We prove the analog of Cram\'er's short intervals theorem for primes in
arithmetic progressions and prime ideals, under the relevant Riemann
Hypothesis. Both results are uniform in the data of the underlying structure.
Our approach is based mainly on the inertia property of the counting functions
of primes and prime ideals.Comment: minor change to Proposition 

Grenié, L.

Molteni, G.

Perelli, A.

English

arXiv

Loïc Grenié

Giuseppe Molteni

Alberto Perelli

Crossref

Mathematika

PRIMES AND PRIME IDEALS IN SHORT INTERVALS

We prove the analog of Cramer's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals

GRENIE, L.

MOLTENI, G.

PERELLI, ALBERTO

Archivio istituzionale della ricerca - Università di Genova

Primes and prime ideals in short intervals

We prove the analog of Cram\ue9r's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals

L. Greni&#233

G. Molteni

A. Perelli

AIR Universita degli studi di Milano

PRIMES AND PRIME IDEALS IN SHORT INTERVALS
LOI¨C GRENIE´, GIUSEPPE MOLTENI and ALBERTO PERELLI
Mathematika 63(2), 364-371 (2017).
Electronically published on February 9, 2017.
DOI: http://dx.doi.org/10.1112/S0025579316000310
Abstract. We prove the analog of Crame´r’s short intervals theorem for primes in arithmetic
progressions and prime ideals, under the relevant Riemann hypothesis. Both results are uniform
in the data of the underlying structure. Our approach is based mainly on the inertia property
of the counting functions of primes and prime ideals.
Mathematics Subject Classification (2010): 11N13, 11R44
Keywords: primes in arithmetic progressions, prime ideals, short intervals, Riemann Hy-
pothesis.
1. Introduction
A famous theorem of Crame´r [2] states that, assuming the Riemann hypothesis, there is
always a prime between x and x + h provided x ≥ x0 and c1
√
x log x ≤ h ≤ x, with suitable
constants x0, c1 > 0. Actually, under the same assumptions we have that
pi(x+ h)− pi(x) ≥ c2 h
log x
with a suitable c2 > 0, and also that
pi(x+ h)− pi(x) ∼ h
log x
provided x ≥ h =∞(√x log x). Here f(x) =∞(g(x)) means that f(x)/g(x)→∞ as x→∞.
Apart from the explicit values of the involved constants, this is still the best known result
about primes in short intervals, under the Riemann hypothesis. Sharper results can be obtained
assuming various forms of the pair-correlation conjecture for the zeta zeros; see, e.g., Heath-
Brown [7], Languasco et al [13] and the literature quoted there. A simple proof of Crame´r’s
theorem can be obtained from a suitable smoothed explicit formula for ψ(x); see the footnote
of Ingham [9, p. 256].
In this paper we show that rather general theorems of Crame´r’s type follow, under the
appropriate Riemann hypothesis, from two results often ready in the literature, namely a short
intervals mean-square estimate and a Brun–Titchmarsh-type theorem. Indeed, the latter result
implies that the relevant counting function satisfies a suitable inertia property, which is then
played against the short intervals mean-square bound to get a contradiction if the interval is
not too short. We illustrate our approach in the case of primes in arithmetic progressions and of
prime ideals, since apparently these results do not appear in the literature. In the first case all
the ingredients are already known, so we proceed directly to the proof of Crame´r’s theorem for
arithmetic progressions. In the case of algebraic number fields we first deal with the required
