Abstract
Let \({{\mathcal {H}}}= \{h_1, \ldots , h_k\}\) be a fixed set of k distinct non-negative integers. We show that Bombieri–Vinogradov type theorems for a certain class of functions f in arithmetic progressions can be extended to the product \(f(n) \mu ^2(n+h_1) \cdots \mu ^2(n+h_k)\), up to almost the same level of distribution as that for f. As an application of this result, we show equidistribution of tuples of squarefree integers in arithmetic progressions with a level of distribution up to 2 / 3. This generalizes a result of Orr (J Number Theory 3:474–497, 1971). We also formulate arithmetic progression analogues of the Chowla conjecture on correlations of the Möbius function. We are able to prove this for some special cases, thereby generalizing a result of Siebert and Wolke (Math Z 122(4):327–341, 1971). We also show the relevance of such conjectures to the twin prime problem.
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Acknowledgements
I am grateful to Professor Ram Murty for encouraging this work, and Professors J. Friedlander, H. Iwaniec and Greg Martin for their valuable remarks when these results were presented in a number theory session at the Mathematical Congress of the Americas (MCA), Montreal. I also thank Arindam Roy for comments on a previous version of this paper. I especially thank the referee for very helpful and detailed comments which have substantially improved the readability of this paper.
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Vatwani, A. Variants of equidistribution in arithmetic progression and the twin prime conjecture. Math. Z. 293, 285–317 (2019). https://doi.org/10.1007/s00209-018-2177-z
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DOI: https://doi.org/10.1007/s00209-018-2177-z