## Explicit bounds for primes in residue classes (1996)

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Venue: | Math. Comp |

Citations: | 17 - 1 self |

### BibTeX

@ARTICLE{Bach96explicitbounds,

author = {Eric Bach and Jonathan Sorenson},

title = {Explicit bounds for primes in residue classes},

journal = {Math. Comp},

year = {1996},

pages = {173--5}

}

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### Abstract

Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree-1 prime p of K such that p = σ, satis-

### Citations

309 |
Approximate formulas for some functions of prime numbers
- Rosser, Schoenfeld
- 1962
(Show Context)
Citation Context ...ters in the sum. First, we require two lemmas. Lemma 4.1. If n ≥ 2,then � ϕ(n/d) ≤ ϕ(n)(e γ log log n +2). d|n d>1 Proof. Since n = � d|n ϕ(n/d), the sum is n − ϕ(n) =ϕ(n)[n/ϕ(n) − 1]. From (3.41) of =-=[30]-=- we conclude that n/ϕ(n) ≤ eγ log log n+3forn≥2, which completes the proof. Lemma 4.2. Let E/K be a cyclic extension of degree n, with(primitive) character χ and conductor f. Let∆be the absolute value... |

289 |
Algebraic Number Theory
- Lang
- 1994
(Show Context)
Citation Context ...show that if χ and χ ′ generate the same subgroup of the character group, they must have the same conductor. (Here it is essential to interpret the conductor as a “cycle” or “ray modulus.” See, e.g., =-=[24]-=-.) By the definition of conductors, if p ≡ 1modf,thenχ(p) = 1; because χ ′ is a power of χ, wehave χ ′ (p) = 1. Therefore f ′ , the conductor of χ ′ , is a multiple of f. By the same argument, f is a ... |

98 |
Effective versions of the Chebotarev density theorem. Algebraic number fields
- Lagarias, Odlyzko
- 1975
(Show Context)
Citation Context ... many prime ideals with each possible Artin symbol, and estimates their density. It then becomes a problem to estimate the least such prime ideal. This was done (assuming ERH) by Lagarias and Odlyzko =-=[23]-=-, and Lagarias, Montgomery, and Odlyzko [22]. Oesterlé [29] has stated an explicit version of this theorem: if E/K is a Galois extension of number fields, then the least prime ideal of K with a given ... |

69 |
Explicit bounds for primality testing and related problems
- Bach
- 1990
(Show Context)
Citation Context ...s suggests an interesting arena for more computational experiments. The ERH is supported by computational evidence and probabilistic arguments. For the first, we refer the reader to the references in =-=[5]-=- and the recent work of Rumely [31]. An example of the second, based on ideas of Cramér, appears in [6].sEXPLICIT BOUNDS FOR PRIMES IN RESIDUE CLASSES 1719 As possible applications of our results, we ... |

45 |
Bounds for discriminants and related estimates for class numbers, regulators, and zeros of zeta functions: a survey of recent results
- Odlyzko
- 1990
(Show Context)
Citation Context ... of infinite class field towers [15] imply that n/ log∆ �= o(1) (see Hasse [16, p. 46]). Alternatively, we can bound n in terms of log ∆; for this purpose, the discriminant bounds surveyed by Odlyzko =-=[28]-=- are useful. 4. Technical estimates In this section we fill in the missing details from the proof of Theorem 3.1 by giving estimates for p(x), r(x), d(x), i(x), and by evaluating � χ S(x, ˆχ)χ(µ) by r... |

42 | Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression
- Heath-Brown
- 1992
(Show Context)
Citation Context ... very large. Thus, a search for p through the sequence m, m+q, m+2q,... should terminate quickly. Linnik [21] proved that p ≤ q O(1) ;the sharpest known estimate of the exponent is due to Heath-Brown =-=[17]-=-: p = O(q 11/2 ). (No explicit version of Linnik’s theorem seems to be known.) However, the available data on primes in progressions [36] suggest this exponent is too large. In an attempt to obtain re... |

39 | Searching for primitive roots in finite fields
- Shoup
- 1992
(Show Context)
Citation Context ...ctions. If G is a proper subgroup of the multiplicative group modulo p, theleastprime outside G is O(log p) 2 [4] assuming ERH. Under the same assumption, the least primitive root mod p is O(log p) 6 =-=[32]-=-. In both cases, using the theorems of this paper would lead to bounds of O(p log p) 2 . It is also of interest to ask if the growth rates in our estimates are best possible. We believe they are not, ... |

35 | Bericht uber Neuere Untersuchungen und Problems aus der Theorie der algebraischen Zahlkdrper - HASSE - 1970 |

