On the Möbius function in all short intervals

  • Kaisa Matomäki

    University of Turku, Finland
  • Joni Teräväinen

    University of Oxford, UK; University of Turku, Finland
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We show that, for the Möbius function , we have

for any . This improves on a result of Motohashi and Ramachandra from 1976, which is valid for . Motohashi and Ramachandra’s result corresponded to Huxley’s exponent for the prime number theorem in short intervals. The main new idea leading to the improvement is using Ramaré’s identity to extract a small prime factor from the -sum. The proof method also allows us to improve on an estimate of Zhan for the exponential sum of the Möbius function as well as some results on multiplicative functions and almost primes in short intervals.

Cite this article

Kaisa Matomäki, Joni Teräväinen, On the Möbius function in all short intervals. J. Eur. Math. Soc. 25 (2023), no. 4, pp. 1207–1225

DOI 10.4171/JEMS/1205