Ricerche di Matematica
Vol. LII , fasc. 1◦ , (2003), 1-I BOZZE

On the symmetry of primes in almost all short intervals
G. COPPOLA (*)

Abstract. – In this paper we study the symmetry of primes in almost all
short intervals; by elementary methods (based on the Large Sieve) we give, forh  x1/3 logc x (c > 0, suitable), a non-trivial estimate for the mean-square (overN < x ≤ 2N ) of an average of “symmetry sums”; these sums control the symmetryof the von-Mangoldt function in short intervals around x. We explicitly remarkthat our results are out of reach of the classic analytic methods based on explicit
formulas and complex integrals
Key Words: Missing.
A.M.S. Classification: 11N37, 11N36.

1. Introduction and statement of the results
In this paper we study the “symmetry” of prime numbers in “almostall” short intervals.
As usual, we mean by “short interval” an interval of the type [x, x + H],
with H → ∞ and H = o(x), as x → ∞; and we say that something is true for“almost all” short intervals of the kind [x − h, x + h], if, given N < x ≤ 2N ,with N → ∞, and given h = h(N ) = o(N ), with h(N ) monotone increasing
and h → ∞ as N → ∞, the property holds for all x ∈ [N, 2N ], with anumber of (eventual) exceptions which is o(N ) (their Lebesgue measure issuch). In the sequel, however, our variable x will assume integer values (ifnot explicitly stated).
(*)Missing address

File: 679.tex
6.luglio.2003

2G. Coppola

We mean, by studying this symmetry, to study sums of the kindΛ(n) sgn(n − x)|n−x|≤hand other sums like this (see our theorems).
(Here Λ(n) is the von-Mangoldt function, deﬁned byΛ(n) :=log p0

if n = pk , p a prime and k ∈ Notherwise

and the sign function is given by sgn(t) := t/|t|, ∀t ∈ R − {0}, sgn(0) := 0.)
In fact, Kaczorowski and Perelli were the ﬁrst (as far as the Author isaware of) to introduce this type of sums (but with a particular weight function, G(n, x, T ), see [6]), in connection with the problem of the estimationof the “Selberg integral”, i.e.
2 2N   dx
Λ(n)−HJ(N, H) :=Nx<n≤x+H(here [x − H, x + H] is always a short interval, but the length H is slightly
smaller then our h).
Actually, it can be easily shown that the Prime Number Theorem (abbrev. PNT) x<n≤x+H Λ(n) ∼ H holds for almost all the short intervals[x, x + H] if and only if J(N, H) = o(N H 2 ), whereas the Brun-Titchmarsh
inequality gives (if H = N θ , θ ∈]0, 1[) the “trivial” estimate O(N H 2 ).
By classical estimates involving the Riemann-von Mangoldt explicit formula (see [5]), it is possible to prove that J(N, H) = o(N H 2 ), whenever
H = N 1/6−ε (with ε = ε(N ) → 0, as N → ∞), see [9].
In their work [6] Kaczorowski & Perelli give a link (for H a power of N ;the Author & Vitolo [4] did the same for H a power of log N ) between theSelberg integral and the mean-square of a variant of our “symmetry sums”(see [6], Theorem 2). This sum arises as a main term in the remainders of
their new form of the Riemann-von Mangoldt explicit formula [5].
In particular, if an estimate gaining L1+ε (ε > 0 enough small) w.r.t.
the trivial one (that is, O(N H 2 ), see before) for the mean-square of thissymmetry sum would be available, it would be possible to get J(N, H) =o(N H 2 ), whence the “almost-all short intervals” PNT (quoted before).
We remark that the present work is still far from getting such a result;
this would have, also, great consequences on the zeros of the Riemann-zetafunction, as exploited in [10].

