Abstracts of papers by Alessandro Zaccagnini

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Abstracts of papers by Alessandro Zaccagnini
Abstracts of papers by Alessandro Zaccagnini
Alessandro Zaccagnini
Dipartimento di Matematica, Università di Parma,
Parco Area delle Scienze, 53/A, Campus Universitario
43124 Parma, Italy.
[email protected]
December 1, 2011
This file contains full references and abstracts of my papers. Chronologically,
these deal with the following topics:
• Large gaps between primes in arithmetic progressions [1].
• An additive problem of Hardy and Littlewood and its variants [2, 3, 4].
• The Selberg integral [5, 6].
• An elementary talk on Goldbach’s problem [7].
• An “inverse” problem related to the Selberg integral [8].
• Another additive problem of the Hardy–Littlewood type [9].
√
• An elementary talk on the continued fraction for 2 [10].
• An elementary talk on properties of prime numbers and their relations to
cryptography [11].
• A review paper on the Selberg integral and related topics [12].
• An elementary paper that deals with various approximations to π [13].
• A paper on the Mertens product over primes in arithmetic progressions [14].
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• A paper dealing with an additive problem involving primes and powers of 2
[15].
• An elementary paper on the use of hand-held non-programmable calculators
[16].
• A paper on an average result concerning Hardy–Littlewood numbers [17].
• A paper on the averages of the error term of the Mertens product over primes
in arithmetic progressions [18].
• An elementary paper on the computation of areas without calculus [19].
• A paper on the numerical computation of the constants appearing in the
Mertens product over primes in arithmetic progressions [20].
• A paper on protocols in Peer-to-Peer networks [21].
• A paper on identities satified by the constants appearing in the Mertens
product over primes in arithmetic progressions [22].
• A paper on extremely small gaps between prime numbers, and short intervals that do not contain any prime [23].
• A paper on the numerical computation of the Mertens and Meissel–Mertens
constants for primes in arithmetic progressions [24].
• An elementary paper on the Euclidean Algorithm, its relation to the Fibonacci sequence and continued fractions [25].
• A paper on a diophantine problem with primes and powers of 2 [26].
• A paper on the Montgomery-Hooley inequality in short intervals [27].
• A paper on the asymptotic number of representations of an even integer as
a sum two primes [28].
The following “papers” only appear on the web.
• These “papers” deal with properties of prime numbers: [29], [30], [31].
• Properties of prime numbers and their relation to cryptography [32].
• Properties of prime numbers: in particular, small and large gaps between
consecutive primes. [33].
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Papers
1
Abstract of paper [1]
Let
G(x; q, a) :=
max
(pn+1 − pn ),
pn ≤x
pn ≡pn+1 ≡a mod q
where (a, q) = 1 and pn , pn+1 are consecutive primes in the arithmetic progression
a mod q. We prove the following result: let C be any fixed positive number,
C < 1. Then, uniformly for
log log log log x
1 ≤ q ≤ exp exp C log log x
log log log x
we have:
log log log log x
G(x; q, a) ≥ eγ + oC (1) φ(q) log x log log x
,
(log log log x)2
where γ is Euler’s constant. The o-symbol may depend on C.
2
Abstract of paper [2]
Let k ≥ 2 be an integer, and set Ek (X) := |{n ≤ X : n 6= mk , n is not a sum of
a prime and a k-th power}|. We prove that there exists δ = δ(k) > 0 such that
Ek (X) k X 1−δ , by means of a suitable application of the circle method, essentially a variant of Montgomery & Vaughan’s method (Acta Arithmetica 1975).
The proof is similar to the one given by Brünner, Perelli & Pintz (Acta Math. Hungarica 1989) in the case k = 2, the main new difficulty being in the treatment of
the singular series.