ingredients; see in particular Proposition 1 below, which is of some independent interest. In
both cases our results are uniform in the data of the underlying structure. However, in the
second case the inertia method gives Proposition 3, which in the uniformity aspect is weaker
1
2than Theorem 2, proved here by the classical smoothed explicit formula approach. This is due
to the lack, in the current literature, of sharp uniform bounds of Brun–Titchmarsh type for
number fields. We shall discuss this issue later on in the paper.
As usual, for (a, q) = 1 we write
pi(x; q, a) =
∑
p≤x
p ≡ a (mod q)
1
and let ϕ(q) denote Euler’s function. Moreover, given an algebraic number field K of degree
nK , we denote by dK the absolute value of its discriminant, by P the prime ideals of the ring
OK of the integers of K, by N(P) their norm and write
piK(x) =
∑
N(P)≤x
1.
Finally, given an integer q ≥ 1 and a number field K, we denote by GRH and DRH the Riemann
hypothesis for the Dirichlet L-functions associated with the characters χ (mod q) and for the
Dedekind zeta function ζK(s), respectively. With this notation, our main results are as follows.
Theorem 1. Let (a, q) = 1 and assume GRH. Then there exist absolute constants x0, c1, c2 >
0 such that for x ≥ x0 and c1ϕ(q)
√
x log x ≤ h ≤ x we have
pi(x+ h; q, a)− pi(x; q, a) ≥ c2 h
ϕ(q) log x
.
Clearly, under the same assumptions the same argument also gives
pi(x+ h; q, a)− pi(x; q, a) ∼ h
ϕ(q) log x
provided x ≥ h =∞(ϕ(q)√x log x).
Theorem 2. Assume DRH for the number field K. Then there exist absolute constants
x0, c1, c2 > 0 such that for x ≥ x0 and c1(nK log x+ log dK)
√
x ≤ h ≤ x we have
piK(x+ h)− piK(x) ≥ c2 h
log x
.
As before, the same proof shows also that
piK(x+ h)− piK(x) ∼ h
log x
(1)
provided x ≥ h = ∞((nK log x + log dK)√x). Note that Theorem 2 represents an instance of
Lang’s [12] “recipe” asserting that, broadly speaking, when extending to a number field K the
classical results known for Q one should replace log x by nK log x+ log dK . Note also that if K
is a cyclotomic field then the quality of the K-uniformity in Theorem 2 is comparable to the
q-uniformity in Theorem 1.
We conclude by remarking that the technique in the proof of the above theorems works for
rather general counting functions, giving individual short intervals results as soon as suitably
sharp short intervals mean-value and inertia-type results are available.
32. Proofs
As customary, we prove Theorem 1 for the ψ-function and then the required result is recovered
by elementary arguments since h/ϕ(q) is large enough. Let X be sufficiently large, q, h ≤ X,
(a, q) = 1 and write
∆(x, h) = ψ(x+ h; q, a)− ψ(x; q, a)− h
ϕ(q)
.
The required mean-square bound follows from a result of Prachar [15] under GRH (see also
Goldston and Yıldırım [5]), namely∫ 2X
X
|∆(x, h)|2dx hX log2(qX), (2)
where the constant in the -symbol is absolute. Let now h/ϕ(q) > X1/10. From the well-
known Brun–Titchmarsh theorem, see Montgomery and Vaughan [14], we deduce that if there
exists x ∈ (X, 2X) such that
|∆(x, h)| > 1
4
h
ϕ(q)
, (3)
then
|∆(x, h)| > c h
ϕ(q)
(4)
for all x ∈ (x− c′h, x + c′h), with certain absolute constants c, c′ > 0. Inequalities (3) and (4)
express the inertia property of the ψ-function (see also Bazzanella and Perelli [1, Theorem 1]).
Let now
E(X, h) =
{
x ∈ [X, 2X] : |∆(x, h)| > 1
4
h
ϕ(q)
}
and suppose that E(X, h) 6= ∅. Then from (2)–(4) we get
h
( h
ϕ(q)
)2

