = `N};
the so-called Integer Chebyshev Problem [4].
In particular, Aparicio [1] proved the following theorem about the struc-
ture of polynomials in TN .
Theorem 1.1. Given any sufficiently large positive integer N , for all
P ∈ TN it holds
(x(1− x))bλ1Nc(2x− 1)bλ2Nc(5x2 − 5x+ 1)bλ3Nc | P (x)
in Z[x], where
λ1 ∈ [0.1456, 0.1495], λ2 ∈ [0.0166, 0.0187], λ3 ∈ [0.0037, 0.0053]
are some constants.
It is known that CN converges to a limit C, as N → +∞ (see [8, Chap-
ter 10]). Furthermore, Pritsker [11, Theorem 3.1] showed that
C ∈ ]0.85991, 0.86441[,
and this is the best estimate of C known to date.
As a consequence of Pritsker’s result, the Gelfond–Shnirelman–Nair me-
thod cannot lead to a lower bound better than
ψ(x) ≥ 0.86441 · x,
which is quite far from what is expected by the Prime Number Theorem.
To deal with this problem, Bazzanella [2, 3] suggested to study the poly-
nomials PN such that |I(PN )| is nonzero and minimal, or, without loss of
generality, such that I(PN ) is positive and minimal.
We recall the following elementary lemma about the existence of solutions
of some linear diophantine equations.
Lemma 1.2. Fix some integers c1, . . . , ck. Then the diophantine equation
k∑
i= 1
cixi = 1
has a solution x1, . . . , xk ∈ Z if and only if gcd{c1, . . . , ck} = 1. Moreover,
if a solution exists, then there exist infinitely many solutions.
640 Carlo Sanna
On the one hand, because of the above considerations, we known that if
I(PN ) > 0 then I(PN ) ≥ 1/dN . On the other hand, I(PN ) = 1/dN if and
only if
N−1∑
n= 0
dN
n+ 1 · an = 1,
and it is easy to see that each of the coefficients dN/(n + 1) is an integer
and
gcd
{
dN
n+ 1 : n = 0, . . . , N − 1
}
= 1.
Hence, by Lemma 1.2, there exist infinitely many polynomials PN such that
I(PN ) = 1/dN , so that (1.1) holds with the equality.
This leads to define the following set of polynomials
SN := {P (x) ∈ Z[x] : deg(P ) < N, I(P ) = 1/dN}.
Bazzanella proved some results about the roots of the polynomials in
SN . In particular, regarding the multiplicity of the roots x = 0 and x = 1,
he gave the following theorem [2, Theorem 1], which is vaguely similar to
Theorem 1.1.
Theorem 1.3. For each positive integer N , there exists P (x) ∈ SN such
that
xbN/2c | P (x)
in Z[x]. Moreover, the exponent bN/2c cannot be improved, i.e., there exist
infinitely many positive integers N such that
xbN/2c+1 - P (x)
for all P (x) ∈ SN . The same results hold if the polynomial xbN/2c is replaced
by (1− x)bN/2c.
Actually, what Bazzanella proved is that the maximum nonnegative in-
teger K(N) such that there exists a polynomial P (x) ∈ SN divisible by
xK(N), respectively by (1− x)K(N), is given by
K(N) = min{pm − 1 : p prime, m ≥ 1, pm > N/2},
so that Theorem 1.3 follows quickly.
Despite the similarity between Theorems 1.1 and 1.3, note that the state-
ment of Theorem 1.1 holds “for all P (x) ∈ TN”, while Theorem 1.3 only
says that “there exists P (x) ∈ SN”. However, this distinction is unavoid-
able, indeed: On the one hand, TN is a finite set, even conjectured to be a
singleton for any sufficiently large N [4, §5 Q2]. On the other hand, SN is
an infinite set and if P (x) ∈ SN then (dN + 1)P (x) − 1 ∈ SN , hence the
elements of SN have no common nontrivial factor in Z[x].
A factor of integer polynomials with minimal integrals 641
The purpose of this paper is to move another step further in the direction
of a stronger analog of Theorem 1.1 for the set of polynomials SN . For we
prove the following theorem.
