Duality between prime factors and prime numbers in arithmetic progressions

Abstract: In 1977, I noticed a duality between the largest and smallest prime factors of the integers involving the Mobius function, and used this to establish the following result as a consequence of the Prime Number Theorem for Arithmetic Progressions: If $k$ and $\ell$ are positive integers, with $1\le \ell\le k$ and $(\ell, k)=1$, then $$ \sum_{n\ge 2, \, p(n)\equiv\ell(mod\,k)}\frac{\mu(n)}{n}=\frac{-1}{\phi(k)}, $$ where $\mu(n)$ is the Mobius function, $p(n)$ is the smallest prime factor of $n$, and $\phi(k)$ is the Euler function. In the last decade, several authors have obtained analogues of (1) in the setting of algebraic number fields by using the Chebotarev Density Theorem. Also in 1977, I proved higher order duality identities involving the $k$-th largest and smallest prime factors, facilitated by the Mobius function and $\omega(n)$, the number of distinct prime factors of $n$. In this talk we will exploit the second order duality between the second largest prime factor and the smallest prime factor, to show that if $\ell$ and $k$ are as above, then $$ \sum_{n\ge 2,\, p(n)\equiv\ell(mod\,k)}\frac{\mu(n)\omega(n)}{n}=0. $$ The proof of (2) is more complicated owing to the weight $\omega(n)$, and also because it relies on the distribution of the second largest prime factor which is more subtle compared to the distribution of the largest prime factor. All results are established quantitatively. This is joint work with my PhD student Jason Johnson. Recently, another PhD student of mine, Sroyon Sengupta, has extended the Alladi-Johnson results to algebraic number fields using the Chebotarev Density Theorem. Towards the end of the talk, we will briefly mention further joint work with Sengupta on consequences of such dualities involving the $k-th$ largest and smallest prime factors, when $k\ge 3$.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)