Pascadi 2025 exponents distribution primes smooth numbers

../

corollary_1_4: The number of primes p <= x with p + 2 also prime is at most (3.203 + o(1)) Pi_2(x) as x tends to infinity, where Pi_2(x) is the Hardy-Littlewood prediction; the paper says this improves the constant 3.229.

corollary_1_6: For every epsilon > 0 there is C > 0 such that, for x >= 2 and y in [(log x)^C, x^(1/C)], the integers n <= x with n and n + 1 both y-smooth number <<_epsilon x rho(u)^(1+5/8-epsilon), where u = log x / log y.

corollary_7_2: For coprime a, c with ad - bc nonzero and small coefficients, the n <= x with an + b y_1-smooth and cn + d y_2-smooth number <<_epsilon Psi(x, y_1) rho(u_2)^(5/8-epsilon), for (log x)^C <= y_1 <= y_2 <= x with y_2 <= y_1^C.

definition_1_1: A sequence (lambda_q) on q <= Q is triply-well-factorable of level Q when, for every split Q = Q_1 Q_2 Q_3 with each Q_i >= 1, it is a triple Dirichlet convolution of 1-bounded sequences supported on q_i <= Q_i.

theorem_1_3: For fixed nonzero a, primes up to x are equidistributed in residue classes a modulo q on average over q <= Q, with a saving of any power of log x, against triply-well-factorable weights of level Q <= x^(5/8-epsilon) or upper-bound well-factorable linear sieve weights of level Q <= x^(3/5-epsilon).

theorem_1_5: For fixed nonzero a and y in [(log x)^C, x^(1/C)] with C large in terms of a, A and epsilon, the y-smooth numbers up to x are equidistributed in the classes a modulo q, summed in absolute value over q <= x^(5/8-epsilon), with saving (log x)^(-A) relative to Psi(x, y).

theorem_7_1: A divisor-weighted form of Theorem 1.5 for y_2-smooth moduli q ~ Q with Q <= x^(5/8-epsilon), in the classes a_1 times the inverse of a_2 modulo q_0 q, whose bound carries the extra factor Psi(Q, y_2) / (phi(q_0) Q) e^(O_k(u_2)).

---

Alexandru Pascadi, On the exponents of distribution of primes and smooth numbers. arXiv preprint (2025). arXiv:2505.00653. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2505.00653), every other right reserved. The copy read for this card is arXiv:2505.00653v2 (29 June 2025), 42 pages; page numbers below are that edition's.

Pascadi shows unconditionally that both primes with triply-well-factorable weights and smooth numbers have exponent of distribution 5/8 - epsilon, removing the dependence on Selberg's eigenvalue conjecture in earlier work of Lichtman and the author which built on Maynard and Drappeau. Theorem 1.3 gives the prime result, part (i) at level x^(5/8-epsilon) for triply-well-factorable weights (Definition 1.1) and part (ii) at x^(3/5-epsilon) for the upper-bound well-factorable linear sieve weights, improving the previous 66/107 (Lichtman) and 7/12 (Maynard); Theorem 1.5 gives the same 5/8 - epsilon exponent for smooth numbers Psi(x, y; a, q) with y in [(log x)^C, x^(1/C)], improving the author's earlier unconditional 66/107 - epsilon (the 3/5 - epsilon of Fouvry-Tenenbaum and Drappeau came before it), and Theorem 7.1 refines it for smooth moduli. Applications include Corollary 1.4, bounding the number of twin primes up to x by (3.203 + o(1)) Pi_2(x), improving 3.229, and Corollary 1.6 on counts of consecutive smooth numbers. The method combines Linnik's dispersion method and Deshouillers-Iwaniec-style bounds for sums of Kloosterman sums with the author's large sieve inequality for exceptional Maass forms (for additively structured sequences) and a large sieve inequality of Watt.

Source: https://arxiv.org/abs/2505.00653.

Bears on. #158: indirect only, through Theorem 1.3. An exponent of distribution here controls a weighted sum over all moduli up to the level for one fixed residue class (p. 2); the paper does not mention the problem.

Results. Labels and pages are those of v2.

• Definition 1.1 (p. 2): triply-well-factorable weights of level Q.
• Theorem 1.3 (p. 2): primes in progressions to moduli up to x^(5/8-epsilon) with triply-well-factorable weights, and up to x^(3/5-epsilon) with upper-bound well-factorable linear sieve weights.
• Corollary 1.4 (p. 3): at most (3.203 + o(1)) Pi_2(x) twin primes up to x.
• Theorem 1.5 (p. 3): y-smooth numbers in progressions to moduli up to x^(5/8-epsilon), summed in absolute value, for y in [(log x)^C, x^(1/C)].
• Corollary 1.6 (p. 3): consecutive y-smooth pairs n, n+1 up to x number <<_epsilon x rho(u)^(1+5/8-epsilon).
• Theorem 7.1 (p. 39): the smooth-number estimate on smooth moduli with divisor weights.
• Corollary 7.2 (p. 40): smooth values of factorable quadratic polynomials, from which Corollary 1.6 follows.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.