On the symmetry of primes in almost all short intervals

3

Instead, we will conﬁne our attention in proving a (much weaker) related result; in fact (see our Theorem 1 for the exact statement) we willbound a mean-square of an average of “symmetry sums”, calculated in shortxx
intervals, going from [x − h, x + h] to the trivial [ 2h− 12 , 2h+ 12 ].
However, there is an arithmetical interest for such a result, due to theintrinsic elementary nature of the arguments used; our proof is only based,in fact, on the Large Sieve.
Also, due to the low cancelation of the remainders in the (quoted) explicit formulas for Λ, we see that the classical methods (using complex
integrals and Dirichlet series) are unuseful to prove our results.
Our present paper continues, from our previous papers [2] and [3], thestudy of the symmetry in almost all short intervals; in particular, it takes
inspiration much from Lemma 2 of [3].
In [3] we study the symmetry of the divisor function d(n), while in [2]we have studied the prime-divisors ω(n).
In all our papers we use the properties of the function χq (x) (see thefollowing) in the classic environment of the Large-Sieve.
We will explore the set of arithmetical functions f for which the method
works in future papers.
Here we conﬁne to f = Λ, the von-Mangoldt function (now on L :=log N ).
Theorem 1. – Let A > 2 and let N, h be natural numbers, such thath = h(N ) → ∞ and h  N L−A as N → ∞. Then2  x 
Λ(d) sgn d −  N h2 L−A ,jx∼N j≤2h |d− x |≤ h
jj

provided h ≥ N 1/3 LB , where B = B(A) > 0 can be explicitly computed.
(Actually, a good value is B = A + 6.)
We explicitly remark that it’s non-trivial w.r.t. the same without the“sgn”.
Actually, the same methods (see the proof of our Theorem 1) allow us toprove that the same is true for such “symmetry sums” of, also, PK numbers,
not only primes (here PK stands for an integer product of exactly K prime
factors, counted with their multiplicities).

4G. CoppolaAs Λ serves as a substitute for the characteristic function of primes, ΛKserves as a substitute for the characteristic function of PK numbers:ΛK (n) := µ(d) log

Kd|nnd.

As regards these generalized von-Mangoldt functions ΛK , we have the following
Theorem 2. – Let A > 2 and N, h ∈ N, with h = h(N ) → ∞ as
N → ∞. Assume K > 1 is a (ﬁxed) natural number and h  N L1−K−A .
Then2  x ΛK (d) sgn d −  N h2 L−A ,jx∼N j≤2h |d− x |≤ hjj

provided h ≥ N 1/3 LB , where B = B(A, K) > 0 can be explicitly computed.
Also this bound is non-trivial w.r.t. the one without the “sgn”.
Let’s deﬁne the function χd (x) (not a Dirichlet character!):χd (x) :=x+hd

+x−hd −2xd=−sgn(n − x);|n−x|≤hn≡0(modd)

as a function of x(∈ N) with period d ∈ N has a ﬁnite Fourier expansion.
We explicitly remark that in [3] we use the “ﬂipping” property of thedivisors d to apply the Large Sieve to the divisor function (through χd (x)).
(In the sequel we’ll talk about “moduli” and “divisors” meaning thesame.)
The moduli d we will use in Theorem 1 proof are “low” and “high”; by
low we meanthat we can treat them by the Large Sieve (here “we can go√
beyond N ”, due to the low mean-square of χd (x) Fourier coeﬃcients); by ithigh we mean the Large Sieve’s out of reach, but by a weak ﬂipping (insteadof the ﬂipping, not applicable) we make them become sporadic terms, which
can be estimated by the “Fragmented Large Sieve”.

On the symmetry of primes in almost all short intervals

52. Proof of Theorem 1
First of all,sgn(n − x)|n−x|≤hΛ(d) =d|nsgn(n − x) log n|n−x|≤hn−xsgn(n − x) log 1 +=x|n−x|≤hh2;xwriting the ﬁrst sum in terms of χd (x) this is equivalent in saying
Λ(d)χd (x) d≤x+hh2;
x

and this will be our start point to prove Theorem 1; in fact, we will beable to treat all the relevant ranges of d, except for the one that will be in
our ﬁnal statement (see the sum on d in Theorem 1), but we know that thewhole range of summation on d gives a small contribution (even individually,rather than on average over x !); so we will get the Theorem.
In order to distinguish between “low” and “high” modules d, we thenintroduce a (free) parameter M = M (N, h) → ∞ as N → ∞Σ1 (x) :=d≤M√χd (x)Λ(d)Σ2 (x) :=NM√χd (x)Λ(d) .