∫ 2X
X
|∆(x, h)|2dx hX log2(qX),
thus, h  √Xϕ(q) log(qX). Hence, with suitable absolute constants in the -symbols and
provided x is sufficiently large,
ψ(x+ h; q, a)− ψ(x; q, a) h
ϕ(q)
if x ≥ h ϕ(q)√x log(qx). Theorem 1, and the statement after it, therefore follow. 
As anticipated in the Introduction, in the number fields case we first present the proof of a
weaker form of Theorem 2 in the uniformity aspect, obtained by the inertia approach. Write
ψK(x) =
∑
N(Pm)≤x
logN(P) =
∑
n≤x
ΛK(n) say, and ∆K(x, h) = ψK(x+ h)− ψK(x)− h,
and let L = logX. The analog of (2) is given by the following result.
Proposition 1. Assume DRH for the number field K. Then there exist absolute constants
c,X0 > 0 such that for X ≥ X0 and 2 ≤ h ≤ X we have∫ 2X
X
|∆K(x, h)|2dx ≤ cX(h+ L2)(nKL+ log dK)2.
4Proof. Let 2 ≤ T ≤ x; the constants in the O- and -symbols below are absolute. Denoting
by NK(T ) the number of zeros ρ = β + iγ of ζK(s) with 0 ≤ β ≤ 1 and |γ| ≤ T , using the
notation in the Introduction, we unconditionally have
NK(T ) =
nK
pi
T log T +
T
pi
log
(
dK
(2pie)nK
)
+O
(
log(dKT
nK )
)
; (5)
see Kadiri and Ng [10]. Moreover, using (5) in the explicit formula in Lagarias and Odlyzko [11]
(specialized to the case of ζK(s)) we have, again unconditionally, that
ψK(x) = x−
∑
|γ|≤T
xρ
ρ
+RK(x, T ), (6)
with
RK(x, T ) = O
( x
T
(nK log x+ log dK) log x
)
.
Proposition 1 follows now from (4) and (6) by the classical arguments in Saffari and Vaughan [16,
Lemmas 5 and 6] (notice a misprint in (6.20) there, where h2 on the right-hand side should be
replaced by h); here is a brief sketch. Arguing as in Lemma 6 of [16] we have (see (6.21) there)∫ 2X
X
|∆K(x, h)|2dx X
h
∫ 3h/X
h/3X
(∫ 3X
X
|ψK(x+ θx)− ψK(x)− θx|2dx
)
dθ. (7)
Choosing T = X in (6), the contribution of RK(x, T ) to the right-hand side of (7) is
 XL2(nKL+ log dK)2, (8)
while the contribution of the remaining part of the explicit formula is, thanks to (5),
 X2
( h
X
)2
NK
(
X
h
)
max
2≤t≤X/h
(
NK(t+ 1)−NK(t)
) hX(nKL+ log dK)2. (9)
Proposition 1 follows from (7)–(9). 
Proposition 1 represents another instance of Lang’s “recipe” reported in the Introduction. As
far as we know, such a phenomenon has not been established in the case of Brun–Titchmarsh-
type bounds, and actually it is not clear to us how the right extension should look like in this
case; we briefly discuss this issue at the end of the section. Hence, we use the following simple
but uniform bound, which however is unlikely to be sharp in the range needed here.
Proposition 2. Let K be a number field and 2 ≤ h ≤ x. Then
piK(x+ h)− piK(x) ≤ 4nK h
log h
. (10)
Proof. Again we use the notation in the Introduction. Let {kj}j∈J , kj ≥ 1, be the exponents
of the prime powers in the interval (x, x + h]; clearly, |J | ≤ h + 1. Since it is well known that
for 1 ≤ k ≤ nK there are at most nK/k prime ideals of OK with norm pk, we have
piK(x+ h)− piK(x) ≤ nK
∑
j∈J
1
kj
(
pi((x+ h)1/kj)− pi(x1/kj)).
But (x + h)1/k − x1/k ≤ x1/kh/(kx); hence, applying to pi((x + h)1/kj) − pi(x1/kj) the Brun–
Titchmarsh theorem when kj = 1 (Montgomery and Vaughan [14] with modulus q = 1) and
the trivial bound ≤ h/(kj
√
x) + 1 when kj ≥ 2, we get
piK(x+ h)− piK(x) ≤ 2nK h
log h
+ nK
∑
j∈J,kj≥2
1
kj
( h√
x
1
kj
+ 1
)
.
5Since clearly
∑
j∈J,kj≥2 1/kj ≤ log(|J | + 1) ≤ log(h + 2), Proposition 2 follows by a simple
computation. 
