C. Hooley, On Artin's conjecture, J. Reine Angew. Math., vol. 22, 1967, pp. 209--220.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....n, 1) is the number of primes up to x not dividing a that are congruent to 1 modulo n. The Chebotarev Density Theorem provides us with an asymptotic formula for #(x, n, d) This was the main ingredient in the famous proof of Artin Conjecture subject to the Riemann Hypothesis due to C. Hooley [4]. The following result is due to Lagarias and Odlyzko [6] Here we state the version that was used in [9, page 376] Lemma 2.1 (Chebotarev Density Theorem. With the above notations, there exist absolute constants A and B such that if n B(log x) #(x, n, d) k n,d Li(x) O x exp( A ....

C. Hooley, On Artin's Conjecture. J. Reine Angew. Math.226 (1967), 207--220.

....bounds on ord p (b) that hold for a full density subset of the primes In [3] Erd os and Murty proved that if b 6= 0; 1, then there exists a 0 so that ord p (b) is at least p exp( log p) for a full density subset of the primes. However, we expect the typical order to be much larger. In [6] Hooley proved that the Generalized Riemann Hypothesis (GRH) implies Artin s conjecture. Moreover, if f : R is an increasing function tending to in nity, Erd os and Murty showed [3] that GRH implies that the order of b modulo p is greater than p=f(p) for full density subset of the primes. ....

C. Hooley. On Artin's conjecture. J. Reine Angew. Math., 225:209-220, 1967.

....which a 1 , a r generate a primitive root (mod p) In the case r = 1, the Artin s Conjecture for primitive roots predicts the probability for a prime p to have a given number a as a primitive root. Fo r examp e , i f a = 2, then Artin Conjecture states that #2# . 1. 2) Hooley [7] has shown that if the generalized Riemann hypothesis holds for the Dedekind zeta function of the fields Q(# l , 2 ) with l prime, then the asymptotic formula in (1.2) holds. The idea of considering higher rank analogue to the Artin Conjecture is due to Rajiv Gupta and Maruti Ram Murty who in ....

....are primes in Theorem 4.1, then the estimate of Corollary 4.2 does not hold anymore. Indeed if K = Q(# 21 , 5 , 40 ) then [K : Q] #(21) 21 3 giving a counterexample to (4.31) We are now ready to express the density as an Euler product. The case r =1 has been dealt with by C. Hooley in [7]. We report it here for completeness: Lemma 4.3. Let p be a prime, nm = Q(# m ,p ) Q]and let A = 4.32) be Artin s constant, then we have: A if p if p 1(mod4) 4.33) Proof. If p 1 (mod 4) then nm = m#(m) for every m and the result follows from the definition ....

C. Hooley, On Artin's conjectures,J.ReineAngew.Math. 225(1967), 209--220. MR 34:7445

....multiplicative orders of any fixed integer g 2. 5 Remarks We remark that it is likely to be true that L(x, y) #(u)#(x) in the stated range for y. The slightly weaker estimate L(x, y) u#(u)#(x) is likely to be provable assuming the Generalised Riemann Hypothesis, using the tools that Hooley [10] has used to prove Artin s conjecture on the Generalised Riemann Hypothesis. Studying other arithmetic properties of l(p) for example, the number of prime and integer divisors, is of interest as well. A recent paper on this subject is [17] also see [13] Finally, having in mind applications to ....

C. Hooley, `On Artin's conjecture,' J. Reine Angew. Math. 225 (1967), 209--220.

....best we can hope is that the period is q 1. This occurs when q is prime and 2 is a primitive root modulo q. The search for primes q such that 2 is a primitive root is related to a large body of contemporary number theory. It is believed that there are infinitely many primes q with this property [5]. Itowever, finding such primes (and even finding large primes at all) is problematic. In this paper we consider two fundamental questions about FCSR sequences: 1. ttow can we guarantee the output sequence has large period 2. What are the statistical properties of large period FCSR sequences ....

C. Hooley, On Artin's conjecture. J. Reine Angew. Math. vol. 22, 1967 pp. 209- 220.

....we can hope is that the period is q 1. This occurs when q is prime and 2 is a primitive root modulo q.Thesearchforprimesq such that 2 is a primitive root is related to a large body of contemporary number theory. It is believed that there are infinitely many primes q with this property [5]. However, finding such primes (and even finding large primes at all) is problematic. In this paper we consider two fundamental questions about FCSR sequences: 1. How can we guarantee the output sequence has large period 2. What are the statistical properties of large period FCSR sequences ....

