Refinements of Harman's sieve yield Bombieri-Vinogradov-type mean value theorems for primes to moduli up to x^{9/17} (bilinear) and x^{17/32} (trilinear), plus new almost-all-q bounds on pi(x;q,a).
On the exponents of distribution of primes and smooth numbers
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Any classical sifting problem equipped with a Bombieri-Vinogradov error term can be made fully effective by adjusting the sieve bounds to avoid Siegel-zero complications, preserving the original asymptotic form.
Under GRH, every large even integer is two primes plus six powers of 2, and unconditionally at least 25% of odd numbers are prime plus a power of 2.
citing papers explorer
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Primes in arithmetic progressions to large moduli and refinements of Harman's sieve
Refinements of Harman's sieve yield Bombieri-Vinogradov-type mean value theorems for primes to moduli up to x^{9/17} (bilinear) and x^{17/32} (trilinear), plus new almost-all-q bounds on pi(x;q,a).
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An effective Bombieri-Vinogradov error term for sifting problems
Any classical sifting problem equipped with a Bombieri-Vinogradov error term can be made fully effective by adjusting the sieve bounds to avoid Siegel-zero complications, preserving the original asymptotic form.
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An update on the Linnik--Goldbach problem
Under GRH, every large even integer is two primes plus six powers of 2, and unconditionally at least 25% of odd numbers are prime plus a power of 2.