A Computational and Statistical Analysis of the Relationship Between Euler’s Totient Function and the Sum-of-Divisors Function
DOI:
https://doi.org/10.70882/josrar.2026.v3i5.309Keywords:
Euler’s Totient Function, Sum-of-Divisors Function, Number Theory, Statistical Analysis, Pearson Correlation, Linear Regression, MATLAB, Arithmetic Functions, Prime FactorizationAbstract
This study examines the statistical relationship between Euler’s Totient Function, ϕ(n), and the Sum-of-Divisors Function, σ(n), using computational and statistical methods implemented in MATLAB. The analysis was conducted on the first 1,000 positive integers, for which both arithmetic functions were computed. Descriptive statistics, covariance, Pearson’s correlation coefficient, statistical significance testing, linear regression, and graphical analysis were employed to investigate the relationship between the two functions. The results showed that the mean value of ϕ(n) was 304.1920 whereas the mean value of σ(n) was 823.0810. The Pearson's correlation coefficient was calculated and there was a positive but relatively weak-to-moderate linear association, r=0.3443. There was a significant relation between the two (p < 0.001). A simple linear regression was used to get the model as σ(n)=0.8737+ϕ(n)=557.3142 and the coefficient of determination R 2 = 0.1185. This means that there is about 11.85% of the variance in the Sum-of-Divisors Function accounted for by the simple linear relationship with Euler's Totient Function. The results demonstrate that there is a positive and statistically significant relationship between ϕ(n) and σ(n); however, the relationship is not strongly linear. This may be attributed to the distinct arithmetic properties measured by the two functions and their different dependencies on the prime factorization and divisor structure of integers. Overall, the study highlights the value of computational and statistical methods implemented in MATLAB for exploring relationships between classical arithmetic functions.
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Copyright (c) 2026 Hind Mahdi Zwain (Author)
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