## Sieve Methods

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@MISC{Charles_sievemethods,

author = {Denis Xavier Charles},

title = {Sieve Methods},

year = {}

}

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### Abstract

Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum

### Citations

296 |
Approximate formulas for some functions of prime numbers
- Rosser, Schoenfeld
- 1962
(Show Context)
Citation Context ...teger D such that: 1. 1 ≤ D ≤ (logN) 20h ; 2. gcd(iD + 1,N) = 1, for 1 ≤ i ≤ h. Proof : For h = 1 we can take D = q−1, where q is the least prime not dividing N. Since ∑p≤D log p ≤ logN, we have from =-=[RS62]-=- Theorem 10, that either D ≤ 100 or 0.84D ≤ logN. Since D ≤ N we have D ≤ (logN) 20 , for all N ≥ 3. If N ≤ (logN) 20h , then D = N satisfies the conditions of the theorem, so we can assume N > (logN)... |

154 |
Sieve methods
- Halberstam, Richert
- 1974
(Show Context)
Citation Context ...ements in the error term of this distribution, using known results regarding the Riemann Zeta function. The second chapter deals with Brun’s Combinatorial sieve as presented in the modern language of =-=[HR74]-=-. We apply the general sieve to give a simpler proof of a theorem of Rademacher [Rad24]. The bound obtained by this simpler proof is slightly inferior, but still sufficient for applications such as th... |

41 |
On the Representation of a Large Even Integer as the Sum of a Prime and the Product of at Most Two
- Chen
- 1973
(Show Context)
Citation Context ... sieve, to name a few. Many beautiful results have been proved using these sieves. The Brun-Titchmarsh theorem and the extremely powerful result of Bombieri are two important examples. Chen’s theorem =-=[Che73]-=-, namely that there are infinitely many primes p such that p + 2 is a product of at most two primes, is another indication of the power of sieve methods. Sieve methods are of importance even in applie... |

41 |
On the large sieve
- Montgomery, Vaughan
- 1973
(Show Context)
Citation Context ...∑ 1≤a,b≤q a⊥q,b⊥q 1≤a≤q a⊥q ∣S ( a q ( a S q ) . µ(d) d Z ( q d ,h )∣ ∣∣∣ 2 . ( a S q ) S )∣ ∣∣∣ 2 . ) S ( ) b q ( b q ) e ∑ e 1≤h≤q ( ) (b − a)h q ( ) (b − a)h q86 4. THE LARGE SIEVE THEOREM 4.2.2. =-=[Mon68]-=- Let Z(q,h) and Z be defined as before, and let x ≥ 1. For each prime p ≤ x let H(p) be the union of ω(p) distinct residue classes modulo p. Let an be complex numbers that satisfy Then for each q ≤ x,... |

40 |
Further results on the construction of mutually orthogonal Latin squares and falsity of Euler’s conjecture
- Bose, Shrikhande, et al.
- 1960
(Show Context)
Citation Context ...1 2⎠. 2 3 1 Euler conjectured that there are no mutually orthogonal Latin squares of order n, where n ≡ 2 mod 4. The conjecture was disproved for the case n = 10, and later Bose, Parker and Shrikande =-=[BPS60]-=- showed that for every higher n > 6 the conjecture was false. Let ⊥(n) be the number of orthogonal latin squares of order n. Chowla, Erdős and Straus [CES60] building on this and some previous results... |

34 |
Shifted primes without large prime factors
- Baker, Harman
- 1998
(Show Context)
Citation Context ...e., if then there is a prime factor exceeding x θ . 1 − 1 2e 1 4 > θ, Among known improvements to this result, the best one is that the largest prime factor exceeds x θ for θ = 0.677 (see [BakHar95], =-=[BakHar98]-=-, and also [Ho73]). 3.4. A theorem of Hooley Chebyhev proved that if Px is the largest prime factor of ∏n≤x(n2 +1), then Px x → ∞. Hooley [Ho67] (see also [Ho76]) improved the previous best known resu... |

32 |
Applications of Sieve Methods to the Theory of Numbers
- Hooley
- 1976
(Show Context)
Citation Context ...x θ for θ = 0.677 (see [BakHar95], [BakHar98], and also [Ho73]). 3.4. A theorem of Hooley Chebyhev proved that if Px is the largest prime factor of ∏n≤x(n2 +1), then Px x → ∞. Hooley [Ho67] (see also =-=[Ho76]-=-) improved the previous best known result of Px x > (logx)A1 logloglogx by Erdős [Erd52] to Px > x 11 10 using the Selberg sieve. In this section we shall outline the proof given by Hooley in [Ho76]. ... |

