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Zerofree regions for Dirichlet Lfunctions and the least prime in an arithmetic progression
 Proc. Lond. Math. Soc
, 1992
"... The classical theorem of Dirichlet states that any arithmetic progression a(mod q) in which a and q are relatively prime contains infinitely many prime numbers. A natural question to ask is then, how big is the first such prime, P (a, q) say? In one direction we have trivially ..."
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The classical theorem of Dirichlet states that any arithmetic progression a(mod q) in which a and q are relatively prime contains infinitely many prime numbers. A natural question to ask is then, how big is the first such prime, P (a, q) say? In one direction we have trivially
Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
Zeros of families of automorphic Lfunctions close to 1
 Pacific J. Math
"... For many Lfunctions of arithmetic interest, the values on or closeto theedgeof theregion of absoluteconvergenceare of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to nonvanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet Lfunctions ( ..."
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For many Lfunctions of arithmetic interest, the values on or closeto theedgeof theregion of absoluteconvergenceare of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to nonvanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet Lfunctions (e.g., because of the Dirichlet classnumber formula) and the symmetric square Lfunctions of classical automorphic forms. For analytic purposes, in the absence of the Generalized Riemann Hypothesis, it is very useful to have an upperbound, on average, for the number of zeros of the Lfunctions which are very close to 1. We prove a very general statement of this typefor forms on GL(n)/Q for any n � 1, comparableto the logfree density theorems for Dirichlet characters first proved by Linnik. 1. Introduction.
COUNTING CONGRUENCE SUBGROUPS
"... Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence ..."
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Abstract. Let Γ denote the modular group SL(2, Z) and Cn(Γ) the number of congruence
On the Least Prime in Certain Arithmetic Progressions
"... We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators. * The second author is partly supported by an N.S.F. gr ..."
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We find infinitely many pairs of coprime integers, a and q, such that the least prime j a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators. * The second author is partly supported by an N.S.F. grant 1. Introduction For any x ? x 0 and for any positive valued function g(x) define R(x) = e fl log x log 2 x log 4 x=(log 3 x) 2 ; L(x) = exp(log x log 3 x= log 2 x) and E g (x) = exp \Gamma log x=(log 2 x) g(x) \Delta : Here log k x is the kfold iterated logarithm, fl is Euler's constant, and x 0 is chosen large enough so that log 4 x 0 ? 1. The usual method used to find large gaps between successive prime numbers is to construct a long sequence S of consecutive integers, each of which has a "small" prime factor (so that they cannot be prime); then, the gap between the largest prime before S and the next, is at least as long as S. Similarly if one wishes to find an arithm...
On the infinitude of elliptic Carmichael numbers
, 1999
"... ABSTRACT. In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all ell ..."
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ABSTRACT. In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author. 1.
PRIME SPLITTING IN ABELIAN NUMBER FIELDS AND LINEAR COMBINATIONS OF DIRICHLET CHARACTERS
"... Abstract. Let X be a finite group of primitive Dirichlet characters. Let ξ =∑ χ∈X aχχ be a nonzero element of the group ring Z[X]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ X and that satisfies∑ χ∈X aχχ(q) 6 = 0. Our main result is a nontrivial upper bound on q ..."
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Abstract. Let X be a finite group of primitive Dirichlet characters. Let ξ =∑ χ∈X aχχ be a nonzero element of the group ring Z[X]. We investigate the smallest prime q that is coprime to the conductor of each χ ∈ X and that satisfies∑ χ∈X aχχ(q) 6 = 0. Our main result is a nontrivial upper bound on q valid for certain special forms ξ. From this, we deduce upper bounds on the smallest unramified prime with a given splitting type in an abelian number field. For example: • Let p be a prime, and let χ be a Dirichlet character modulo p. Suppose that χ has order n and that d is a divisor of n with d> 1. The least prime q for which χ(q) is a primitive dth root of unity satisfies q n,ε pλ+ε, where λ = n