### Citations

380 |
Introduction to analytic and probabilistic number theory
- Tenenbaum
- 1995
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Citation Context ...me factors of n counted with multiplicity and z is a fixed number between 0 and 1. We take S = ∅ and h(n) = zΩ(n) and fq(a) = zΩ((a,q)). For large x we know from a result of A. Selberg (see Tenenbaum =-=[17]-=-) that A(x) ∼ x(log x) z−1 Γ(z) . From Theorem 2.4 and the above, we deduce that for fixed u≥max(e2/(1−z), e100) and large x there exists y ∈ (x/4, x) and an arithmetic progression a (mod ) with ≤ ... |

259 |
Sieve methods
- Halberstam, Richert
- 1974
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Citation Context ...ot have a prime factor p ∈ P with p ≤ z. Sieve theory is concerned with estimating S(B,P, z) under certain natural hypotheses for B,P and u := log x/ log z. The fundamental lemma of sieve theory (see =-=[7]-=-) implies (for example when B is the set of integers 4 ANDREW GRANVILLE AND K. SOUNDARARAJAN in an interval) that∣∣∣∣∣∣S(B,P, z)− x ∏ p∈P,p≤z ( 1− 1 p )∣∣∣∣∣∣ ( 1 + o(1) u log u )u x ∏ p∈P,p≤z ( 1− ... |

186 |
Multiplicative Number Theory,
- Davenport
- 1967
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Citation Context ...unction with modulus q between √ T and T has a zero in the region σ > 1 − c1/ log q, and |t| ≤ T . Further if this exceptional Siegel zero exists then it is real, simple and unique (see Chapter 14 in =-=[2]-=-). We call the modulus of such an exceptional character a Siegel modulus. Below Q there are log logQ Siegel moduli. Denote these by ν1, ν2, . . . , ν, and for each select a prime divisor v1, . . . ... |

46 |
Das asymptotische Verhalten von Summen uber multiplikative Funktionen,
- Wirsing
- 1961
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Citation Context ...; that is∑ p≤x p∈P log p p = (α + o(1)) log x, as x → ∞. Let A be the set of integers not divisible by any prime in P and let a(n) = 1 if n ∈ A and a(n) = 0 otherwise. Wirsing proved (see page 417 of =-=[18]-=-) that A(x) ∼ e γα Γ(1− α)x ∏ p≤x p∈P ( 1− 1 p ) .(6.8) 36 ANDREW GRANVILLE AND K. SOUNDARARAJAN Let h be the multiplicative function defined by h(p) = 0 if p ∈ P and h(p) = 1 if p /∈ P and take fq(a)... |

43 |
Primes in short intervals
- Maier
- 1985
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Citation Context ... sequences” in arithmetic progressions and short intervals. Our discussions are motivated by a general result of K. F. Roth [15] on irregularities of distribution, and a particular result of H. Maier =-=[11]-=- which imposes restrictions on the equidistribution of primes. If A is a subset of the integers in [1, x] with |A| = ρx then, as Roth proved, there exists N ≤ x and an arithmetic progression a (mod q)... |

30 |
On a Theorem of
- Serre
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Citation Context ...primes admitting in K at least one prime ideal divisor of residual degree 1. It is well known that E(K) ≥ 1/[K : Q] and also we know that E(K) ≤ 1 − 1/ [K : Q] (see the charming article of J-P. Serre =-=[16]-=-). We now describe what the natural associated multiplicative functions h and fq should be. Define δ(n) = 1 when n is the norm of some integral ideal in K and δ(n) = 0 otherwise. Clearly n ∈ A(i) for ... |

28 |
A large sieve density estimate near σ
- Gallagher
- 1970
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Citation Context ...1 and 5.2 we require knowledge of the distribution of primes in certain arithmetic progressions. We begin by describing such a result, which will be deduced as a consequence of a theorem of Gallagher =-=[5]-=-. For 1 ≤ j ≤ J := [log z/(2 log 2)], consider the dyadic intervals Ij = (z/2j , z/2j−1]. Let Pj denote a subset of the primes in Ij , and let πj denote the cardinality of Pj . We let Q denote the set... |

28 | The Spectrum of Multiplicative Functions
- Granville, Soundararajan
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Citation Context ...plicative functions satisfying 0 ≤ f(n) ≤ 1 for all n (see Theorem 3.1). Along with our general formalism, this forms the main new ingredient of our paper and is partly motivated by our previous work =-=[6]-=- on multiplicative functions and integral equations. In Section 7 we present a simple analogue of such oscillation results for a wide class of integral equations which has the flavor of a classical “u... |