25 |
The Theory of the Riemann Zetafunction, (2 nd Ed
- Titchmarsh, Heath-Brown
- 1986
(Show Context)
Citation Context ... d>y d2s } 1 = ∑ 1≤n ns ( ) ∑ µ(d) . 1 ns ( ∑ d>y d 2 \n d>y d 2 \n ) µ(d) , then as s → 0 this sum equals Σ2. Thus we need a way of evaluating this sum when s → 0. The following result (Lemma (3.12) =-=[Tit86]-=- p60) will help us do just that.1.5. THE ERROR TERM IN THE DISTRIBUTION OF SQUAREFREE NUMBERS 21 LEMMA 1.5.5. [Tit86] Let 〈an〉 be a sequence of real numbers, such that as σ → 1 from above, |an| ∑ n≥1... |

19 |
The large sieve
- Gallagher
- 1967
(Show Context)
Citation Context ... ∑ −K≤n≤K ane(nx), where K is a positive integer and an ∈ . Notation : We write ‖t‖ to mean the distance from t to the nearest integer, i.e., ‖t‖ = minn |t − n| = ∣ ⌊ t + 1 ⌋ ∣ 2 −t ∣. THEOREM 4.1.3 (=-=[Gal67]-=-). If S(x) = ∑−K≤n≤K ane(nx) and x1,··· ,xR are real numbers such that then Proof : For any u we can write Using this we have ∑ 1≤r≤R ‖xr − xs‖ ≥ δ > 0 for r ̸= s, |S(xr)| 2 ≤ (δ −1 + 2πK) S 2 (xr) = ... |

10 | On the maximal number of pairwise orthogonal Latin squares of a given order
- Chowla, Erdős, et al.
- 1960
(Show Context)
Citation Context ...= 10, and later Bose, Parker and Shrikande [BPS60] showed that for every higher n > 6 the conjecture was false. Let ⊥(n) be the number of orthogonal latin squares of order n. Chowla, Erdős and Straus =-=[CES60]-=- building on this and some previous results, established that ⊥(n) > 1 1 3n 91 for large enough n. The proof involves an interesting use of the Brun Sieve, and we shall give an account of this. The ex... |

10 |
Einige Sätze über quadratfreie Zahlen
- Estermann
- 1931
(Show Context)
Citation Context ...d≤x,ab>y cd< x(x+2) y2 ( 2c −1 p ( −1 2d p ca 2 − db 2 = 2,db 2 ≤ x. ) ≡ 1 ) ≡ 1 mod p, for all p\d, mod p, for all p\c. Estermann studied these congruences and for the case cd not a square he proved =-=[Est31]-=-: M(x;c,d,2) = O(lnx), in fact that M(x;c,d,2) ≤ 4(ln(x + 2) + 1). If cd is a square then since the equation (1.4) implies c ⊥ d we can set c = l 2 , d = m 2 to obtain: M(x;c,d,2) = ≤ ∑ l2a2−m2b2 =2 ∑... |

10 | Questions of Method - Hurley, M - 1980 |

10 |
On an elementary method in the theory of primes, Norske Vid
- Selberg
- 1947
(Show Context)
Citation Context .... cϕ(k) x + O k ( x 0.97) ≤ ∑ n≤x gcd(n,k)=1 1 ≤ c′ ϕ(k) x + O k ( x 0.975) ,CHAPTER 3 Selberg’s Sieve Around 1946 Atle Selberg introduced a new method for finding upper bounds to the sieve estimate =-=[Sel47]-=-. The method usually gives much better bounds than the Brun’s sieve. To obtain lower bounds one can couple the Selberg sieve with the Buchstab identities. After developing the basic ideas of this siev... |

10 |
Concerning the number of mutually orthogonal Latin squares
- Wilson
- 1974
(Show Context)
Citation Context ... u1 ̸≡ − n1 ̸ k mod p, p ≤ k, p \k, u1 ̸≡ n2 mod p, p ≤ k, and u1 < n 159 200 . So here both n and m are odd. The argument then proceeds similarly. Better estimates for ⊥(n) are known— for example in =-=[Wil74]-=- a bound ⊥(n) ≥ n 1 17 −2 is proved (for large enough n). The current best estimate seems to be ⊥(n) ≥ n 1 14.8 [Be83].2.4. A THEOREM OF SCHINZEL 49 2.4. A Theorem of Schinzel In this section we will... |