32 | The area-time complexity of binary multiplication
- Brent, Kung
- 1981
(Show Context)
Citation Context ...sEXPLICIT BOUNDS FOR PRIMES IN RESIDUE CLASSES 1719 As possible applications of our results, we mention the following. Brent and Kung’s construction of n-bit multipliers with low area-time complexity =-=[9]-=- uses a prime congruent to 1 mod n. Bach and Shallit’s generalization of the “p±1” factoring method [7] requires a prime in a certain residue class, with prescribed splitting behavior. A similar devic... |

31 |
Finding irreducible polynomials over finite fields
- Adleman, Lenstra
- 1986
(Show Context)
Citation Context .... Bach and Shallit’s generalization of the “p±1” factoring method [7] requires a prime in a certain residue class, with prescribed splitting behavior. A similar device was used by Adleman and Lenstra =-=[2]-=- to construct irreducible polynomials over finite fields. Finally, our results allow one to estimate the least prime p for which ( p ) takes a prescribed value; such primes are useful in n primality t... |

31 |
The least quadratic nonresidue
- Ankeny
- 1952
(Show Context)
Citation Context ... this here, except to note some cases in which our bounds do not lead to efficient constructions. If G is a proper subgroup of the multiplicative group modulo p, theleastprime outside G is O(log p) 2 =-=[4]-=- assuming ERH. Under the same assumption, the least primitive root mod p is O(log p) 6 [32]. In both cases, using the theorems of this paper would lead to bounds of O(p log p) 2 . It is also of intere... |

25 | Numerical computations concerning the ERH
- Rumely
- 1993
(Show Context)
Citation Context ...r more computational experiments. The ERH is supported by computational evidence and probabilistic arguments. For the first, we refer the reader to the references in [5] and the recent work of Rumely =-=[31]-=-. An example of the second, based on ideas of Cramér, appears in [6].sEXPLICIT BOUNDS FOR PRIMES IN RESIDUE CLASSES 1719 As possible applications of our results, we mention the following. Brent and Ku... |

24 |
On the least prime in an arithmetic progression, I. The basic theorem
- Linnik
- 1944
(Show Context)
Citation Context ...authors have observed that if gcd(m, q) = 1, then the least prime p congruent to m mod q is not very large. Thus, a search for p through the sequence m, m+q, m+2q,... should terminate quickly. Linnik =-=[21]-=- proved that p ≤ q O(1) ;the sharpest known estimate of the exponent is due to Heath-Brown [17]: p = O(q 11/2 ). (No explicit version of Linnik’s theorem seems to be known.) However, the available dat... |

22 | Die Theorie der algebraischen Zahlkörper - Hilbert |

22 |
Odlyzko, ‘A bound for the least prime ideal in the Chebotarev density theorem’, Invent
- Lagarias, Montgomery, et al.
- 1979
(Show Context)
Citation Context ...symbol, and estimates their density. It then becomes a problem to estimate the least such prime ideal. This was done (assuming ERH) by Lagarias and Odlyzko [23], and Lagarias, Montgomery, and Odlyzko =-=[22]-=-. Oesterlé [29] has stated an explicit version of this theorem: if E/K is a Galois extension of number fields, then the least prime ideal of K with a given Artin symbol must have norm no larger than 7... |

20 |
Factoring with cyclotomic polynomials
- Bach, Shallit
- 1989
(Show Context)
Citation Context ... the following. Brent and Kung’s construction of n-bit multipliers with low area-time complexity [9] uses a prime congruent to 1 mod n. Bach and Shallit’s generalization of the “p±1” factoring method =-=[7]-=- requires a prime in a certain residue class, with prescribed splitting behavior. A similar device was used by Adleman and Lenstra [2] to construct irreducible polynomials over finite fields. Finally,... |

18 |
Shafarevich.On the class field tower
- Golod, R
- 1964
(Show Context)
Citation Context ...log log ···log n), where the log is iterated r times. To be able to disregard the 2n term, some condition on E seems necessary, because Golod and Shafarevich’s examples of infinite class field towers =-=[15]-=- imply that n/ log∆ �= o(1) (see Hasse [16, p. 46]). Alternatively, we can bound n in terms of log ∆; for this purpose, the discriminant bounds surveyed by Odlyzko [28] are useful. 4. Technical estima... |

18 |
Versions effectives du théorème de Chebotarev sous l’hypothése de Riemann généralisée, Astérisque 61
- Oesterlé
- 1979
(Show Context)
Citation Context ...imates their density. It then becomes a problem to estimate the least such prime ideal. This was done (assuming ERH) by Lagarias and Odlyzko [23], and Lagarias, Montgomery, and Odlyzko [22]. Oesterlé =-=[29]-=- has stated an explicit version of this theorem: if E/K is a Galois extension of number fields, then the least prime ideal of K with a given Artin symbol must have norm no larger than 70(log |∆|) 2 , ... |