C. Hooley, On Artin's conjecture. J. Reine Angew. Math. vol. 22, 1967 pp. 209-220.

....a primitive prime (to the base 2) Examples of primitive primes include 101, 107, 131, 139 etc. There is a famous conjecture by E. Artin that there are infinitely many such primes, and that they have a natural density. However, neither of those two assertions has been proved yet, although Hooley [8] deduced the Artin conjecture from the generalised Riemann hypothesis. Following [21] we introduce R p = F 2 [X] X 1) In the sequel we will suppose that the prime p is primitive. If this is the case then R p = F 2 p 1 F 2 . We can pass from R p to F 2 p 1 in both directions very ....

C. Hooley. On Artin's Conjecture. J. Reine Angew. Math., 225:209--220, 1967.

....a 6= 1 is a primitive root Our construction, therefore, really nds a needle in a haystack which explains one of the reasons why such a construction was not easy to come by. mod p for in nitely many primes p. The Artin conjecture is known to hold under the Generalized Riemann Hypothesis (GRH) [14]. In fact, several unconditional results are also known concerning the Artin conjecture. For instance, the conjecture is known to be true for most values of a, and in particular for one of the three numbers 2; 3; 5 [5, 13] see the expository paper [19] for details. This implies that we can ....

Christopher Hooley. On Artin's conjecture. J. Reine Angew. Math., 225:209-220, 1967.

....this expression whether the density is positive. However, one can show that the in nite sum is equal to a positive rational multiple c A of Artin s constant A = 1 X n=1 (n) n (n) Y l prime (1 1 l(l 1) 0:3739558136: Typeset by A M S T E X 26 HANS ROSKAM In 1967 Hooley [5] proved that the density (1) is indeed equal to the sum (2) if the generalized Riemann hypothesis (GRH) holds true. Furthermore, he explicitly determined c in terms of the prime factorization of . Over arbitrary number elds, there are two ways in which Artin s conjecture can be generalized. ....

C. Hooley, On Artin's conjecture, J. reine u. angew. Math. 225 (1967), 209-220.

.... : p xg of the set of primes p for which a is a primitive root modulo p inside the set of all rational primes exists; moreover it equals a positive rational multiple of what is now known as Artin s constant A = 1 X n=1 (n) n (n) Y l prime (1 1 l(l 1) 0:3739558136: In 1967 Hooley [3] proved Artin s conjecture under the assumption of the generalized Riemann hypothesis and explicitly determined the rational multiple in terms of the arithmetic structure of a. This paper deals with the following generalization of this conjecture. Fix a number eld K with ring of integers O and ....

C. Hooley, On Artin's conjecture, J. reine u. angew. Math. 225 (1967), 209-220.

....F p has order p 1. However, we do expect the equality P 1 h=1 (T h ) 1 to hold. In other words, we expect that for most primes p, the order of a modulo p is almost maximal. These conjectures are still open, but have been proved under the assumption of the generalized Riemann hypothesis [5,7,12]. To adapt Artin s conjecture to our situation, we rst need an upper bound for the order of 2 (O=pO) The trivial upper bound for this order is the exponent PRIME DIVISORS OF LINEAR RECURRENCES AND ARTIN S CONJECTURE 69 e(p) of (O=pO) The value of e(p) as function of the prime p ....

....[2] Under some mild assumptions on the polynomial f , they conjectured that the set T 1 has positive density. The generalized Artin conjecture has been proved for integers 62 f0; 1g, or, equivalently, for linear polynomials f , under the assumption of the generalized Riemann hypothesis [5,7,12]. Under the same assumption, we proved in the second chapter of this thesis that part of the conjecture holds for quadratic polynomials f . Unfortunately, we are not able to prove the generalized Artin conjecture for a single polynomial of degree at least 3, not even under the assumption of the ....