9 |
The number of primes in a short interval
- Heath-Brown
- 1988
(Show Context)
Citation Context ...hese estimates. COROLLARY 1.5.3. The number of squarefree numbers in the interval [x,··· ,x + √ x] is asymptotic to 6√ x π 2 . The corresponding problem for primes seems to be far more difficult, see =-=[HB88]-=-. It turns out that if the Riemann Hypothesis holds then M(y) = O( √ y), and using this in the above proof we get the following theorem: THEOREM 1.5.4. Assuming the Riemann Hypothesis, κ(x) = 6 11 x +... |

8 |
On the greatest prime factor of a quadratic polynomial
- Hooley
- 1967
(Show Context)
Citation Context ...e factor exceeds x θ for θ = 0.677 (see [BakHar95], [BakHar98], and also [Ho73]). 3.4. A theorem of Hooley Chebyhev proved that if Px is the largest prime factor of ∏n≤x(n2 +1), then Px x → ∞. Hooley =-=[Ho67]-=- (see also [Ho76]) improved the previous best known result of Px x > (logx)A1 logloglogx by Erdős [Erd52] to Px > x 11 10 using the Selberg sieve. In this section we shall outline the proof given by H... |

7 |
The Brun–Titchmarsh Theorem on average, Analytic Number Theory
- HARMAN
- 1996
(Show Context)
Citation Context ... − θ) > 0 i.e., if then there is a prime factor exceeding x θ . 1 − 1 2e 1 4 > θ, Among known improvements to this result, the best one is that the largest prime factor exceeds x θ for θ = 0.677 (see =-=[BakHar95]-=-, [BakHar98], and also [Ho73]). 3.4. A theorem of Hooley Chebyhev proved that if Px is the largest prime factor of ∏n≤x(n2 +1), then Px x → ∞. Hooley [Ho67] (see also [Ho76]) improved the previous bes... |

7 | K , Fenner D Fecal incontinence in US women: a population-based study - MY |

6 |
Some series involving Euler’s function
- Ward
- 1927
(Show Context)
Citation Context ... for which M < n ≤ M + N, kn + l is prime, and kn + l > z. Then ω(p) = 1 whenever p ≤ z and p ̸ \k. Thus by Theorem (4.2.3) we have where Taking z = where √ 2 3N and using Lemma (4.3.1), we have From =-=[War27]-=- we have as v → ∞. By partial summation we find that π(x + y;k,l) − π(x;k,l) ≤ L −1 + π(z), L = ∑ (N + q≤z q⊥k 3 2 qz)−1 µ2(q) ϕ(q) . π(x + y;k,l) − π(x;k,l) < kN ϕ(k)J + √ (N), J = ∑(1 + qz q≤z −1 ) ... |

6 | OM , Smilgin-Humphreys MM , Cunningham C , Mortensen NJ Patterns of fecal incontinence after anal surgery. Dis Colon Rectum. 2004; 47: 1643- 9 - Lindsey |

5 |
On the prime factors
- Erdös, Graham, et al.
- 1975
(Show Context)
Citation Context ...ey Chebyhev proved that if Px is the largest prime factor of ∏n≤x(n2 +1), then Px x → ∞. Hooley [Ho67] (see also [Ho76]) improved the previous best known result of Px x > (logx)A1 logloglogx by Erdős =-=[Erd52]-=- to Px > x 11 10 using the Selberg sieve. In this section we shall outline the proof given by Hooley in [Ho76]. The exponent 11 10 has since been improved to θ < 1.202···, where θ is the solution to 2... |

5 | The square sieve and consecutive square-free - Heath-Brown - 1984 |

5 |
On the distribution of square-free numbers
- Montgomery, Vaughan
- 1981
(Show Context)
Citation Context ... µ(d) d2m≤x = ∑ µ(d) d2≤x µ(d) ⌊ x d2 ⌋ .1.5. THE ERROR TERM IN THE DISTRIBUTION OF SQUAREFREE NUMBERS 19 ( x Let S2(x,y) = ∑d≤y µ(d)δ d2 ) , where δ(z) = z − ⌊z⌋ − 1 2 and M(y) = ∑n≤y µ(n). Then In =-=[MV81]-=- (see p.255) the following bound is proved: µ κ(x) = x ∑ d2≤x 2 (d) d2 − S2(x, √ x) − 1 2 M(√x). S2(x,y) = O(x 2 7 + y 1 2 x 1 7 +ε ), and this implies that S(x, √ x) = O(x 11 28 ). Now consider: µ(d)... |

