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The Pseudosquares Prime Sieve
"... Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It use ..."
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Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Riemann Hypothesis, we have p ≤ 2(log n) 2, but it is conjectured that p ∼ 1 log nlog log n. Thus, log2 the conjectured complexity of our prime sieve is O(n log n) arithmetic operations in O((log n) 2) space. The primes generated by our algorithm are proven prime unconditionally. The best current unconditional bound known is p ≤ n 1/(4√e−ɛ) 1.132, implying a running time of roughly n using roughly n 0.132 space. Existing prime sieves are generally faster but take much more space, greatly limiting their range (O(n / log log n)operationswithn 1/3+ɛ space, or O(n) operationswithn 1/4 conjectured space). Our algorithm found all 13284 primes in the interval [10 33,10 33 +10 6] in about 4 minutes on a1.3GHzPentiumIV. We also present an algorithm to find all pseudosquares Lp up to n in sublinear time using very little space. Our innovation here is a new, spaceefficient implementation of the wheel datastructure. 1
Research Statement
"... My research centers on computational number theory. It is the main concern of my thesis, and has also been the basis for several fruitful collaborations. My thesis [Gal00a] deals with an analytic algorithm for computing π(x), the number of primes below x. This algorithm, first described by Lagarias ..."
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My research centers on computational number theory. It is the main concern of my thesis, and has also been the basis for several fruitful collaborations. My thesis [Gal00a] deals with an analytic algorithm for computing π(x), the number of primes below x. This algorithm, first described by Lagarias and Odlyzko, is based on numerical integration of a function related to the Riemann zeta function, ζ(s), in combination with summation of a “kernel function ” evaluated at prime powers “near ” x [LO84]. Later, with the development by Odlyzko and Schönhage of a fast method for computing ζ(s) [OS88], Lagarias and Odlyzko showed their algorithm could find π(x) in O(x1/2+ɛ) time [LO87]. Although asymptotically this is the fastest known algorithm for computing π(x), several unresolved technical issues have prevented any practical implementation until now. My thesis addresses these issues. Results include: • Design of a kernel function superior to the kernel proposed in [LO87]; • Choice of a quadrature algorithm, with a careful analysis of quadrature error; • Development of a new sieving algorithm requiring O(x 1/3+ɛ) bits of memory for efficient operation, in contrast to previously known sieving methods, which require O(x 1/2+ɛ) bits; • An improved method for computing the Riemann zeta function to arbitrarily high accuracy.
Article electronically published on December 19, 2003 PRIME SIEVES USING BINARY QUADRATIC FORMS
"... Abstract. We introduce an algorithm that computes the prime numbers up to N using O(N/log log N) additions and N 1/2+o(1) bits of memory. The algorithm enumerates representations of integers by certain binary quadratic forms. We present implementation results for this algorithm and one of the best p ..."
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Abstract. We introduce an algorithm that computes the prime numbers up to N using O(N/log log N) additions and N 1/2+o(1) bits of memory. The algorithm enumerates representations of integers by certain binary quadratic forms. We present implementation results for this algorithm and one of the best previous algorithms. 1.