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3
Abstract of paper [3]
Let k ≥ 2 be a fixed integer and p denote a prime. For any n ∈ N such that the
polynomial xk − n is irreducible over Q let
Rk (n) :=
∑
Λ(h),
ρk (n, p) := |{m mod p : mk ≡ n
mod p }|
h+mk =n
and
ρk (n, p) − 1
ρk (n, p)
1
Sk (n) := ∏ 1 −
= ∏ 1−
1−
.
p−1
p
p
p
p
One expects that for the integers n that we are considering one has Rk (n) ∼
n1/k Sk (n) as n → ∞. Let L := log N,
R∗k (n) :=
∑
Λ(h),
and
h+mk =n
N−Y ≤h≤N
Y /2≤mk ≤3/2Y
Pk∗ (n) :=
1
k
1
∑
m k −1 .
h+m=n
N−Y ≤h≤N
Y /2≤m≤3/2Y
Building on previous work by Perelli & Pintz in the case k = 2, we prove
Theorem 1 Let k ≥ 3, ε, A > 0, N 7/12+ε ≤ Y ≤ N and max Y 1−1/k+ε , N 1/2+ε ≤
H ≤ Y . Then
?
∑
|R∗k (n) − Pk∗ (n)Sk (n)|2 ε,A,k HY 2/k L−A ,
N≤n≤N+H
where the ? means that the sum is over n ∈ N such that xk − n is irreducible over
Q.
Theorem 2 Assume the Generalized Riemann Hypothesis and let k ≥ 2, ε > 0.
Then
Ek (N) := |{N ≤ n ≤ 2N : n 6= p + mk }| k,ε N 1+ε−2/(kK)
where K := 2k−1 .
Theorem 2 is obtained by means of Hardy & Littlewood’s circle method, and
Weyl’s inequality, by a suitable treatment of the singular series.
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4
Abstract of paper [4]
In their third paper of the “Partitio Numerorum” series, Hardy & Littlewood conjectured that for k = 2, 3 and for n → ∞, n 6= mk , there is an asymptotic formula for rk (n) := |{(m, p) : n = mk + p, p is a prime}| and that, in particular,
rk (n) → ∞ if n → ∞, n 6= mk . For k ≥ 2 let Ek (X) := |{n ≤ X : rk (n) = 0}| be
the number of exceptions to the “weak” conjecture rk (n) ≥ 1 for n ≥ n0 (k), n not
a power. We prove that there exists δ = δ(k) > 0 such that Ek (X) X 1−δ , that
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Ek (X + H) − Ek (X) H(log X)−A for X 12 (1−1/k)+ε ≤ H ≤ X, and also give estimates for the number of integers for which the asymptotic formula actually holds.
Furthermore, we give explicit estimates for δ(k) under the Generalized Riemann
Hypothesis. This paper contains in part [2, 3] and a sketch of the circle method,
as used in these problems.
5
Abstract of paper [5]
We give an alternative proof of the well-known estimate J(X, h) = o Xh2 for the
Selberg integral
J(X, h) :=
Z 2X X
ψ(t) − ψ(t − h) − h2 dt
when h ≥ X 1/6+ε , by means of Heath-Brown’s identity.
6
Abstract of paper [6]
We prove the estimate J(x, h) = o xh2 (log x)−2 for the Selberg integral
Z 2x h 2
J(x, h) :=
π(t) − π(t − h) − logt dt,
x
when h ≥ x1/6−ε(x) , provided that ε(x) → 0 as x → +∞. The proof depends on
an identity of Linnik and Heath-Brown which yields a suitable Dirichlet series
decomposition for the quantity that we want to estimate. This is in a form that can
be attacked by means of mean value theorems for Dirichlet series.
5
7
Sunto dell’articolo [7]
Si tratta del testo di una conferenza divulgativa tenuta in occasione dell’allestimento della Mostra “Oltre il Compasso” a Parma fra il 1◦ .10 ed il 5.11.1998. Si
vuole mostrare come argomentazioni essenzialmente elementari derivate in ultima
analisi dal Crivello di Eratostene permettono di “indovinare” la formula asintotica
che si congettura corretta per il numero di rappresentazioni di un intero pari grande come somma di due numeri primi. La stessa tecnica è applicata a problemi
analoghi. In coda si danno riferimenti a letture ed approfondimenti ulteriori.