4 |
On integers all of whose prime factors are small
- HALBERSTAM
- 1970
(Show Context)
Citation Context ...ly orthogonal Latin squares. The formulation of Brun’s sieve in [HR74] also includes a proof of the important Buchstab identity. We use it to derive some bounds on the distribution of smooth numbers (=-=[Hal70]-=-). The third chapter deals with the development and the applications of Selberg’s upper bound method. The proof by van Lint and Richert [vLR65] of the Brun-Titchmarsh theorem is given as the chief app... |

4 |
On primes in arithmetic progressions
- Lint, Richert
- 1965
(Show Context)
Citation Context ...erive some bounds on the distribution of smooth numbers ([Hal70]). The third chapter deals with the development and the applications of Selberg’s upper bound method. The proof by van Lint and Richert =-=[vLR65]-=- of the Brun-Titchmarsh theorem is given as the chief application. Hooley’s improvement of bounds on prime factors in a problem studied by Chebyschev is also outlined here. The last chapter is a study... |

4 | Sieve methods - Selberg - 1971 |

4 | Hannestad YS , Daltveit AK , Hunskaar S Age- and type-dependent effects of parity on urinary incontinence: the Norwegian EPINCONT study. Obstet Gynecol. 2001; 98: 1004- 10 - Rortveit - 1999 |

4 | Ringa V, Fritel X, Varnoux N, Zins M, Bréart G. Negative impact of urinary incontinence on quality of life, a cross-sectional study among women aged 49-61 years enrolled in the GAZEL cohort. Neurourol Urodyn - Saadoun |

3 |
The distribution of square-free numbers
- Baker, Pintz
- 1985
(Show Context)
Citation Context ...y and Vaughan went on to estimate the sums involved more precisely to show that κ(x) = 1 ζ(2) x + O(x 9 28 +ε). Subsequently the exponent of the error term was reduced to 7 22 by various authors (see =-=[BakPin85]-=-). 1.6. Pairs of squarefree numbers The famous twin prime problem asks whether there are infinitely many primes p such that p + 2 is also prime. Although this problem is still open, the analogous ques... |

3 | On some applications of Brun’s method - Erdős - 1949 |

3 |
On the largest prime factor of p
- Hooley
- 1973
(Show Context)
Citation Context ...a prime factor exceeding x θ . 1 − 1 2e 1 4 > θ, Among known improvements to this result, the best one is that the largest prime factor exceeds x θ for θ = 0.677 (see [BakHar95], [BakHar98], and also =-=[Ho73]-=-). 3.4. A theorem of Hooley Chebyhev proved that if Px is the largest prime factor of ∏n≤x(n2 +1), then Px x → ∞. Hooley [Ho67] (see also [Ho76]) improved the previous best known result of Px x > (log... |

3 |
On the frequency of pairs of square-free numbers with a given difference
- MIRSKY
- 1949
(Show Context)
Citation Context ...lthough this problem is still open, the analogous question for the squarefree numbers can be settled rather easily using the methods we have seen so far. For a more general version of this result see =-=[Mir49]-=-. Let κ2(x) = ∣ ∣ {n(n + 2) | µ(n) 2 = µ(n + 2) 2 = 1,n ≤ x} ∣ ∣. THEOREM 1.6.1. κ2(x) = ∏ p Proof : Let s(n) = ∑ d 2 \n µ(d). Using this we have ( 1 − 2 p2 ) x + O(x 2 3 ln 4 3 x). κ2(x) = ∑ s(n)s(n ... |

3 | The risk of lower urinary tract symptoms five years after the first delivery. Neurourol Urodyn - Viktrup |

3 | Herbison GP Obstetric practice and the prevalence of urinary incontinence three months after delivery. Br J Obstet Gynaecol - RM - 1996 |

3 | Eckford S , Swithinbank L , Abrams P The Bristol Female Lower Urinary Tract Symptoms questionnaire: development and psychometric testing - Donovan |

3 | The impact of confounder selection criteria on effect estimation - RM, Greenland - 1989 |

3 | Berkshire perineal management trial: three year follow up. Br Med J - Sleep, West - 1987 |