Proposition 3. Theorem 2 holds with c1n
1/2
K (nK log x + log dK)
√
x ≤ h ≤ x in place of
c1(nK log x+ log dK)
√
x ≤ h ≤ x.
Proof. We argue along the lines of Theorem 1. Indeed, for X sufficiently large and, e.g.,
h/nK > X
1/10, from Proposition 2 we have that if there exists x ∈ (X, 2X) with
|∆K(x, h)| > 1
4
h, then |∆K(x, h)| > ch
for all x ∈ (x − c′h/nK , x + c′h/nK), with certain absolute constants c, c′ > 0. Playing this
against Proposition 1, we therefore obtain that
h3
nK
 (nKL+ log dK)2hX.
Hence h n1/2K (nKL+ log dK)
√
X, and Proposition 3 follows. 
The sharper result stated in Theorem 2 is obtained using the direct approach by the smoothed
explicit formula. We follow the general lines of the proofs in Dudek [3] and Dudek et al [4],
where explicit versions of Ingham’s approach to Crame´r’s theorem are developed. Integrating
the infinite explicit formula for ψK(x) from 2 to x, see Grenie´ and Molteni [6, (1.3a) and
Lemmas 3.2 and 3.3], we obtain∫ x
2
ψK(t)dt =
x2
2
−
∑
ρ
xρ+1
ρ(ρ+ 1)
− cKx+ c′K +O(nKx log x),
where ρ runs over the non-trivial zeros of ζK(s) and cK , c
′
K are certain constants depending onK;
we are not concerned with their values since cK and c
′
K simply disappear after the manipulations
leading to the next displayed equation. Introducing the weight w(n) = max(1− |x−n|
h
, 0) as in
the last row of [3, p. 773] and arguing as on p. 774 there, we get
W (x, h) :=
∑
x−h<n<x+h
ΛK(n)w(n)
= h− 1
h
∑
ρ
(x+ h)ρ+1 − 2xρ+1 + (x− h)ρ+1
ρ(ρ+ 1)
+O
(
nK
x
h
log x
)
.
Now we split the sum over the ζK-zeros into the subsums Σ1 and Σ2 cutting at T = x/h, and
use DRH and (5) as in the proof of [3, Theorem 1.2], thus obtaining
W (x, h) = h+O((nK log x+ log dK)
√
x) +O
(
nK
x
h
log x
)
. (11)
From (11), we obtain the behavior of the unweighted sum, observing that for every 0 < ε < 1,
−1
ε
(
(1− ε)W (x, (1− ε)h)−W (x, h)) ≤ ψK(x+ h)− ψK(x− h)
≤ 1
ε
(
(1 + ε)W (x, (1 + ε)h)−W (x, h))
since ΛK(n) ≥ 0. Theorem 2 and the assertion after it follow at once. 
We conclude with a brief discussion on the Brun–Titchmarsh theorem for number fields and
its relevance to this paper. Note that the dependence on the data of K in Proposition 2, where
bounded h are allowed, is essentially best possible. Indeed, if a prime p ∈ (x, x+2], say, splits in
OK into the product of nK prime ideals of norm p, then clearly piK(x+ 2)−piK(x) ≥ nK , while
Proposition 2 gives piK(x+2)−piK(x) ≤ cnK , with some absolute c > 0. Note that, although the
6constant in the classical Brun–Titchmarsh theorem is of great interest, the absolute constant
in front of nK in (10) plays essentially no role in this paper. For larger h the dependence on K
in (10) is unsatisfactory, as indeed the prime ideal theorem, or (1), shows.
The bounds of Brun–Titchmarsh type are usually obtained by the Selberg sieve. Apparently,
an application of the Selberg sieve to piK(x + h) − piK(x), see, e.g., Hinz and Loedemann [8],
brings into play the residue νK of the Dedekind zeta function ζK(s). It is well known that νK
depends on several invariants of K, and even under DRH its dependence on such invariants is
not completely under control. This adds some difficulties to the problem of obtaining sharp
versions of Proposition 2. Perhaps one can prove that
piK(x+ h)− piK(x) ≤ ch/ log(h/dK),
but this is weaker than what is obtainable for an abelian extension K/Q, namely
