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I am looking for references on conjectures or heuristics concerning cancellations of the Möbius function in very short intervals, namely how small is it believed one can take $H$ so that

$$\sum_{X \le n \le X+H} \mu(n) = o(H)$$

holds for every $X$ (rather than almost every $X$), especially if there are conjectures that go beyond $H = O(X^{1/2+\epsilon})$ from the Riemann Hypothesis.

In the case of short sums of the Von Mangoldt function (that is, counting primes in small intervals) the expected heuristic is known to fail for $H = (\log X)^A$ and conjectured to be true for $H = X^{\epsilon}$. I am looking for something similar for the mobius function.

Thanks in advance

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    $\begingroup$ You can take a look at cs.uleth.ca/~nathanng/RESEARCH/mobiusshort.pdf . Certainly one should expect cancellation in every interval of length $X^{\epsilon}$ (maybe even log powers). You can see the $X^{\epsilon}$ conjecture is implicit in Conjecture B of Ng. $\endgroup$
    – Lucia
    May 6, 2018 at 1:08

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