Theorem 1.4. For each positive integer N , there exist infinitely many
P (x) ∈ SN such that
(x(1− x))bN/3c | P (x)
in Z[x]. Moreover, the exponent bN/3c cannot be improved, i.e., there exist
infinitely many positive integers N such that
(x(1− x))bN/3c+1 - P (x),
for all P (x) ∈ SN .
We leave the following informal question to the interested readers:
Question. Let {QN (x)}N≥1 be a sequence of “explicit” integer polynomi-
als such that for each positive integer N it holds QN (x) | P (x) in Z[x],
for some P (x) ∈ SN . In light of Theorems 1.3 and 1.4, three examples
of such sequences are given by {xbN/2c}N≥1, {(1 − x)bN/2c}N≥1, and
{(x(1− x))bN/3c}N≥1.
How big can be
δ := lim inf
N→+∞
deg(QN )
N
?
Can δ be arbitrary close to 1, or even equal to 1?
Note that the sequences of Theorem 1.3 give δ = 1/2, while the sequence
of Theorem 1.4 gives δ = 2/3.
2. Preliminaries
In this section, we collect a number of preliminary results needed to prove
Theorem 1.4. The first is a classic theorem of Kummer [7] concerning the
p-adic valuation of binomial coefficients.
Theorem 2.1. For all integers u, v ≥ 0 and any prime number p, the p-
adic valuation of the binomial coefficient
(u+v
v
)
is equal to the number of
carries that occur when u and v are added in the base p.
Now we can prove the following lemma.
Lemma 2.2. For any positive integer N , and for all integers u, v ≥ 0 with
u+ v < N , we have that
(2.1) dN
(u+ v + 1)
(u+v
u
)
is an integer.
642 Carlo Sanna
Proof. We have to prove that for any prime number p ≤ N the p-adic
valuation of the denominator of (2.1) does not exceed νp(dN ) = blogpNc.
Write u+ v + 1 in base p, that is
u+ v + 1 =
s∑
i= i0
dip
i,
where i0 := νp(u+ v + 1) and di0 , . . . , ds ∈ {0, . . . , p− 1}, with di0 , ds > 0.
Hence, the expansion of u+ v in base p is
(2.2) u+ v =
s∑
i= i0+1
dip
i + (di0 − 1)pi0 +
i0−1∑
i= 0
(p− 1)pi.
In particular, by (2.2), we have that u+v written in base p has exactly s+1
digits, of which the i0 least significant are all equal to p− 1. Therefore, in
the sum of u and v in base p there occur at most s−i0 carries. Since, thanks
to Theorem 2.1, we know that i1 := νp
((u+v
v
))
is equal to the number of
carries occurring in the sum of u and v in base p, it follows that i1 ≤ s− i0.
In conclusion,
νp
(
(u+ v + 1)
(
u+ v
v
))
= i0 + i1 ≤ s ≤ blogpNc,
where the last inequality holds since u+ v + 1 ≤ N . 
We recall the value of a well-known integral (see, e.g., [6, §11.1.7.1,
Eq. 2]).
Lemma 2.3. For all integers u, v ≥ 0, it holds∫ 1
0
xu(1− x)v dx = 1
(u+ v + 1)
(u+v
v
) .
We conclude this section with a lemma that will be fundamental in the
proof of Theorem 1.4.
Lemma 2.4. Let N and m be integers such that N ≥ 1 and 0 ≤ m ≤
(N − 1)/2. The following statements are equivalent:
(1) There exist infinitely many P (x) ∈ SN such that (x(1−x))m | P (x)
in Z[x].
(2) For each prime number p ≤ N , there exists an integer hp such that
hp ∈ [m,N −m− 1] and
νp
(
(hp +m+ 1)
(
hp +m
m
))
= blogpNc.
A factor of integer polynomials with minimal integrals 643
Proof. Let P (x) ∈ Z[x] be such that deg(P ) < N and
(x(1− x))m | P (x)
in Z[x]. Hence,
P (x) = (x(1− x))m
N−m−1∑
h=m
bhx
h−m,
for some bm, . . . , bN−m−1 ∈ Z. Then, by Lemma 2.3, it follows that
I(P ) =
N−m−1∑
h=m
bh
∫ 1
0
xh(1− x)m dx =
N−m−1∑
h=m
bh
(h+m+ 1)
(h+m
m
) .
