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10
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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Cited by 17 (1 self)
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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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Cited by 6 (0 self)
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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 (This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv).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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Cited by 5 (0 self)
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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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Cited by 2 (0 self)
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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
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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Cited by 1 (1 self)
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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.
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.
unknown title
"... i iiTo my advisor, Eli Stein, for showing me the importance of good exposition; To my friends, for supporting this experiment; And to the readers of my blog, for their feedback and contributions. Contents ..."
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i iiTo my advisor, Eli Stein, for showing me the importance of good exposition; To my friends, for supporting this experiment; And to the readers of my blog, for their feedback and contributions. Contents
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).
EVASIVENESS AND THE DISTRIBUTION OF PRIME NUMBERS
, 2010
"... Abstract. A Boolean function on N variables is called evasive if its decisiontree complexity is N. A sequence Bn of Boolean functions is eventually evasive if Bn is evasive for all sufficiently large n. We confirm the eventual evasiveness of several classes of monotone graph properties under widely ..."
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Abstract. A Boolean function on N variables is called evasive if its decisiontree complexity is N. A sequence Bn of Boolean functions is eventually evasive if Bn is evasive for all sufficiently large n. We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla’s conjecture on Dirichlet primes implies that (a) for any graph H, “forbidden subgraph H” is eventually evasive and (b) all nontrivial monotone properties of graphs with ≤ n 3/2−ǫ edges are eventually evasive. (n is the number of vertices.) While Chowla’s conjecture is not known to follow from the Extended Riemann Hypothesis (ERH, the Riemann Hypothesis for Dirichlet’s L functions), we show (b) with the bound O(n 5/4−ǫ) under ERH. We also prove unconditional results: (a ′ ) for any graph H, the query complexity of “forbidden subgraph H ” is ` ´ n −O(1); (b) for some constant c> 0, all nontrivial monotone
An Overview of Sieve Methods
"... We provide an overview of the power of Sieve methods in number theory meant for the nonspecialist. ..."
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We provide an overview of the power of Sieve methods in number theory meant for the nonspecialist.