Upper bounds for prime k-tuples of size log N and oscillations

Christian Elsholtz

doi:10.1007/s00013-003-4780-3

Upper bounds for prime k-tuples of size log N and oscillations

Christian Elsholtz^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We prove the estimate E_k ≪ N/exp((1/4 + o(1)) log N log log log N/log log N), for the number E_k (N) of k-tuples (n + a ₁, . . ., n + a_k) of primes not exceeding N, for k of size c₁ log N and N sufficiently large. A bound of this strength was previously known in the special case n - 2ⁱ (1 ≦ i <log n/log 2) only, (Vaughan, 1973). For general a_i this is an improvement upon the work of Hofmann and Wolke (1996). The number of prime tuples of this size has considerable oscillations, when varying the prime pattern.

Original language	English
Pages (from-to)	33-39
Number of pages	7
Journal	Archiv der Mathematik
Volume	82
Issue number	1
DOIs	https://doi.org/10.1007/s00013-003-4780-3
Publication status	Published - Jan 2004

ASJC Scopus subject areas

Mathematics(all)

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abstract = "We prove the estimate Ek ≪ N/exp((1/4 + o(1)) log N log log log N/log log N), for the number Ek (N) of k-tuples (n + a 1, . . ., n + ak) of primes not exceeding N, for k of size c1 log N and N sufficiently large. A bound of this strength was previously known in the special case n - 2i (1 ≦ i i this is an improvement upon the work of Hofmann and Wolke (1996). The number of prime tuples of this size has considerable oscillations, when varying the prime pattern.",

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AB - We prove the estimate Ek ≪ N/exp((1/4 + o(1)) log N log log log N/log log N), for the number Ek (N) of k-tuples (n + a 1, . . ., n + ak) of primes not exceeding N, for k of size c1 log N and N sufficiently large. A bound of this strength was previously known in the special case n - 2i (1 ≦ i i this is an improvement upon the work of Hofmann and Wolke (1996). The number of prime tuples of this size has considerable oscillations, when varying the prime pattern.

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Upper bounds for prime k-tuples of size log N and oscillations

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