C. Hooley, On Artin's conjecture, J. reine u. angew. Math. 225 (1967), 209-220.

.... 1= 2 ) 9 19 f 0 ( La deuxi eme condition du th eor eme 3 donne une condition pour qu il y ait un nombre in ni de p tel que L(s; p; ait une fronti ere naturelle pour tout 2 M . Mais ceci d epend d un analogue de la conjecture d Artin (on peut esp erer que les methodes de [13] [7] 9] s appliquent aussi dans ce cas) Conjecture d Artin Chebotarev. Soit f(X) un polyn ome irr eductible tel que f(0) 6= 1 n est pas un carr e, alors pour chaque type de factorisation de densit e positive, il y a un nombre in ni de nombres premiers p tel que chaque facteur de f(X) mod p ....

C. Hooley, On Artin's conjecture, J. Reine Angew. Math. 225 (1967), 209-220.

....in the following form: #C(a) x 0 (a) #x # x 0 (a) # a (x) # x (C(a) log 2 x) 1) 3 where # a (x) is the number of primes up to x for which a is a primitive root. It is known that A contains the odd powers of all but at most two prime numbers [9] and of all prime numbers under the ERH [10]; many other relevant results can be found in [16] For any a # IN, clearly a 2 ## A. Theorem 1 For any prime power q = p k # A and any su#ciently large integer N there is an integer n = r 1) 2 with N # n # M , where M = 3C(q)N log N and C(q) is as in (1) such that the Gauss ....

....IF q , and therefore the polynomial f is zero. If e # e, then f(0) 1. Thus e # = e. But then the monomial x 2t occurs in f with nonzero coe#cient, where t = min(E # E # ) This contradiction proves the claim. ## We note that under the ERH, from the asymptotic formula for # q (x) of [10], one can get a slightly better estimate for n, namely apparently one can take M = cN log log log q, provided that N q C with some absolute constants c and C. Unfortunately C does not seems to be e#ectively computable and it is not clear how to use this better bound in order to design a faster ....

C. Hooley, `On Artin's conjecture', J. Reine Angew. Math., 225 (1967), 209--220.

....already implemented in current software systems such as Maple and Pari. For example, a FCSR based on the prime number q = 2 128 2 5 2 4 2 2 Gamma 1 needs only 2 bits of memory and has maximal period T = q Gamma 1: Heilbronn (revising Artin s conjecture) conjectured, and Hooley [17] proved, that if an extension of the Riemann hypothesis to the Dedekind zeta function over certain Galois fields is true, then the number N(n) of primes q n for which ord q (2) q Gamma 1 is N(n) A Delta n ln 2 (n) O n ln 2 ln 2 (n) ln 2 2 (n) where A ( 3739558136 to ....

C. Hooley, On Artin's conjecture. J. Reine Angew. Math. vol. 22, 1967 pp. 209-220.

....holds in the following form: 9C(a) x 0 (a) 8x x 0 (a) a (x) x= C(a) log 2 x) 1) where a (x) is the number of primes up to x for which a is a primitive root. It is known that A contains the odd powers of all but at most two prime numbers [3] and of all prime numbers under the ERH [4]; many other relevant results can be found in [10] For any a 2 IN; clearly a 2 62 A. Theorem 1. For any prime power q = p k 2 A and any sufficiently large integer N there is an integer n = r Gamma 1) 2 with N n M , where M = 3C(q)N log N and C(q) is as in (1) such that the Gauss period ....

...., and therefore the polynomial f is zero. If e 0 e, then f(0) Gamma1. Thus e 0 = e. But then the monomial x 2t occurs in f with nonzero coefficient, where t = min(E [ E 0 ) This contradiction proves the claim. ut We note that under the ERH, from the asymptotic formula for a (x) of [4], one can get a slightly better estimate for n, namely apparently one can take M = cN log log log q, provided that N q C with some absolute constants c and C. Unfortunately C does not seems to be effectively computable and it is not clear how to use this better bound in order to design a ....

C. Hooley, "On Artin's conjecture", J. Reine Angew. Math., 225 (1967), 209--220.

....modulo p all the elements of Gamma. We let N Gamma (x) # Phi p x; p = 2 supp( Gamma) j Gamma p = F p Psi : 1) The statement that N hai (x) 1 as x 1 (when a is an integer 6= 0, Sigma1 and not a perfect square) is known as the Artin Conjecture for primitive roots. in 1967 Hooley [3] proved that the Generalized Riemann Hypothesis (GRH) for the Dedekind zeta functions of certain Kummer extensions implies the strong form of the Artin Conjecture: N hai (x) ffi hai x log x : 4) Furthermore, Holley proved that if a = b h with b 2 Z not a power, then ffi hai = c a Y jh ....

