Journal of the Ramanujan Mathematical Society

Stephan Baier ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, 1 Dr. Homi Bhaba Road, Colaba, Mumbai-400005, IN

Abstract

We give a short alternative proof using Heath-Brown’s square sieve of a bound of the author for the large sieve with square moduli.

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Stephan Baier ¹

Affiliations
1 Department of Mathematics and Statistics, Queen’s University, University Ave, Kingston, Ontario, K7L 3N6, CA

Abstract

Extending a method of D. Wolke [10], we establish a general result on the large sieve with sparse sets 𝒮 of moduli which are in a sense well-distributed in arithmetic progressions. We then use this result together with Fourier techniques to obtain large sieve bounds for the case when 𝒮 consists of sqares. These bounds improve a recent result by L. Zhao [11].

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Stephan Baier ¹

Affiliations
1 Jacobs University Bremen, School of Engineering and Science, P.O. Box 750 561, 28725 Bremen, DE

Abstract

For an elliptic curve E over Q and an integer, r let π^r_E (x) be the number of primes p ≤ x of good reduction such that the trace of the Frobenius morphism of E/F_p equals r. We consider the quantity π^r_E (x) on average over certain sets of elliptic curves. More in particular, we establish the following: If A,B > x^1/2+ε and AB > x^3/2+ε, then the arithmetic mean of π^r_E (x) over all elliptic curves E : y² = x³ + ax + b with a, b ∈ Z, |a| ≤ A and |b| ≤ B is ∼ Cr √ x/ log x, where C_r is some constant depending on r. This improves a result of C. David and F. Pappalardi. Moreover, we establish an “almost-all” result on π^r_E (x).

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