Matomaki 2016 multiplicative functions short intervals

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corollary_1: States that for each epsilon > 0 there is C(epsilon) > 0 such that, for all large enough X, the interval [X, X + C(epsilon) sqrt(X)] contains at least sqrt(X)(log X)^{-4} numbers that are X^epsilon-smooth.

corollary_2: States that for every integer h >= 1 there is delta(h) > 0 with |(1/X) sum_{n <= X} lambda(n)lambda(n+h)| <= 1 - delta(h) for all large enough X, lambda being Liouville's function, and that the same holds for every completely multiplicative f into [-1,1] that is negative somewhere.

corollary_3: States that a multiplicative f into the reals has a positive proportion of sign changes if and only if f(n) < 0 for some integer n > 0 and f(n) is nonzero for a positive proportion of the integers n.

corollary_4: States that if a multiplicative f into the reals satisfies f(n) < 0 for some integer n and f(n) is nonzero for a positive proportion of n, then for any psi(x) tending to infinity almost every interval [x, x + psi(x)] contains a sign change of f.

corollary_5: States that if a completely multiplicative f into the reals satisfies f(n) < 0 for some integer n > 0 and f(n) is nonzero for a positive proportion of n, then some constant C > 0 gives a sign change of f in [x, x + C sqrt(x)] for all large enough x.

corollary_6: States that for psi(x) tending to infinity and fixed u > 0, for almost all x the number of x^{1/u}-smooth integers in [x, x + psi(x)] is asymptotically rho(u) psi(x), rho being the Dickman-de Bruijn function.

theorem_1: States that for multiplicative f into [-1,1], all 2 <= h <= X and all delta > 0, the average of f over [x, x+h] is within delta + C'(log log h)/log h of its average over [X, 2X] for all but CX((log h)^{1/3}/(delta^2 h^{delta/25}) + 1/(delta^2 (log X)^{1/50})) integers x in [X, 2X], with absolute constants C, C' > 1.

theorem_2: States that for multiplicative f into [-1,1] and every 10 <= h <= x, the sum of f(n_1)f(n_2) over x <= n_1 n_2 <= x + h sqrt(x) with sqrt(x) <= n_1 <= 2 sqrt(x), divided by h sqrt(x) log 2, equals the square of the mean of f over [sqrt(x), 2 sqrt(x)] up to O((log log h)/log h + (log x)^{-1/100}).

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Kaisa Matomäki, Maksym Radziwiłł, Multiplicative functions in short intervals. Annals of Mathematics 183 (2016), 1015-1056. doi:10.4007/annals.2016.183.3.6. The print carries "©2016 Department of Mathematics, Princeton University." in the footer of its first page (the text layer renders the symbol as a circled c), every other right reserved.

Theorem 1 is the paper's central estimate: for any multiplicative f taking values in [-1,1] and any 2 <= h <= X, the average of f over [x, x+h] differs from its average over [X, 2X] by at most delta + C' (log log h)/log h for all but at most C X ((log h)^{1/3}/(delta^2 h^{delta/25}) + 1/(delta^2 (log X)^{1/50})) integers x in [X, 2X], for every delta > 0, with absolute constants C, C' > 1 (one can take C' = 20000), so h, delta and f may all vary. Theorem 2 is a bilinear variant valid in every interval [x, x + h sqrt(x)] with 10 <= h <= x, which removes the uncontrolled large-prime-factor contribution. The proof relates short averages to long averages using Dirichlet-polynomial decompositions and Halász-type mean value machinery. Consequences include cancellation in sums of the Möbius function in almost all intervals [x, x+psi(x)] with psi growing arbitrarily slowly, X^epsilon-smooth numbers in [X, X + C(epsilon) sqrt(X)] (Corollary 1), the bound |sum_{n<=X} lambda(n) lambda(n+h)| <= (1 - delta(h)) X (Corollary 2), a characterization of multiplicative functions with a positive proportion of sign changes (Corollary 3), sign changes in almost all short intervals and, for completely multiplicative f, in all square-root-length intervals (Corollaries 4 and 5), and the asymptotic count rho(u) psi(x) of x^{1/u}-smooth integers in almost all intervals [x, x+psi(x)] (Corollary 6). For problem 1201 Theorem 1 is the uniform almost-all-short-interval estimate that Chojecki's note applies, in a half-open form, to the indicator of the smooth integers to deduce the lower-density form of the problem.

Source: https://doi.org/10.4007/annals.2016.183.3.6.

Bears on. #1201: the paper does not mention the problem. Chojecki's note (its card) applies a half-open form of Theorem 1 to the indicator of the integers with no prime factor above X^β, β = 1-ε/2, and deduces the problem's statement with lower density in place of density.

Results. Labels and pages are those of the Annals print named above. Read depth: claims checked for each page below; no proof was checked independently.

• Theorem 1 (pp. 1015--1016; proof Section 9, pp. 1043--1044): for multiplicative f: N -> [-1,1], absolute constants C, C' > 1 such that for every 2 <= h <= X and δ > 0 the average of f over [x, x+h] is within δ + C'(log log h)/log h of its average over [X, 2X] for all but at most CX((log h)^{1/3}δ^{-2}h^{-δ/25} + δ^{-2}(log X)^{-1/50}) integers x in [X, 2X]; one can take C' = 20000.
• Theorem 2 (p. 1016; proof pp. 1047--1048): for multiplicative f: N -> [-1,1] and every 10 <= h <= x, the sum of f(n_1)f(n_2) over x <= n_1 n_2 <= x + h sqrt(x) with sqrt(x) <= n_1 <= 2 sqrt(x), divided by h sqrt(x) log 2, equals the square of the mean of f over [sqrt(x), 2 sqrt(x)] up to O((log log h)/log h + (log x)^{-1/100}).
• Corollary 1 (p. 1016; proof pp. 1049--1050): for each ε > 0 some C(ε) > 0 gives at least sqrt(X)(log X)^{-4} X^ε-smooth numbers in [X, X + C(ε) sqrt(X)] for all large enough X.
• Corollary 2 (p. 1017; proof pp. 1051--1052): for every integer h >= 1 some δ(h) > 0 gives |(1/X) sum_{n<=X} λ(n)λ(n+h)| <= 1 - δ(h) for all large enough X > 1, and the same for every completely multiplicative f: N -> [-1,1] with f(n) < 0 for some n > 0.
• Corollary 3 (p. 1018; proof p. 1051): a multiplicative f: N -> R has a positive proportion of sign changes if and only if f(n) < 0 for some integer n > 0 and f(n) != 0 for a positive proportion of n.
• Corollary 4 (p. 1018; proof pp. 1050--1051): under the same hypotheses (f(n) < 0 for some integer n), for any ψ(x) -> ∞ almost every interval [x, x+ψ(x)] contains a sign change of f.
• Corollary 5 (p. 1019; proof pp. 1052--1053): for completely multiplicative f: N -> R under the hypotheses of Corollary 3, some C > 0 gives a sign change in [x, x + C sqrt(x)] for all large enough x.
• Corollary 6 (p. 1019; proof p. 1048): for ψ(x) -> ∞ and u > 0, for almost all x the number of x^{1/u}-smooth integers in [x, x+ψ(x)] is asymptotically ρ(u)ψ(x).

Theorems 3 and 4 (pp. 1020--1021), the variants on integers with prime factors in prescribed ranges from which Theorems 1 and 2 are deduced, have no page here.

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