We give essentially elementary arguments based on Eratosthenes’ sieve to find
the expected asymptotic formula for the number of representations of a large even
number n as a sum of two primes and apply the same method to similar additive
problems involving prime numbers. The argument is used to explain the irregularities observed in the number of representations, as n grows to infinity.
8
Abstract of paper [8]
We find zero-free regions and density theorems for the Riemann zeta-function
depending on non-trivial estimates for
Z 2x
J(x, θ) :=
|ψ(t) − ψ(t − θt) − θt|2 dt,
x
uniformly in some range
of3 2θ. A Corollary of our main result is the following:
x θ
, uniformly for x−β ≤ θ ≤ 1, where F is a positive,
Assume that J(x, θ) = O F(xθ)
increasing function which is unbounded as x → +∞, such that F(x) = O (xε ) for
every ε > 0, and that β ∈ (0, 1). Then the Riemann zeta-function does not vanish
in the region
log F(t)
σ > 1 −C
,
logt
where C is some absolute, positive constant. The proof is based on Turán’s power
sum method.
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9
Abstract of paper [9]
This paper contains the proof that “almost all” positive integers can be represented
as a sum of a prime and a value of a polynomial f ∈ Z[x], provided that f satisfies
a necessary arithmetical condition.
10
Sunto dell’articolo [10]
Si tratta del testo di una conferenza divulgativa che prende a pretesto il fatto che
il rapporto fra le dimensioni dei lati di un
√ foglio di carta nel formato A4 è uno dei
convergenti della frazione continua di 2 per parlare di ricorrenze, geometria, il
metodo di Newton ed altro, usando solo argomentazioni elementari. In coda, si
danno spunti per letture ed approfondimenti ulteriori.
We take as our text the fact that the ratio of the sides of a sheet of paper√in the
common A4 format is one of the convergents of the continued fraction for 2 for
a journey through geometry, recurrences, Newton’s method, . . . , using essentially
only elementary arguments.
11
Sunto dell’articolo [11]
Si studiano alcune proprietà elementari dei numeri primi e se ne discute la rilevanza alle applicazioni crittografiche.
We study some elementary properties of prime numbers and discuss their relevance to applications in modern cryptography.
12
Abstract of paper [12]
This is a paper containing a review of results that appeared in [6] and [8].
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13
Sunto dell’articolo [13]
In questo articolo elementare si danno due dimostrazioni della formula di Archimede–Viète per π e le dimostrazioni delle formule usate da Machin, Shanks ed
altri per calcolare π con centinaia di cifre a partire dal Settecento.
14
Abstract of paper [14]
We study the Mertens product over primes in arithmetic progressions, and find a
uniform version of previous results on the asymptotic formula, improving at the
same time the size of the error term, and giving an alternative, simpler value for
the constant appearing in the main term.
15
Abstract of paper [15]
We prove that, for any fixed positive integer k, a suitable asymptotic formula for
the number of representations of an even integer
N ∈ [1, X] as
the sum of two
3/5
10
primes and k powers of 2 holds with at most Ok X (log X)
exceptions.
16
Sunto dell’articolo [16]
In questo articolo elementare, rivolto agli insegnanti delle Scuole Medie inferiori e superiori, si tratta dell’uso della calcolatrice tascabile non programmabile, delle sue limitazioni intrinseche e delle sue potenzialità, per favorirne un uso
consapevole da parte degli studenti.
17
Abstract of paper [17]
Here we give an average result on the main term for the asymptotic formula of the
Hardy–Littlewood numbers in short intervals.
8
18
Abstract of paper [18]
We give estimates for the error term of the Mertens product over primes in arithmetic progressions of the Bombieri–Vinogradov and Barban–Davenport–Halberstam type.
