Prescribing the binary digits of primes

We present a new result on counting primes p < N = 2 n for which r (arbitrarily placed) digits in the binary expansion of p are specified. Compared with earlier work of Harman and Katai, the restriction on r is relaxed to r < c ( n /log n ) 4/7 . This condition results from the estimates of Ga...

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Bibliographic Details
Published inIsrael journal of mathematics Vol. 194; no. 2; pp. 935 - 955
Main Author Bourgain, Jean
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.03.2013
Subjects
Algebra
Analysis
Applications of Mathematics
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
Pigeon Hole Principle
Binary Expansion
Circle Method
Principal Character
Binary Digit
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ISSN0021-2172
1565-8511
DOI10.1007/s11856-012-0104-2

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Summary:We present a new result on counting primes p < N = 2 n for which r (arbitrarily placed) digits in the binary expansion of p are specified. Compared with earlier work of Harman and Katai, the restriction on r is relaxed to r < c ( n /log n ) 4/7 . This condition results from the estimates of Gallagher and Iwaniec on zero-free regions of L -functions with ‘powerful’ conductor.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-012-0104-2