C. Hooley, On Artin's conjecture, J. Reine Angew. Math. 225 (1967), 209--220.

....holds in the following form: 9C(a) x 0 (a) 8x x 0 (a) a (x) x= C(a) log 2 x) 1) where a (x) is the number of primes up to x for which a is a primitive root. It is known that A contains the odd powers of all but at most two prime numbers [9] and of all prime numbers under the ERH [10]; many other relevant results can be found in [16] For any a 2 IN; clearly a 2 62 A. Theorem 1. For any prime power q = p k 2 A and any sufficiently large integer N there is an integer n = r Gamma 1) 2 with N n M , where M = 3C(q)N log N and C(q) is as in (1) such that the Gauss period ....

...., and therefore the polynomial f is zero. If e 0 e, then f(0) Gamma1. Thus e 0 = e. But then the monomial x 2t occurs in f with nonzero coefficient, where t = min(E [ E 0 ) This contradiction proves the claim. ut We note that under the ERH, from the asymptotic formula for q (x) of [10], one can get a slightly better estimate for n, namely apparently one can take M = cN log log log q, provided that N q C with some absolute constants c and C. Unfortunately C does not seems to be effectively computable and it is not clear how to use this better bound in order to design a ....

C. Hooley, `On Artin's conjecture', J. Reine Angew. Math., 225 (1967), 209--220.

....q. By Theorem 3.6 an sequence is generated whenever q is chosen so that ord q ( jR= q) Gammaf0gj. The search for primes q such that is a primitive root, is related to a large body of contemporary number theory. It is believed that there are infinitely many primes q with this property [4]. 4 Cracking d Fold Summation Ciphers As mentioned in the introduction, our analysis has important consequences for the summation cipher [9] In this cipher, two m sequences a 1 and a 2 are combined using addition with carry . The resulting sequence is used as a pseudo one time pad. These ....

C. Hooley, On Artin's conjecture. J. Reine Angew. Math. vol. 22, 1967 pp. 209-220.

....p. Zeros of zeta functions of radical extensions are especially interesting because of their connection with Artin s conjecture that any integer a not equal to Gamma1; 0; or a perfect square is a primitive root for infinitely many primes (and even for a positive fraction of all primes) Hooley [9], 10] proved this conjecture under the assumption of the Generalized Riemann Hypothesis. Unfortunately, our results do not help to obtain an unconditional proof, because a much wider zero free region appears to be needed (cf. 10] and in any event, real zeros of i K (s) for K = Q[ p p a] ....

C. Hooley. On Artin's conjecture. J. reine angew. Math., 225:209--220, 1969.

....p ) We will establish various estimates for such a sum. If f(1) 1 and f(x) 0 for #= 1, then the famous Artin Conjecture for primitive roots states that (1) # i p = 1 ##(x) where # is the Artin constant, # = l prime # = 0.373955813619202 . In 1967, C. Hooley (see [5]) proved the Artin Conjecture as a consequence of the Generalized Riemann Hypothesis. The weaker form of the Artin Conjecture states that any fixed integer b 1 that is not a perfect square is a primitive root for infinitely many primes. Heath Brown [4] Gupta and Murty [3] see also [9] solved ....

....choosing z = log log log x, say. The estimate assuming GRH is proven in a similar way, therefore we omit it. Proof of Theorem 2. a) We write f(m)Hm (x) First note that since Hm (x) 0 if m 0, i p # 1 # x by a similar argument as in the Theorem of Hooley ([5], pages 211 212) So, if we choose y = x , we get that the above sum is . Further, if z goes to infinity as x goes to infinity, f(m) #(x, m, 1) m z #(m) log x = o(#(x) by the Brun Titchmarsh Theorem and the hypothesis that converges. Finally # m#(x) o ....

Hooley C., On Artin's Conjectures, J. Reine Angew. Math. 226 (1967), 207--220.

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C. Hooley, On Artin's conjecture, J. Reine Angew. Math., vol. 22, 1967, pp. 209--220.

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C. Hooley, On Artin's conjecture, J. Reine Angew. Math. 225 (1967), 209#220.

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