19
Sunto dell’articolo [19]
In questo articolo elementare, rivolto agli insegnanti delle Scuole Medie inferiori
e superiori, si calcolano i valori delle aree sottese dal grafico di alcune funzioni
elementari senza fare ricorso al Teorema fondamentale del calcolo integrale, usando opportune decomposizioni del dominio, e qualche identità algebrica. L’articolo
contiene le dimostrazioni di tutte le identità e dei limiti notevoli utilizzati.
20
Abstract of paper [20]
This paper reports on the explicit numerical computation with at least 100 decimal
digits of the constants appearing in the Mertens product over primes in arithmetic
progressions, for progressions with moduli up to 100.
21
Abstract of paper [21]
This paper suggests an alternative protocol for the management of reputation in
Peer-to-Peer networks based on DHT (Distributed Hash Tables).
22
Abstract of paper [22]
We give new identities for the constants in the Mertens product over primes in the
arithmetic progressions a mod q, extending previous work by Uchiyama, Grosswald, Williams and Moree.
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23
Abstract of paper [23]
We give a new estimate for the integral moments of primes in short intervals of the
type (p, p + h], and prove that for every λ > 1/2 there exists a positive proportion
of primes p ≤ X such that the interval (p, p + λ log X] contains at least a prime
number. We improve Cheer and Goldston’s result (1987) on the size of real numbers λ > 1 with the property that there is a positive proportion of integers m ≤ X
such that the interval (m, m + λ log X] contains no primes.
24
Abstract of paper [24]
We give explicit numerical values with 100 decimal digits for the Mertens constant
involved in the asymptotic formula for ∑ p≤x 1/p and, as a by-product, for the
p≡a mod q
Meissel-Mertens constant defined as ∑ p≡a mod q log(1 − 1/p) + 1/p , for q ∈ {3,
. . . , 100} and (q, a) = 1.
25
Sunto dell’articolo [25]
Partendo dal semplice problema concreto del calcolo efficiente del massimo comun divisore fra due interi positivi, si descrive l’Algoritmo di Euclide e se ne
fa un’analisi di complessità parziale, scoprendo che il numero di iterazioni necessarie è massimo se gli interi dati sono termini consecutivi della successione
dei numeri di Fibonacci. Si interpreta il calcolo come la frazione continua del
rapporto fra gli interi dati e si generalizza alle frazioni continue infinite, concludendo con la scoperta che le frazioni continue periodiche hanno valore irrazionale
quadratico.
26
Abstract of paper [26]
We refine a recent result of Parsell (2003) on the values of the form λ1 p1 + λ2 p2 +
µ1 2m1 + · · · + µs 2ms , where p1 , p2 are prime numbers, m1 , . . . , ms are positive
integers, λ1 /λ2 is negative and irrational and λ1 /µ1 , λ2 /µ2 ∈ Q.
10
27
Abstract of paper [27]
We study a short-interval version of a result due to Montgomery and Hooley.
Write
2
q
h
S(x, h, Q) = ∑ ∑ ψ(x + h; q, a) − ψ(x; q, a) −
ϕ(q) q≤Q a=1
(a,q)=1
and κ = 1 + γ + log 2π + ∑ p (log p)/p(p − 1). Denote the expected main term
by M(x, h, Q) = hQ log(xQ/h) + (x + h)Q log(1 + h/x) − κhQ. Let ε, A > 0 be
arbitrary, x7/12+ε ≤ h ≤ x and Q ≤ h. There exists a positive constant c1 such that
!
3/5
(log
2h/Q)
S(x, h, Q) − M(X, h, Q) h1/2 Q3/2 exp −c1
+ h2 log−A x.
(log log 3h/Q)1/5
Now assume GRH and let ε > 0, x1/2+ε ≤ h ≤ x and Q ≤ h. There exists a positive
constant c2 such that
h 1/4+ε
Q2 + hx1/2 logc2 x.