Now we have P (x) ∈ SN if and only if I(P ) = 1/dN , i.e., if and only if
N−m−1∑
h=m
dN
(h+m+ 1)
(h+m
m
) · bh = 1.
Therefore, thanks to Lemma 2.2 and Lemma 1.2, we get infinitely many
P (x) ∈ SN if and only if
gcd
{
dN
(h+m+ 1)
(h+m
m
) : h = m, . . . , N −m− 1} = 1.
At this point, recalling that νp(dN ) = blogpNc for each prime number p,
the equivalence of (1) and (2) follows easily. 
3. Proof of Theorem 1.4
We are ready to prove Theorem 1.4. Put m := bN/3c, s := blogpNc,
and pick a prime number p ≤ N . In light of Lemma 2.4, in order to prove
the first part of Theorem 1.4 we have to show the existence of an integer
hp ∈ [m,N −m− 1] such that
(3.1) νp
(
(hp +m+ 1)
(
hp +m
m
))
= s.
Let us write N = `ps+r, for some ` ∈ {1, . . . , p−1} and r ∈ {0, . . . , ps−1}.
We split the proof in three cases:
Case ` ≥ 2. It is enough to take hp := `ps−m−1. In fact, on the one hand,
it is straightforward that (3.1) holds. On the other hand, since ` ≥ 2, we
have
hp = `ps −m− 1 ≥
2
3(`+ 1)p
s −m− 1 > 23N −m− 1 ≥ m− 1,
while clearly hp ≤ N −m− 1, hence hp ∈ [m,N −m− 1], as desired.
644 Carlo Sanna
Case m < ps−1. It holds
ps
3 ≤
N
3 < m+ 1 ≤ p
s−1,
hence p = 2. Now it is enough to take h2 := 2s−m− 1. In fact, on the one
hand, it is again straightforward that (3.1) holds. On the other hand, since
m < 2s−1, we have
h2 = 2s −m− 1 > 2s − 2s−1 − 1 = 2s−1 − 1 ≥ m,
while obviously h2 ≤ N −m− 1, hence h2 ∈ [m,N −m− 1], as desired.
Case ` = 1 and m ≥ ps−1. This case requires more effort. We have
ps−1 ≤ m ≤ N3 =
ps + r
3 <
2ps
3 < p
s,
hence the expansion of m in base p is
m =
s−1∑
i= 0
dip
i,
for some d0, . . . , ds−1 ∈ {0, . . . , p− 1}, with ds−1 > 0.
Let i1 be the least nonnegative integer not exceeding s such that
(3.2) di ≥
p− 1
2 , ∀ i ∈ Z, i1 ≤ i < s.
Moreover, let i2 be the greatest integer such that i1 ≤ i2 ≤ s and
di =
p− 1
2 , ∀ i ∈ Z, i1 ≤ i < i2.
Note that, by the definitions of i1 and i2, we have
(3.3) di >
p− 1
2 , ∀ i ∈ Z, i2 ≤ i < s.
Clearly, it holds
(3.4) m =
∑
i2≤ i< s
dip
i +
∑
i1≤ i< i2
p− 1
2 p
i +
∑
0≤ i< i1
dip
i.
Define now
(3.5) hp :=
∑
i2≤ i< s
dip
i +
∑
i1≤ i< i2
p− 1
2 p
i +
∑
0≤ i< i1
(p− di − 1)pi.
Note that (3.5) is actually the expansion of hp in base p, that is, all the
coefficients of the powers pi belong to the set of digits {0, . . . , p−1}. At this
point, looking at (3.4) and (3.5), and taking into account (3.3), it follows
A factor of integer polynomials with minimal integrals 645
easily that in the sum of hp and m in base p there occur exactly s − i2
carries. Therefore, by Theorem 2.1 we have
(3.6) νp
((
hp +m
m
))
= s− i2.
