Abstract
We obtain the expected asymptotic formula for the number of primes p < N = 2n with r prescribed (arbitrary placed) binary digits, provided r < cn for a suitable constant c > 0. This result improves on our earlier result where r was assumed to satisfy \(r < c{\left( {\frac{n}{{\log n}}} \right)^{4/7}}\).
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References
E. Bombieri, Le Grand Crible dans la Théorie Analytique des Nombres, Astérisque 18 (1987).
J. Bourgain, Prescribing the binary digits of primes, Israel J. Math. 194 (2013), 935–955.
J. Bourgain, Monotone Boolean functions capture their primes, J. Anal. Math., to appear.
H. Davenport, Multiplicative Number Theory, Springer-Verlag, New York-Berlin, 1980.
P. X. Gallagher, Primes in progressions to prime-power modulus, Invent. Math. 16 (1972), 191–201.
H. Iwaniec, On zeros of Dirichlet L-series, Invent. Math. 23 (1974), 97–104.
H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society, Providence, RI, 2004.
G. Harman and I. Katai, Primes with preassigned digits. II, Acta Arith. 133 (2008), 171–184.
A. E. Ingham, A note on Fourier transforms, J. London Math. Soc. 9 (1934), 29–32.
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This work was partially supported by NSF grants DMS-1301619 and DMS-0835373.
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Bourgain, J. Prescribing the binary digits of primes, II. Isr. J. Math. 206, 165–182 (2015). https://doi.org/10.1007/s11856-014-1129-5
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DOI: https://doi.org/10.1007/s11856-014-1129-5