Summing : a faster elementary algorithm

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.