25 |
Limitations to the equi-distribution of primes I
- Friedlander, Granville
- 1989
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Citation Context ... usual (that is, ≥ (1+ δA)(log x)A−1 primes for some δA > 0) and also intervals (x, x + (log x)A) containing significantly fewer primes than usual. Adapting his method J. Friedlander and A. Granville =-=[3]-=- showed that there are arithmetic progressions containing significantly more (and others with significantly fewer) primes than usual. A weak form of their result is that, for every A ≥ 1 there exist l... |

25 |
Remark concerning integer sequences,
- Roth
- 1964
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Citation Context ...vestigate the limitations to the equidistribution of interesting “arithmetic sequences” in arithmetic progressions and short intervals. Our discussions are motivated by a general result of K. F. Roth =-=[15]-=- on irregularities of distribution, and a particular result of H. Maier [11] which imposes restrictions on the equidistribution of primes. If A is a subset of the integers in [1, x] with |A| = ρx then... |

15 | Discrepancy in arithmetic progressions
- Matousek, Spencer
- 1996
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Citation Context ...nsity 0 or 1, there must be an arithmetic progression in which the number of elements of A is a little different from the average. Following work of A. Sarkozy and J. Beck, J. Matousek and J. Spencer =-=[12]-=- showed that Roth’s theorem is best possible, in that there is a *Le premier auteur est partiellement soutenu par une bourse du Conseil de recherches en sciences naturelles et en génie du Canada. The... |

12 |
Oscillation theorems for primes in arithmetic progressions and for sifting functions
- Friedlander, Granville, et al.
- 1991
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Citation Context ... Hypothesis. We have also been able to extend the uniformity with which Friedlander and Granville’s result (1.1) holds, obtaining results which previously Friedlander, Granville, Hildebrand and Maier =-=[4]-=- established conditionally on the Generalized Riemann Hypothesis. We will describe these in Section 5. This paper is structured as follows: In Section 2 we describe the framework in more detail, and s... |

12 |
On the distribution of reduced residues
- Montgomery, Vaughan
- 1986
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Citation Context ... numbers chosen with probability φ(q)/q. Indeed when φ(q)/q → 0 AN UNCERTAINTY PRINCIPLE FOR ARITHMETIC SEQUENCES 5 this follows from work of C. Hooley [10]; and of H. L. Montgomery and R. C. Vaughan =-=[13]-=- who showed that #{n ∈ [m,m + h) : (n, q) = 1} has Gaussian distribution with mean and variance equal to hφ(q)/q, as m varies over the integers, provided h is suitably large. This suggests that #{n ∈ ... |

10 | Irregularities in the distribution of primes in short intervals - Hildebrand, Maier - 1989 |

8 | Halving an estimate obtained from Selberg's upper bound method, Acta Arith 25 - Hall - 1974 |

7 |
On the difference of consecutive numbers prime to n
- HOOLEY
- 1963
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Citation Context ... are expected to be distributed much like random numbers chosen with probability φ(q)/q. Indeed when φ(q)/q → 0 AN UNCERTAINTY PRINCIPLE FOR ARITHMETIC SEQUENCES 5 this follows from work of C. Hooley =-=[10]-=-; and of H. L. Montgomery and R. C. Vaughan [13] who showed that #{n ∈ [m,m + h) : (n, q) = 1} has Gaussian distribution with mean and variance equal to hφ(q)/q, as m varies over the integers, provide... |

4 | Sums of two squares in short intervals
- Balog, Wooley
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Citation Context ...repancy exhibited is much larger in (1.1) (being within a constant factor of the main term), but the modulus q is much closer to x (but not so close as to be trivial). Recently A. Balog and T. Wooley =-=[1]-=- proved that the sequence of integers that may be written as the sum of two squares also exhibits “Maier type” irregularities in some intervals (x, x+(log x)A) for any fixed, positive A. While previou... |

4 |
On the norms of algebraic integers, Mathematika 22
- Odoni
- 1975
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Citation Context ...bounds are y−τ(1+o(1)). Thus we note that the asymptotic, suggested by probability considerations, ϑ(x + y)− ϑ(x) = y + O(y 12+ε), fails sometimes for y ≤ exp((log x) 12−ε). A. Hildebrand and Maier =-=[14]-=- had previously shown such a result for y ≤ exp((log x) 13−ε) (more precisely they obtained a bound y−(1+o(1))τ/(1−τ) in the range 0 < τ < 1/3), and were able to obtain our result assuming the valid... |