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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
Explicit bounds for primes in residue classes
 Math. Comp
, 1996
"... 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 degree1 prime p of K su ..."
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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 degree1 prime p of K such that p = σ, satis
ANALYTIC PROBLEMS FOR ELLIPTIC CURVES
, 2005
"... Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the dist ..."
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Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an interesting dichotomy for the occurence of the possible groups. (This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv). Contents
Fast Integer Multiplication Using Modular Arithmetic
 In Fortieth Annual ACM Symposium on Theory of Computing
, 2008
"... We give an O(N ·log N ·2 O(log ∗ N)) algorithm for multiplying two Nbit integers that improves the O(N · log N · log log N) algorithm by SchönhageStrassen [SS71]. Both these algorithms use modular arithmetic. Recently, Fürer [Für07] gave an O(N · log N · 2 O(log ∗ N)) algorithm which however uses ..."
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We give an O(N ·log N ·2 O(log ∗ N)) algorithm for multiplying two Nbit integers that improves the O(N · log N · log log N) algorithm by SchönhageStrassen [SS71]. Both these algorithms use modular arithmetic. Recently, Fürer [Für07] gave an O(N · log N · 2 O(log ∗ N)) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from Fürer’s algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a padic version of Fürer’s algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar. 1
ON AN ANALOGUE OF TITCHMARSH’S DIVISOR PROBLEM FOR HOLOMORPHIC CUSP FORMS
"... A central question in analytic number theory is to understand the average value of arithmetical functions ξ(n) at primes, which amounts to estimating the sum ..."
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A central question in analytic number theory is to understand the average value of arithmetical functions ξ(n) at primes, which amounts to estimating the sum
A GEOMETRIC VARIANT OF TITCHMARSH DIVISOR PROBLEM
"... Abstract. We formulate a geometric analogue of the Titchmarsh Divisor Problem in the context of abelian varieties. For any abelian variety A defined over Q, we study the asymptotic distribution of the primes of Z which split completely in the division fields of A. For all abelian varieties which con ..."
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Abstract. We formulate a geometric analogue of the Titchmarsh Divisor Problem in the context of abelian varieties. For any abelian variety A defined over Q, we study the asymptotic distribution of the primes of Z which split completely in the division fields of A. For all abelian varieties which contain an elliptic curve we establish an asymptotic formula for such primes under the assumption of GRH. We explain how to derive an unconditional asymptotic formula in the case that the abelian variety is a CM elliptic curve. 1.
Approximating reals by sums of two rationals
, 2008
"... We generalize Dirichlet’s diophantine approximation theorem to approximating any real number α by a sum of two rational numbers a1 q1 a2 q2 with denominators 1 ≤ q1, q2 ≤ N. This turns out to be related to the congruence equation problem xy ≡ c (mod q) with 1 ≤ x, y ≤ q 1/2+ǫ. ..."
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We generalize Dirichlet’s diophantine approximation theorem to approximating any real number α by a sum of two rational numbers a1 q1 a2 q2 with denominators 1 ≤ q1, q2 ≤ N. This turns out to be related to the congruence equation problem xy ≡ c (mod q) with 1 ≤ x, y ≤ q 1/2+ǫ.
Least Primes in Arithmetic Progressions
"... For a fixed nonzero integer a and increasing function f , we investigate the lower density of the set of integers q for which the least prime in the arithmetic progression a(mod q) is less than qf(q). In particular we conjecture that this lower density is 1 for any f with log x = o(f(x)) and prove ..."
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For a fixed nonzero integer a and increasing function f , we investigate the lower density of the set of integers q for which the least prime in the arithmetic progression a(mod q) is less than qf(q). In particular we conjecture that this lower density is 1 for any f with log x = o(f(x)) and prove this, unconditionally, for f(x) = x=g(x) for any g with log g(x) = o(log x). Under the assumption of a strong form of the prime ktuplets conjecture we prove our conjecture and get strong results on the distribution of values of ß(qlog q; q; a) for any fixed , as q varies. 1. Introduction For given integers a and q; q ? 0; a 6= 0; (a; q) = 1, we define p(q; a) to be the least prime p that is greater than a and congruent to a(mod q). We let p(q) be the largest value of p(q; a) for a in the range 1 a q \Gamma 1; (a; q) = 1 (1) In 1944 Linnik [13] gave the remarkable result that there exists an absolute constant c for which p(q) ø q c , for all positive integers q. Numerous authors have ...
A Titchmarsh divisor problem for elliptic curves, submitted
, 2014
"... Abstract. Let E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that∑ p≤x p≡1 (mod 4) τ(#E(Fp)) ∼ ..."
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Abstract. Let E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that∑ p≤x p≡1 (mod 4) τ(#E(Fp)) ∼
Some Remarks on a Paper of L. Toth
, 2010
"... Consider the functions P(n): = ∑n k=1 gcd(k, n) (studied by Pillai in 1933) and ˜P(n): = n ∏ pn (2 − 1/p) (studied by Toth in 2009). From their results, one can obtain asymptotic expansions for ∑ P(n)/n and n≤x ˜ P(n)/n. We consider two n≤x wide classes of functions R and U of arithmetical function ..."
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Consider the functions P(n): = ∑n k=1 gcd(k, n) (studied by Pillai in 1933) and ˜P(n): = n ∏ pn (2 − 1/p) (studied by Toth in 2009). From their results, one can obtain asymptotic expansions for ∑ P(n)/n and n≤x ˜ P(n)/n. We consider two n≤x wide classes of functions R and U of arithmetical functions which include P(n)/n and ˜P(n)/n respectively. For any given R ∈ R and U ∈ U, we obtain asymptotic expansions for ∑ ∑ ∑ n≤x R(n), n≤x U(n), p≤x R(p − 1) and p≤x U(p − 1).