Binomial Coefficients and the Distribution of the Primes

TL;DR

This paper derives formulas linking binomial coefficients to prime distribution, providing new identities and applications that enhance understanding of prime number patterns and their relation to binomial coefficients.

Contribution

It introduces a novel formula connecting prime factors of binomial coefficients with prime distribution functions and generalizes known logarithmic identities.

Findings

Derived a formula for prime factors in binomial coefficients.

Established identities relating prime distribution to binomial coefficients.

Connected prime distribution with classical logarithmic series.

Abstract

Let omega(n) be the number of distinct prime factors dividing n and m > n natural numbers. We calculate a formula showing which prime numbers in which intervals divide a given binomial coefficient. From this formula we get an identity omega(binom(nk)(mk))=sum_i (pi(k/b(i))- pi(k/a(i))) + O(sqrt(k)). Erdoes mentioned that omega(binom(nk)(mk))= log n^n/(m^m (n-m)^(n-m)) k/log k + o(k/log k). As an application of the above identities, we conclude some well-known facts about the distribution of the primes and deduce for all natural numbers k an expression (also well-known) log k = sum_i a_k(i) which generalizes log 2 = sum_i^(infty) (-1)^(j+1) / j.