N <d≤x+h

Before to apply the Large-Sieve to Σ1 (x) we ﬁrst need to rearrange the innerexponential sum, as follows.
Then, χd (x) deﬁnition (apply the additive characters orthogonality [8]to get the ﬁnite Fourier expansion: see [2] and [3]) gives its Fourier coeﬃcients property cat,bt = 1t ca,bχd (x) =cj,d ed (jx) =j<d=cj,d ed (jx)j<d(j,d)=t

1t|dtcj  ,d/t ed/t (j  x) =j  <(d/t)(j  ,(d/t))=1

whencet|d

Σ1 (x) =t≤M√N

dcr,t et (rx) ,r≤t(r,t)=1

 

t n≤Λ(tn)  ∗
cj,t et (jx) ,√nj≤t
M Nt

6
∗

(G. Coppolaindicates, as usual, the summation over reduced classes, i.e. (j, t) = 1).
Now we can apply the Large-Sieve inequality (see [1])2 ∗2  NM2
αce(jx)|αq |2|cj,t | ,qj,t t√√j<tx∼N q≤M Nj≤tq≤M N

where, by deﬁnition of χq (x)
αq :=

n≤



Λ(qn)
√nM N
qthen we obtain (by the upper bound2|cj,q | j<qh;qn≤X Λ(qn)/n  log X, see [7])2 

Λ(d)χd (x)  N M 2 hL3 .
√x∼N d≤M NHence we know Σ1 (x)-mean-square.
Recalling the deﬁnition of χd (x), we get the series
Σ2 (x) =Λ(d) −Λ(d) √+√j∈N  M N <d≤x+hM N <d≤x+hx <d< x+hj
jx−h <d< xj
j+ O ((d(x − h) + d(x) + d(x + h)) L)(d = x−h, d = xj and d = x+hgive the divisor-functions-remainders; for thejj
moment they will be neglected as this mean-square contributes O(N L5 )).
This series is genuinely ﬁnite, due to the inner constraints on d; in fact

Σ2 (x) =

Λ(d)−Λ(d) √√x+hM N <d≤x+hM N <d≤x+hj< √
M=N




x−hx+hj <d≤ jNx−h <d< xj
jx <d< x+hjj√j≤ x−hM=

x−hx+h√j≤ x−hj <d≤ jM Nsgn d −
sgn d −xΛ(d) + O L
jx−hx+hxΛ(d) + OjM√ <j≤ √NM Nh+1 j hh2+ √ +1 LNM N.

On the symmetry of primes in almost all short intervals

7We isolate the ﬁrst 2h terms of Σ2 (x) (appearing in Theorem 1), say Σ(x)Σ2 (x) := Σ(x) + S(x) + OwhereS(x) := hh2+ √ +1 LNM N√
2h<j≤ x−hMNsgn d −x−hx+hj <d≤ j,xΛ(d) .
jThus (the terms coming from d(x) and d(x ± h) contribute in mean-square
O(N L5 ), see [7]) we will prove that (see at the end for the choice of M )(∗)

N 3/2 7hL .
M2|S(x)| x∼NWe explicitly remark that in S(x) (since j > 2h) the inner sum on d becomessporadic, i.e. present with at most one term (for which the estimate O(1)is non-optimal, since it’s “rather always” 0).
Henceforth, the following argument holds: this sporadic term is present
if and only if the interval ](x − h)/j, (x + h)/j] contains a prime power (and]; the function χj (x)then it’s unique); if so, this prime power is exactly [ x+hj
detects the presence of integers in this interval, also giving (by the sign) theside of x/j (right or left of).
Thenx+h
χj (x)Λ
;S(x) =
jx−h2h<j≤M√N

we use, now, a dyadic dissection of the j-range to get
S(x)  L

2M

where, say,
SJ (x) :=Nχq (x)Λ

q∼Jand∗S (x) :=SJ (x) + S ∗ (x) ,max√
2h<J≤ N −hN −h
√ <q≤ x−h√
2M N
2M Nx+hqχq (x)Λ

,x+hq;

we remark, here, that everything valid for SJ (x) will be also valid for S ∗ (x),in particular the following estimate over the x-exponential sum (see [3]).