piK(x+ h)− piK(x) ≤ ch/ log(h/qK), (12)
where qK is the conductor of K. Bound (12) can be obtained by coupling the classical Brun–
Titchmarsh theorem for arithmetic progressions with the Kronecker–Weber theorem for abelian
extensions of Q. Actually, when (12) is coupled with Proposition 1 we get back, in the abelian
case, a result of the same quality as Theorem 1.
Acknowledgements. We wish to thank Olivier Ramare´ for detecting some inaccuracies in
a previous version. This research was partially supported by grant PRIN2015 Number Theory
and Arithmetic Geometry. The authors are members of the INdAM groups GNSAGA and
GNAMPA.
References
[1] D. Bazzanella, A. Perelli - The exceptional set for the number of primes in short intervals
- J. Number Theory 80 (2000), 109–124.
[2] H. Crame´r - Some theorems concerning prime numbers - Ark. Mat. Astr. Fys. 15 (1920),
1–33.
[3] A. W. Dudek - On the Riemann hypothesis and the difference between primes - Int. J.
Number Theory 11 (2015), 771–778.
[4] A. W. Dudek, L. Grenie´, G. Molteni - Primes in explicit short intervals on RH - Int. J.
Number Theory 12 (2016), 1391–1407.
[5] D. A. Goldston, Y. Yıldırım - Primes in short segments of arithmetic progressions - Can.
J. Math. 50 (1998), 563–580.
[6] L. Grenie´, G. Molteni - Explicit smoothed prime ideals theorems under GRH - Math.
Comp. 85 (2016), 1875–1899.
[7] D. R. Heath-Brown - Gaps between primes, and the pair correlation of zeros of the zeta-
function - Acta Arith. 41 (1982), 85–99.
[8] J. Hinz, M. Loedemann - On Siegel zeros of Hecke-Landau zeta-functions - Mh. Math.
118 (1994), 231–248.
[9] A. E. Ingham - On the difference between consecutive primes - Quart. J. Math. 8 (1937),
255–266.
[10] H. Kadiri, N. Ng - Explicit zero density theorems for Dedekind zeta functions - J. Number
Theory 132 (2012), 748-775.
[11] J. C. Lagarias, A. M. Odlyzko - Effective versions of the Chebotarev density theorem - In
Algebraic Number Fields, ed. by A. Fro¨hlich, 409–464, Academic Press, London, 1977.
[12] S. Lang - On the zeta function of number fields - Invent. Math. 12 (1971), 337–345.
[13] A. Languasco, A. Perelli, A. Zaccagnini - An extension of the pair-correlation conjecture
and applications - Math. Res. Letters. 23 (2016), 201–220.
7[14] H. L. Montgomery, R. C. Vaughan - The large sieve - Mathematika 20 (1973), 119–134.
[15] K. Prachar - Generalisation of a theorem of A. Selberg on primes in short intervals - In
Topics in Number Theory, Coll. Math. Soc. Bolyai 13, ed. by P. Tura´n, 267–280, North-
Holland, Amsterdam, 1976.
[16] B. Saffari, R. C. Vaughan - On the fractional parts of x/n and related sequences. II - Ann.
Inst. Fourier 27 (1977), 1–30.
Lo¨ıc Grenie´, Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Uni-
versita` di Bergamo, viale Marconi 5, 24044 Dalmine (BG), Italy. e-mail: loic.grenie@gmail.com
Giuseppe Molteni, Dipartimento di Matematica, Universita` di Milano, via Saldini 50, 20133
Milano, Italy. e-mail: giuseppe.molteni1@unimi.it
Alberto Perelli, Dipartimento di Matematica, Universita` di Genova, via Dodecaneso 35, 16146
Genova, Italy. e-mail: perelli@dima.unige.it


Primes and prime ideals in short intervals

Abstract

Similar works

Full text

Available Versions

Crossref

Archivio istituzionale della ricerca - Università di Genova

AIR Universita degli studi di Milano