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Primes in arithmetic progressions to large moduli, and Goldbach beyond the square-root barrier
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abstract
We show the primes have level of distribution $66/107\approx 0.617$ using triply well-factorable weights. This gives the highest level of distribution for primes in any setting, improving on the prior record level $3/5=0.60$ of Maynard. We also extend this level to $5/8=0.625$, assuming Selberg's eigenvalue conjecture. As applications of the method, we obtain new upper bounds for twin primes and for Goldbach representations of even numbers $a$. For the Goldbach problem, this is the first use of a level of distribution beyond the square-root barrier, and leads to the greatest improvement on the problem since Bombieri-Davenport from 1966. Our proof optimizes the Deshouillers-Iwaniec spectral large sieve estimates, both in the exceptional spectrum and uniformity in the residue $a$, refining Drappeau-Pratt-Radziwill and Assing-Blomer-Li.
Forward citations
Cited by 5 Pith papers
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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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On the exponents of distribution of primes and smooth numbers
Unconditional level of distribution x^(5/8-o(1)) is proved for primes with triply-well-factorable weights and for smooth numbers, eliminating Selberg's eigenvalue conjecture from previous conditional results.
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Level aspect subconvexity for $\textrm{GL(2)}\times \textrm{GL(2)}$ $\textrm{L}$-functions
L(1/2, f⊗g) ≤ p^{1/2 - 1/524 + ε} for newforms f of prime level p with any nebentypus and fixed eigenforms g, improving the prior δ = 1/1413.
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Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications
A new convolution-type Bombieri-Vinogradov theorem with well-factorable weights improves the lower density for P^+(n)<P^+(n+1) from 0.280 to 0.296 and the upper density for P^+(p-1)>=p^c from (7/2)log(1/c) to a smaller S(c).
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An update on the Linnik--Goldbach problem
Under GRH the Linnik-Goldbach problem is solved with six powers of two; unconditionally more than 25 percent of odd integers are a prime plus a power of two.
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