Furthermore, from (3.4) and (3.5) we get
hp +m+ 1 = 2
∑
i2≤ i< s
dip
i +
∑
0≤ i< i2
(p− 1)pi + 1
= 2
∑
i2≤ i< s
dip
i + pi2 ,(3.7)
hence
(3.8) νp(hp +m+ 1) = i2.
Therefore, putting together (3.6) and (3.8) we obtain (3.1).
It remains only to prove that hp ∈ [m,N − m − 1]. If i2 = s, then
from (3.7) it follows that
hp +m+ 1 = 0 + ps ≤ N,
hence hp ≤ N −m− 1. If i2 < s, then from (3.2) it follows di2 ≥ (p− 1)/2,
hence di2 ≥ 1 and from (3.7) and (3.4) we obtain
hp +m+ 1 ≤ 2
∑
i2≤ i< s
dip
i + di2pi2 ≤ 2m+m = 3m ≤ N,
so that again hp ≤ N − m − 1. If i1 = 0, then by (3.4) and (3.5) we
have immediately that hp = m. If i1 > 0, then by the definition of i1, we
have di1−1 < (p − 1)/2, i.e., di1−1 < p − di1−1 − 1, thus looking at the
expansions (3.4) and (3.5) we get that hp > m. Hence, in conclusion we
have hp ∈ [m,N −m− 1], as desired.
Regarding the second part of Theorem 1.4, take N := 3q, where q > 3 is
a prime number. Put m := bN/3c+ 1 = q + 1, and let h ∈ [m,N −m− 1]
be an integer. On the one hand, it is straightforward that q - h+m+ 1. On
the other, it is also easy to see that in the sum of h and m in base q there
is no carry, hence, by Theorem 2.1, we have that q -
(h+m
m
)
. Therefore,
νq
(
(h+m+ 1)
(
h+m
m
))
= 0 < 1 = blogqNc,
so that, thanks to Lemma 2.4, we have (x(1 − x))m - P (x) in Z[x], for all
P (x) ∈ SN . This completes the proof. 
Acknowledgements. The author is grateful to D. Bazzanella and to the
anonymous referee for many useful suggestions which improve the readabil-
ity of the paper.
646 Carlo Sanna
References
[1] E. Aparicio Bernardo, “On the asymptotic structure of the polynomials of minimal dio-
phantic deviation from zero”, J. Approximation Theory 55 (1988), no. 3, p. 270-278.
[2] D. Bazzanella, “A note on integer polynomials with small integrals”, Acta Math. Hung.
141 (2013), no. 4, p. 320-328.
[3] ———, “A note on integers polynomials with small integrals II”, Acta Math. Hung. 146
(2016), no. 1, p. 71-81.
[4] P. Borwein & T. Erdélyi, “The integer Chebyshev problem”, Math. Comput. 65 (1996),
no. 214, p. 661-681.
[5] P. L. Chebyshev, Collected works. Vol. I, Akad. Nauk SSSR, 1944.
[6] A. Jeffrey & H.-H. Dai, Handbook of mathematical formulas and integrals, 4th ed., Else-
vier/Academic Press, 2008, xlvi+541 pages.
[7] E. E. Kummer, “Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen”, J.
Reine Angew. Math. 44 (1852), p. 93-146.
[8] H. L. Montgomery, Ten lectures on the interface between analytic number theory and har-
monic analysis, Regional Conference Series in Mathematics, vol. 84, American Mathematical
Society, 1994, xii+220 pages.
[9] M. Nair, “A new method in elementary prime number theory”, J. Lond. Math. Soc. 25
(1982), p. 385-391.
[10] ———, “On Chebyshev-type inequalities for primes”, Am. Math. Mon. 89 (1982), p. 126-
129.
[11] I. E. Pritsker, “Small polynomials with integer coefficients”, J. Anal. Math. 96 (2005),
p. 151-190.
Carlo Sanna
Università degli Studi di Torino
Department of Mathematics
Via Carlo Alberto 10
10123 Torino, Italy
E-mail: carlo.sanna.dev@gmail.com
URL: http://orcid.org/0000-0002-2111-7596


A factor of integer polynomials with minimal integrals

Not a member yet

An exponential sum estimate for systems with linear polynomials