S(x, h, Q) − M(x, h, Q) Q
28
Abstract of paper [28]
Let Λ be the von Mangoldt function and R(n) = ∑h+k=n Λ(h)Λ(k) be the counting
function for the Goldbach numbers. Let N ≥ 2 and assume that the Riemann
Hypothesis holds. We prove that
N
∑ R(n) =
n=1
N2
N ρ+1
−2∑
+ O N(log N)3 ,
2
ρ ρ(ρ + 1)
where ρ = 1/2 + iγ runs over the non-trivial zeros of the Riemann zeta-function
ζ(s).
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Papers on the web
29
Sunto dell’articolo [29]
È il primo di una serie di articoli divulgativi in cui si raccolgono proprietà dei numeri primi che di solito non si trovano nei libri di testo. In particolare, si parla del
Teorema Fondamentale dell’Aritmetica, del Crivello di Eratostene e della densità
dei primi nella successione dei numeri naturali. In Appendice si danno risultati
piú complessi.
30
Sunto dell’articolo [30]
È il secondo di una serie di articoli divulgativi in cui si raccolgono proprietà dei
numeri primi che di solito non si trovano nei libri di testo. Qui parliamo di criteri
di primalità ed algoritmi di fattorizzazione, di certificati di primalità, di numeri
primi di forma speciale. Anche qui, risultati piú complessi sono dati in Appendice.
31
Sunto dell’articolo [31]
Si tratta di un breve annuncio del risultato di Goldston, Pintz e Yıldırım a proposito della dimostrazione, lungamente attesa, del fatto che
lim inf
n→∞
pn+1 − pn
= 0,
log pn
dove pn indica l’n-esimo numero primo.
12
32
Sunto dell’articolo [32]
Questo articolo contiene il testo di una conferenza divulgativa tenuta per la “Settimana della cultura scientifica e tecnologica” del 2005. Si parla di due esempi
di crittogrammi presenti nella letteratura, e precisamente “Lo scarabeo d’oro” di
Edgar Allan Poe, e “Viaggio al centro della terra” di Jules Verne. Poi si esamina
il metodo di Cesare, uno dei piú antichi metodi crittografici noti in Occidente, e
le debolezze di questi sistemi crittografici classici. Infine si esaminano un paio di
sistemi crittografici moderni, e se ne studiano le basi matematiche.
33
Sunto dell’articolo [33]
Questo articolo descrive lo stato dell’arte riguardo la questione degli intervalli fra
numeri primi consecutivi, nei due casi di grandi o piccole deviazioni dal comportamento “medio.” Dimostriamo alcuni risultati che, pur non essendo i migliori
oggi noti, sono pur sempre non banali e illustrano bene le tecniche che si usano in
questo campo. Questo articolo ha un livello decisamente superiore e si rivolge a
studenti universitari.
References
[1] A. Zaccagnini. A note on large gaps between consecutive primes in arithmetic progressions. J. Number Theory, 42:100–102, 1992.
[2] A. Zaccagnini. On the exceptional set for the sum of a prime and a k-th
power. Mathematika, 39:400–421, 1992.
[3] A. Perelli and A. Zaccagnini. On the sum of a prime and a k-th power. Izv.
Ross. Akad. Nauk, Ser. Math., 59:185–200, 1995.
[4] A. Zaccagnini. Additive problems with prime numbers. Rend. Sem. Mat.
Univ. Pol. Torino, 53:471–486, 1995. Atti del “Primo Incontro Italiano di
Teoria dei Numeri,” Roma, 3–5 gennaio 1995.
[5] A. Zaccagnini. On the Selberg integral via Heath-Brown’s identity. Riv. Mat.
Univ. Parma, 5:205–212, 1996.
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[6] A. Zaccagnini. Primes in almost all short intervals. Acta Arithmetica,
84:225–244, 1998.
[7] A. Zaccagnini. Variazioni Goldbach: problemi con numeri primi. L’Educazione Matematica, Anno XXI, Serie VI, 2:47–57, 2000. Available in various formats and languages at http://www.math.unipr.it/˜zaccagni/
Talks.html.
[8] A. Zaccagnini. A conditional density theorem for the zeros of the Riemann
zeta-function. Acta Arithmetica, 93:293–301, 2000.