8G. Coppola

We apply here the “fragmented Large-Sieve”, i.e. we prove a LargeSieve type inequality in which the x-summation range is fragmented. Infact, as already seen before, by χq (x) properties we can write  1

SJ (x) =d≤2J n∼ JnΛx+hnd ∗

cj,d ed (jx) .

j≤ddExpanding the square we get2|SJ (x)| =

d1 ,d2 ≤2J n1 ∼J/d1x∼N×Λx∼Nx+hn1 d 1

n2 ∼J/d2Λ

1 1 ∗∗cj1 ,d1 cj2 ,d2 ×
n1 n2 j ≤d j ≤dx+hn2 d 211 e22

 j1
j2−x
d1 d2.

This last sum over x is estimated trivially, when j1 /d1 = j2 /d2 , as O(N L2 );while can be bounded by means of Lemma 2 of [3] when j1 /d1 and j2 /d2
are distinct Farey fractions. In fact, we can break the range of x intoO(N/J) “fragmented” ranges (each of length at most O(J)) in which theΛ-functions are constant (being constant their arguments); then, we argue
as in Lemma 2 of [3], with k := j/d and Ck := cj,d (and we use the classicalestimate for exponential sums, see [8]):




1 1 
N|SJ (x)|  L|Ckr ||Cks | min N,n n
J kr − ksx∼Nn ,n ≤2N 1 2 kr ,ks2

21

2





1  N 1
 L2
N|Ck |2 +|Ckr ||Cks |n nJ kr =kskr − ksn ,n ≤2N 1 2k12which is, by Lemma 2 of [3], with δ ≥ 1/J 2 , at most L2since1L(N + N J)
|Ck |2  N JhL5 ,nnn ,n ≤2N 1 2k1k |Ck |22

 h. Hence we get (∗):
x∼N|S(x)|2 max√ N JhL7 2h<J≤ MN

N 3/2 7hL .
M

Collecting the estimates for Σ1 (x) and S(x), together with the optimal
choice M = N 1/6 L4/3 , we get (with B = A + 6) the theorem.

On the symmetry of primes in almost all short intervals

9REFERENCES[1] E. Bombieri: “Le Grande Crible dans la théorie analytique des nombres”,
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in almost all short intervals”, to appear.
[3] G. Coppola, S. Salerno: “On the symmetry of the divisor function in almost
all short intervals”, to appear.
[4] G. Coppola, A. Vitolo: “On the distribution of primes in intervals of lengthlogΘ N ”, Acta Math. Hungar. 70 (2) (1996), 151-166. MR 96i:11103.
[5] J. Kaczorowski, A. Perelli: “A new form of the Riemann-von Mangoldt
explicit formula”, Boll. Un. Mat. Ital. B 10 (1) (1996), 51-66. MR 97g:11098.
[6] J. Kaczorowski, A. Perelli: “On the distribution of primes in short intervals”, J. Math. Soc. Japan 45 (3) (1993), 447-458. MR 94e:11100.
[7] G. Tenenbaum: “Introduction to analytic and probabilistic number theory”,
Cambridge University Press, Cambridge, 1995. MR 97e:11005b.
[8] I. M. Vinogradov: “The Method of Trigonometrical Sums in the Theory ofNumbers”, Interscience Publishers LTD., London, 1954. MR 15,941b.
[9] A. Zaccagnini: “Primes in almost all short intervals”, Acta Arith. 84 (3)(1998), 225-244. MR 2001k:11105.
[10] A. Zaccagnini: “A conditional density theorem for the zeros of the Riemannzeta-function”, Acta Arith. 93 (3) (2000), 293-301. MR 2001k:11166.

Pervenuto alla Redazione il 4 gennaio 2002