13 |
A divisor problem
- Titchmarsh
- 1930
(Show Context)
Citation Context ...ailable data on primes in progressions [36] suggest this exponent is too large. In an attempt to obtain realistic estimates, several authors have invoked the ERH. From work of Chowla [10], Titchmarsh =-=[34]-=-, Turán [35], and Wang, Hsieh, and Yu [37], we have the bound p = q 2+o(1) , assuming the ERH. In algebraic number theory, Dirichlet’s theorem generalizes to the theorem of Chebotarev [33], which stat... |

12 |
Greatest of the least primes in arithmetic progressions having a given modulus
- Wagstaff
- 1979
(Show Context)
Citation Context ... sharpest known estimate of the exponent is due to Heath-Brown [17]: p = O(q 11/2 ). (No explicit version of Linnik’s theorem seems to be known.) However, the available data on primes in progressions =-=[36]-=- suggest this exponent is too large. In an attempt to obtain realistic estimates, several authors have invoked the ERH. From work of Chowla [10], Titchmarsh [34], Turán [35], and Wang, Hsieh, and Yu [... |

10 |
Statistical evidence for small generating sets
- Bach, Huelsbergen
- 1993
(Show Context)
Citation Context ... by taking the absolute value of each term; one would naturally expect lots of cancellation, which we ignore. Also, simple probabilistic models suggest that the least p ≡ m mod q is O(ϕ(q)(log q) 2 ) =-=[8, 18, 36]-=-. In this case, replacing an ad hoc model with a “name brand” heuristic like the ERH essentially squares the bound. All of this suggests an interesting arena for more computational experiments. The ER... |

9 |
Almost-primes in arithmetic progressions and short intervals
- Heath-Brown
- 1978
(Show Context)
Citation Context ... by taking the absolute value of each term; one would naturally expect lots of cancellation, which we ignore. Also, simple probabilistic models suggest that the least p ≡ m mod q is O(ϕ(q)(log q) 2 ) =-=[8, 18, 36]-=-. In this case, replacing an ad hoc model with a “name brand” heuristic like the ERH essentially squares the bound. All of this suggests an interesting arena for more computational experiments. The ER... |

6 |
Über den Tschebotareffschen Dichtigkeitssatz
- Deuring
- 1935
(Show Context)
Citation Context ...same subgroup of G.) Before giving the proof, we note two facts. Theorem 3.2. If Theorem 3.1 holds for cyclic extensions, then it holds for arbitrary abelian extensions. Proof. This uses a trick from =-=[12]-=- (see also [26]). Let L be the subfield of E fixed by σ. Then E/L is cyclic. Let P be a prime of L satisfying the conditions of the theorem for E/L. ThenPlies above some prime p of K, and we have Np =... |

5 |
A reduction of the Cebotarev density theorem to the cyclic case, Acta Arith
- MacCluer
- 1968
(Show Context)
Citation Context ...f G.) Before giving the proof, we note two facts. Theorem 3.2. If Theorem 3.1 holds for cyclic extensions, then it holds for arbitrary abelian extensions. Proof. This uses a trick from [12] (see also =-=[26]-=-). Let L be the subfield of E fixed by σ. Then E/L is cyclic. Let P be a prime of L satisfying the conditions of the theorem for E/L. ThenPlies above some prime p of K, and we have Np = NP (this is in... |

4 |
obcr die Primzahlen der arithmetischen Progress&men
- Turan
(Show Context)
Citation Context ... on primes in progressions [36] suggest this exponent is too large. In an attempt to obtain realistic estimates, several authors have invoked the ERH. From work of Chowla [10], Titchmarsh [34], Turán =-=[35]-=-, and Wang, Hsieh, and Yu [37], we have the bound p = q 2+o(1) , assuming the ERH. In algebraic number theory, Dirichlet’s theorem generalizes to the theorem of Chebotarev [33], which states that ther... |

3 | The complexity of number theoretic problems - Bach, Giesbrecht, et al. - 1991 |

3 |
Estimates for the Chebyshev Function ψ(x)− θ(x
- Pereira
- 1985
(Show Context)
Citation Context ...>1 Λ(p k )log Using integration by parts, we obtain � x � x nE log(x/t)d(Ψ(t) − θ(t)) = nE 1 Asymptotically, this is 2nE( √ x + O(x 1/3 )). � x pk � . 1 Ψ(t) − θ(t) dt. t From Theorems 2, 4, and 5 in =-=[11]-=-, we easily obtain the explicit bound Ψ(t) − θ(t) < 1.001 √ t +(4/3)t 1/3 for t>0. Hence, p(x) ≤ 2nE(1.001 √ x +2x 1/3 (4.2) ). For values of a near 1 it is better to use the bound p(x) ≤ nE ea (1.001... |