[9] A. Zaccagnini. A note on the sum of a prime and a polynomial. Quart. J.
Math. Oxford, 52:519–524, 2001.
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[12] A. Zaccagnini. Primes in almost all short intervals and the distribution of
the zeros of the Riemann zeta function. Ramanujan Mathematical Society–
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[13] A. Zaccagnini. Quanto vale π? L’Educazione Matematica, Anno XXIV, Serie
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[14] A. Languasco and A. Zaccagnini. A note on Mertens’ formula for arithmetic
progressions. J. Number Theory, 127:37–46, 2007.
[15] A. Languasco, J. Pintz, and A. Zaccagnini. On the sum of two primes and k
powers of two. Bull. London Math. Soc., 39:771–780, 2007.
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[16] A. Zaccagnini. La calcolatrice e le sue limitazioni. L’Educazione Matematica, Anno XXVII, Serie VII, 2:35–45, 2007.
[17] A. Languasco and A. Zaccagnini. On the Hardy–Littlewood problem in short
intervals. International J. Number Theory, 4:715–723, 2008.
[18] A. Languasco and A. Zaccagnini. Some estimates for the average of the
error term of the Mertens product for arithmetic progression. Functiones et
Approximatio Commentarii Mathematici, 38:41–47, 2008.
[19] A. Zaccagnini. È veramente necessario il Teorema fondamentale del calcolo
integrale? Archimede, pages 198–204, 2007.
[20] A. Languasco and A. Zaccagnini. On the constant in the Mertens product
for arithmetic progressions. II. Numerical values. Math. Comp., 78:315–326,
2009.
[21] N. Fedotova, G. Orzetti, L. Veltri, and A. Zaccagnini. Byzantine Agreement
for Reputation Management in DHT-based Peer-to-Peer Networks. In 15th
International Conference on Telecommunications, St Petersburg, 2008.
[22] A. Languasco and A. Zaccagnini. On the constant in the Mertens product
for arithmetic progressions. I. Identities. Functiones et Approximatio Commentarii Mathematici, 42:17–27, 2010.
[23] D. Bazzanella, A. Languasco, and A. Zaccagnini. Primes in logarithmic
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Math., 19:279–284, 2010. http://arxiv.org/abs/0906.2132.
[25] A. Zaccagnini. Algoritmo di Euclide, numeri di Fibonacci e frazioni continue. In P. Vighi, editor, Progettare Lavorare Scoprire, pages 153–162. Dipartimento di Matematica, Università di Parma, 2010. http://www.math.
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and s powers of 2. Acta Arithmetica, 2010. http://arxiv.org/abs/0811.
3663.
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[27] A. Languasco, A. Perelli, and A. Zaccagnini. On the Montgomery-Hooley
theorem in short intervals. Mathematika, 56:231–243, 2010.
[28] A. Languasco and A. Zaccagnini. The number of Goldbach representations
of an integer. Proc. Amer. Math. Soc., 140:795–804, 2012.
[29] A. Languasco and A. Zaccagnini. Alcune proprietà dei numeri primi, I.
Sito web Bocconi-Pristem, 2005. http://matematica.unibocconi.it/
LangZac/zaccagnini.pdf.
[30] A. Languasco and A. Zaccagnini. “Alcune proprietà dei numeri primi, II.
Sito web Bocconi-Pristem, 2005. http://matematica.unibocconi.it/
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consecutivi! Sito web Bocconi-Pristem, 2005. http://matematica.
unibocconi.it/LangZac/risultatoteorianumeri.htm.
[32] A. Zaccagnini. Cryptographia ad usum Delphini. Web publ., 2005. http://
www.math.unipr.it/˜zaccagni/psfiles/papers/CryptoDelph.pdf.
[33] A. Languasco and A. Zaccagnini. Intervalli fra numeri primi consecutivi.
Sito web Bocconi-Pristem, 2005. http://matematica.unibocconi.it/
LangZac/home3.htm.
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