Todd.MolnarAffiliation: University of Florida Title Of Talk: Some Analytic Proofs of Alladi Density Abstract: For relatively prime $(\ell,m)=1$ the following sum was first evaluated in a 1977 paper by K. Alladi: $$ \sum_{\substack{n\geq2\\ \t\tp_{1}(n)\equiv\ell\text{(mod $m$)}}}\frac{\mu(n)}{n}=-\frac{1}{\phi(m)}, $$ where $p_{1}(n)=\min\{p:p|n\}$, $\phi(n)$ is Euler's $\phi\text{-function}$, and $\mu(n)$ is the Moebius function. The term $-1/\phi(m)$ on the right of the above sum has been referred to by some authors as "Alladi density". The original proof of this result utilizes elementary techniques but assumes certain results (such as Walfisz's strong form of the prime number theorem for arithmetic progressions) which at present can only be obtained by analytic methods. Moreover, this proof (and all subsequent demonstrations) makes use of a duality principal and is, in a certain sense, indirect. In this talk, we will present several new analytic proofs of the above result. The first demonstration gives a direct proof which appears to be novel in that it does not rely on the previously mentioned duality.
WARNING: This page contains MATH-JAX
Last update made Tue Mar 10 21:28:05 CDT 2026.
|