3 |
Beweis des Satzes daß jede unbegrenzte arithmetische Progression deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Faktor sind unendlich viele Primzahlen enthält
- Dirichlet
- 1837
(Show Context)
Citation Context ...history, going back to Euler’s statement that every arithmetic progression beginning with 1 contains an infinite number of primes. The generalization to arbitrary progressions was proved by Dirichlet =-=[13, 14]-=-, in work Received by the editor October 4, 1994 and, in revised form, September 5, 1995. 1991 Mathematics Subject Classification. Primary 11N13, 11M26; Secondary 11R44, 11Y35. This research was suppo... |

3 |
Zeta-functions and L-functions
- Heilbronn
- 1967
(Show Context)
Citation Context ... abelian extension, with Galois group G. For each character χ of G, there is an associated intermediate field Eχ (so that K ⊂ Eχ ⊂ E). As1720 ERIC BACH AND JONATHAN SORENSON theorem due to Hecke (see =-=[19]-=-) states that ζE(s), the Dedekind zeta function of E, is a product of Hecke L-functions: ζE(s)= � (2.2) L(s, ˆχ). χ This, together with representations of ζ ′ /ζ and L ′ /L (i.e., (3.11) and (3.12) of... |

2 |
On the least prime in the arithmetical progression
- Chowla
- 1934
(Show Context)
Citation Context ...) However, the available data on primes in progressions [36] suggest this exponent is too large. In an attempt to obtain realistic estimates, several authors have invoked the ERH. From work of Chowla =-=[10]-=-, Titchmarsh [34], Turán [35], and Wang, Hsieh, and Yu [37], we have the bound p = q 2+o(1) , assuming the ERH. In algebraic number theory, Dirichlet’s theorem generalizes to the theorem of Chebotarev... |

1 |
An improvement of an inequality of Minkowski
- Ankeny
- 1951
(Show Context)
Citation Context ...he coefficient 2 can be replaced by any number larger than ψ(1)−log 2π− 4=1.584907.... In some cases, though, the 2n term is superfluous. This is so if E/Q is abelian, for then n/ log∆ = o(1). Ankeny =-=[3]-=- improved this and showed that if the Galois group of E is solvable in r steps, then n/ log∆ = O(1/ log log ···log n), where the log is iterated r times. To be able to disregard the 2n term, some cond... |

1 |
Beweis eines Satzes über die arithmetische Progression
- Dirichlet
(Show Context)
Citation Context ...history, going back to Euler’s statement that every arithmetic progression beginning with 1 contains an infinite number of primes. The generalization to arbitrary progressions was proved by Dirichlet =-=[13, 14]-=-, in work Received by the editor October 4, 1994 and, in revised form, September 5, 1995. 1991 Mathematics Subject Classification. Primary 11N13, 11M26; Secondary 11R44, 11Y35. This research was suppo... |

1 |
Lenstra Jr.. Miller’s primality test
- W
- 1979
(Show Context)
Citation Context ...struct irreducible polynomials over finite fields. Finally, our results allow one to estimate the least prime p for which ( p ) takes a prescribed value; such primes are useful in n primality testing =-=[25]-=- and other contexts. We now give a rough sketch indicating our argument, using the notation of [5]. Suppose for simplicity that (Z/(n)) ∗ /G is cyclic of prime order, and we want a prime belonging to ... |

1 |
A rapidly convergent series for computing ψ(z) and its derivatives
- McCullagh
- 1981
(Show Context)
Citation Context ...omputer. All values were represented as 80-bit floating-point numbers (the long double data type in Turbo C++). Values of the digamma and trigamma functions were computed using methods from McCullagh =-=[27]-=-. In essence, the program consists of three layers; we elaborate below. The bottom layer: applying the technical estimates. First, we wrote a set of functions to compute triples of the form (v0,v1,v2)... |

1 |
Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören
- Tchebotarev
- 1926
(Show Context)
Citation Context ... Titchmarsh [34], Turán [35], and Wang, Hsieh, and Yu [37], we have the bound p = q 2+o(1) , assuming the ERH. In algebraic number theory, Dirichlet’s theorem generalizes to the theorem of Chebotarev =-=[33]-=-, which states that there are infinitely many prime ideals with each possible Artin symbol, and estimates their density. It then becomes a problem to estimate the least such prime ideal. This was done... |

1 |
Two results on the distribution of prime numbers. Zhongguo Kexue Jishu Daxue Xuebao, 1:32–38
- Wang, Hsieh, et al.
- 1965
(Show Context)
Citation Context ...] suggest this exponent is too large. In an attempt to obtain realistic estimates, several authors have invoked the ERH. From work of Chowla [10], Titchmarsh [34], Turán [35], and Wang, Hsieh, and Yu =-=[37]-=-, we have the bound p = q 2+o(1) , assuming the ERH. In algebraic number theory, Dirichlet’s theorem generalizes to the theorem of Chebotarev [33], which states that there are infinitely many prime id... |