منابع مشابه
Average prime-pair counting formula
Taking r > 0, let π2r(x) denote the number of prime pairs (p, p+ 2r) with p ≤ x. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that π2r(x) ∼ 2C2r li2(x) with an explicit constant C2r > 0. There seems to be no good conjecture for the remainders ω2r(x) = π2r(x)−2C2r li2(x) that corresponds to Riemann’s formula for π(x)− li(x). However, there is a heuristic approximate formula f...
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Juggling patterns can be described by a closed walk in a (directed) state graph, where each vertex (or state) is a landing pattern for the balls and directed edges connect states that can occur consecutively. The number of such patterns of length n is well known, but a long-standing problem is to count the number of prime juggling patterns (those juggling patterns corresponding to cycles in the...
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In this paper, we investigate the means of the values of prime counting function $pi(x)$. First, we compute the arithmetic, the geometric, and the harmonic means of the values of this function, and then we study the limit value of the ratio of them.
متن کاملAverage Twin Prime Conjecture for Elliptic Curves
Let E be an elliptic curve over Q. In 1988, N. Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over Fp is prime. This is an analogue of the Hardy–Littlewood twin prime conjecture in the case of elliptic curves. Koblitz’s Conjecture is still widely open. In this paper we prove that Koblitz’s Conjecture is true on average...
متن کاملGenus 2 point counting over prime fields
For counting points of jacobians of genus 2 curves over a large prime field, the best known approach is essentially an extension of Schoof’s genus 1 algorithm. We propose various practical improvements to this method and illustrate them with a large scale computation: we counted hundreds of curves, until one was found that is suitable for cryptographic use, with a state-ofthe-art security level...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2009
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-09-02312-6