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Related Experiment Video

Updated: Oct 14, 2025

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Highly composite polynomials and the maximum order of the divisor function in .

Ardavan Afshar¹

¹Department of Mathematics, University College London, 25 Gordon Street, London, England.

The Ramanujan Journal

|November 1, 2021

Summary

Researchers explored polynomial analogues of highly composite numbers. They developed a novel family of polynomials to efficiently compute the maximum order of the divisor function, improving upon integer-based methods.

Keywords:

Arithmetic of polynomials over finite fields Divisor function Highly composite numbers

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Area of Science:

Number Theory
Algebraic Geometry

Background:

Srinivasa Ramanujan studied highly composite numbers and the maximum order of the divisor function in integers.
Understanding the distribution and properties of numbers with many divisors is crucial in number theory.

Purpose of the Study:

To investigate analogues of highly composite numbers within the domain of polynomials.
To compute the maximum order of the divisor function for these polynomial analogues with improved accuracy.

Main Methods:

Determining a specific family of 'highly composite' polynomials.
Utilizing this polynomial family to calculate the logarithm of the maximum divisor function's order.

Main Results:

A family of highly composite polynomials was identified.
The logarithm of the maximum divisor function was computed for polynomials up to a certain degree with a small constant error.

Conclusions:

The study successfully established polynomial analogues of highly composite numbers.
The developed method offers a more accurate computation of the divisor function's maximum order in polynomials compared to integers, even under the assumption of the Riemann Hypothesis.