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On the exponents of distribution of primes and smooth numbers

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arxiv 2505.00653 v2 pith:UIV2VG5J submitted 2025-05-01 math.NT

Alexandru Pascadi This is my paper · ORCID
classification math.NT
keywords primesnumberssmoothauthorsequencessieveweightsadditively-structured
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abstract

We show that both primes and smooth numbers are equidistributed in arithmetic progressions to moduli up to $x^{5/8 - o(1)}$, using triply-well-factorable weights for the primes (we also get improvements for the well-factorable linear sieve weights). This completely eliminates the dependency on Selberg's eigenvalue conjecture in previous works of Lichtman and the author, which built in turn on results of Maynard and Drappeau. We rely on recent large sieve inequalities for exceptional Maass forms of the author for additively-structured sequences, and on a related result of Watt for multiplicatively-structured sequences. As applications, we prove refined upper bounds for the counts of twin primes and consecutive smooth numbers up to $x$.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Non-abelian amplification and bilinear forms with Kloosterman sums

    math.NT 2025-11 accept novelty 8.0 of 10

    New non-abelian amplification bounds bilinear Kloosterman sums for composite moduli, saving c^{-1/12} for products of two primes of similar size.

  2. Bilinear forms with Kloosterman sums via quadratic characters

    math.NT 2026-07 accept novelty 7.0 of 10

    Bilinear Kloosterman forms save c^{-1/32} at length √c for all moduli via a new link to quadratic character sums, improving prior prime and composite bounds.

  3. Primes in arithmetic progressions to large moduli and refinements of Harman's sieve

    math.NT 2026-02 unverdicted novelty 7.0 of 10

    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).

  4. An improvement on the largest prime factors of consecutive integers

    math.NT 2026-07 conditional novelty 6.0 of 10

    The asymptotic density of n with P^+(n) < P^+(n+1) is shown to be > 0.280, a new record.

  5. An update on the Linnik--Goldbach problem

    math.NT 2026-05 unverdicted novelty 5.0 of 10

    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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