Mangerel 2022 additive functions short intervals gaps conjecture

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Mangerel, Alexander P., Additive functions in short intervals, gaps and a conjecture of Erdős. Ramanujan J. 59 (2022), no. 4, 1023--1090, DOI 10.1007/s11139-022-00623-y. The held PDF is the arXiv preprint, version 1 of 27 August 2021, and the labels cited here are that version's. The arXiv record (https://arxiv.org/abs/2108.12351, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Mangerel develops analogs of the Matomäki-Radziwill theorem for additive functions, approximating the average of an additive function g over a typical short interval (n - h, n] by the corresponding long average. Theorem 1.1 gives, for any additive function g and any integer 10 <= h <= X/100, a bound on the first absolute discrepancy between short and long averages that improves on the trivial Turán-Kubilius bound and decays as h tends to infinity; Theorem 1.4 gives an l^2 (mean-square) analog for the restricted class A_s of additive functions, whose definition rules out the pathology of g(p)/B_g(X) being large on many primes, and rests on a Matomäki-Radziwill variant for divisor-bounded multiplicative functions (Theorem 4.3, from the author's earlier paper). Two families of applications follow. Theorem 1.11 shows that the average gap (1/X) sum |g(n) - g(n-1)| is o(B_g(X)) if and only if the first centered moment (1/X) sum |g(n) - A_g(X)| is o(B_g(X)), with a second-moment version for g in A_s, complementing results of Elliott and Hildebrand. On Erdős's 1946 conjecture (Conjecture 1.6) that an additive function non-decreasing outside a density-zero set B must equal c log n, Corollary 1.7 proves the conjecture for completely additive g satisfying lim F_g(epsilon) = 0 and the stronger sparseness |B(X)| << X/(log X)^(2+delta). Theorem 1.8 shows that any real additive g in A_s with |B(X)| = o(X) is close to lambda(X) log at prime powers, in a weighted mean square, and Theorem 1.9 shows that any real additive g with |B(X)| = o(X) satisfies g(n) = lambda(X) log n - eta(X) + o(B_g(X)) for all but o(X) integers n <= X, with slowly varying parameters. The paper bears on problem 1122 as partial progress on that Erdős conjecture characterizing constant multiples of log n among almost everywhere monotone additive functions.

Source: https://arxiv.org/abs/2108.12351.

Bears on. #1122

Results to transcribe.

• theorem_1_1: For any additive function g and integers 10 <= h <= X/100, the average over n in (X/2, X] of the absolute difference between the short-interval average of g on (n - h, n] and the long average is bounded in terms of B_g(X), improving on the trivial Turán-Kubilius bound by a factor tending to 0 as h tends to infinity.
• theorem_1_4: For additive g in the class A_s and an integer h = h(X) tending to infinity with 10 <= h <= X/100, the mean square of the difference between the short-interval average of g and the long average over n in (X/2, X] is o(B_g(X)^2).
• corollary_1_7: If g is completely additive with lim_{epsilon to 0} F_g(epsilon) = 0 and its set of decrease satisfies |B(X)| << X/(log X)^(2+delta) for some delta > 0, then g(n) = c log n for all n and some constant c, a partial case of Erdős's Conjecture 1.6.
• theorem_1_9: If g is additive with |B(X)| = o(X) then there are slowly varying parameters lambda(X), eta(X) with g(n) = lambda log n - eta + o(B_g(X)) for all but o(X) integers n <= X.
• theorem_1_8: For additive g in A_s with |B(X)| = o(X) there is lambda(X) with |lambda(X)| << B_g(X)/log X such that sum over prime powers p^k <= X of |g(p^k) - lambda log p^k|^2/p^k is o(sum g(p^k)^2/p^k).
• theorem_1_11: For additive g, (1/X) sum_{n<=X} |g(n) - g(n-1)| = o(B_g(X)) if and only if (1/X) sum_{n<=X} |g(n) - A_g(X)| = o(B_g(X)); for g in A_s the same equivalence holds for the corresponding second moments.