3 | Chastang JF , Leclerc A , Zins M , Bonenfant S , Bugel I Socioeconomic, demographic, occupational and health factors associated with participation in a long-term epidemiologic survey: a prospective study of the French GAZEL cohort and its target populatio - Goldberg |

3 | Bréart G. Determinants of hormone replacement therapy among postmenopausal women enrolled - Ringa, Ledésert - 1994 |

3 | Ringa V, Varnoux N, Fauconnier A, Piault S, Bréart G. Mode of delivery and severe stress incontinence. A cross-sectional study among 2625 perimenopausal women - Fritel |

3 | Jesudason V Fecal Incontinence in Wisconsin Nursing Homes Prevalence and Associations - Furner - 2009 |

2 |
Eine Bemerkung zur Abschätzung der Anzahl orthogonaler lateinischer Quadrate mittels
- Thomas
- 1983
(Show Context)
Citation Context ...ent then proceeds similarly. Better estimates for ⊥(n) are known— for example in [Wil74] a bound ⊥(n) ≥ n 1 17 −2 is proved (for large enough n). The current best estimate seems to be ⊥(n) ≥ n 1 14.8 =-=[Be83]-=-.2.4. A THEOREM OF SCHINZEL 49 2.4. A Theorem of Schinzel In this section we will give an application involving a variation of Theorem 2.3.6, where we look at some constant number of constraints. The... |

2 |
Omfordelingen av primtallene i forskjellige talklasser. En vre begrnsning
- Brun
- 1916
(Show Context)
Citation Context ...mbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see =-=[Bru16]-=-,[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum 1 ∑ p, p+2 both prime p converges. This was the first result of its kind, regarding the Twin-prime problem. A slew of s... |

2 |
Le crible d’Eratostne et le thorme de
- Brun
- 1919
(Show Context)
Citation Context ...ater Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],=-=[Bru19]-=-, [Bru22]). Using his formulation of the sieve Brun proved, that the sum 1 ∑ p, p+2 both prime p converges. This was the first result of its kind, regarding the Twin-prime problem. A slew of sieve met... |

2 |
Siev des Eratosthenes
- Brun, Das
- 1922
(Show Context)
Citation Context ...ndre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], =-=[Bru22]-=-). Using his formulation of the sieve Brun proved, that the sum 1 ∑ p, p+2 both prime p converges. This was the first result of its kind, regarding the Twin-prime problem. A slew of sieve methods were... |

2 |
Iwaniec Henryk, On the greatest prime factor of n 2
- Deshouillers
- 1983
(Show Context)
Citation Context ... section we shall outline the proof given by Hooley in [Ho76]. The exponent 11 10 has since been improved to θ < 1.202···, where θ is the solution to 2−θ−2log(2−θ) = 5 4 , by Deshouillers and Iwaniec =-=[DI83]-=- (see also [Dar96]).72 3. SELBERG’S SIEVE THEOREM 3.4.1 ([Ho76]). The largest prime factor of exceeds x 11 10 for all large enough values of x. ∏(n n≤x 2 + 1) Proof : Let Px be the largest prime fact... |

2 |
Über die kleinste quadratfreie Zahl einer arithmetischen
- Erdős
- 1960
(Show Context)
Citation Context ... for helping me with character sums. I thank him for answering my queries in such a way that I gained a new insight into the problem. I 1 This is not a new proof - it is implicit in the work of Erdős =-=[Erd60]-=- 34 PREFACE thank Dr. Alan Selman for his encouragement and advice. I am deeply grateful to Professors Eric Bach, Tom Cusick, Kevin Ford, and Andrew Granville for promptly answering my queries. Their... |

2 | Elementary Methods in the Analytic Theory of Numbers - unknown authors - 1966 |

2 | On the Brun-Titchmarsh theorem - Henryk - 1982 |

2 |
A note on the least prime in an arithmetic progression with a prime difference
- Yoichi
- 1970
(Show Context)
Citation Context ...at we shall see in the next section. The problem is to prove a lower bound on the largest prime divisor of ∏(p p≤x 2 − 1) = ∏(p + 1)∏ (p − 1). p≤x p≤x We will prove the following theorem of Motohashi =-=[Mot70]-=-. THEOREM 3.3.1. Let Px be the largest prime divisor of ∏(p p≤x 2 − 1). Then Px > x θ for any θ < 1 − 1 2e 1 4 . Proof : In this proof q will also stand for primes, and sums or products over q will re... |