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Harald Andrés Helfgott1,2, Lola Thompson3

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This study introduces a novel elementary algorithm for computing the Möbius function, achieving the first exponent improvement since 1985. Space complexity can be reduced at the cost of increased computation time.

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Area of Science:

  • Number Theory
  • Computational Mathematics
  • Algorithm Analysis

Background:

  • The computation of the Möbius function is a fundamental problem in number theory.
  • Existing elementary algorithms for computing the Möbius function have not seen improvements in their time complexity exponent since 1985.
  • Space-time trade-offs in algorithmic computation are a critical area of study.

Purpose of the Study:

  • To present a new elementary algorithm for computing the Möbius function with improved time complexity.
  • To explore the possibility of reducing space consumption for this computation.
  • To analyze the trade-offs between time and space complexity for the proposed algorithm.

Main Methods:

  • Development of a novel elementary algorithm for computing the Möbius function.
  • Bitwise analysis of the algorithm's time complexity, measured as O(x).
  • Investigation of space reduction techniques, potentially utilizing methods from Helfgott (2020).

Main Results:

  • The new elementary algorithm achieves a time complexity of O(x), representing the first improvement in the exponent since 1985.
  • A space-optimized version of the algorithm is presented, reducing space complexity to O(x).
  • The space-optimized version results in an increased time complexity of O(x).

Conclusions:

  • A significant advancement in the computational efficiency of elementary algorithms for the Möbius function has been achieved.
  • The study demonstrates a viable trade-off between time and space complexity, offering flexibility in computational resource allocation.
  • This work provides new tools for number-theoretic computations, with potential implications for related fields.