Happy new year.
The first talk of the year (on Feb 6, 2025) is a ``Ramanujan Special". This year's speaker is Krishnaswami Alladi. Among other things, he is the founding editor of the Ramanujan Journal. Please note that the talk will be later than usual.
Talk Announcement: The 2025 Ramanujan Special
Title: Speaker: Krishnaswami Alladi (University of Florida, USA)
Title: Speaker: Krishnaswami Alladi (University of Florida, USA)
When: Feb 6, 2025, 6:30 PM- 7:30 PM IST (8 AM EST) (Note special time)
(EST= IST - 10:30)
Where: Zoom: Write to sfandnt@gmail.com for a link
Abstract
In 1977, I noticed two duality identities connecting the smallest
and largest prime factors of integers, and vice-versa, the connection being
provided by the Moebius function. Using this duality, I generalized the famous results of Edmund Landau on the Moebius function which are equivalent to the
Prime Number Theorem. When this duality is combined with the Prime Number Theorem for Arithmetic Progressions, this leads to the striking result that
and largest prime factors of integers, and vice-versa, the connection being
provided by the Moebius function. Using this duality, I generalized the famous results of Edmund Landau on the Moebius function which are equivalent to the
Prime Number Theorem. When this duality is combined with the Prime Number Theorem for Arithmetic Progressions, this leads to the striking result that
$$
\sum_{n\ge 2, \, p(n)\equiv\ell(mod\,k)}\frac{\mu(n)}{n}=-\frac{1}{\phi(k)}\tag1
$$
for all positive integers $k$, where $1\le\ell\le k$, $(\ell, k)=1$, $\mu(n)$ is
the Moebius function, $p(n)$ is the smallest prime factor of $n$, and $\phi(k)$
is the Euler function. In the last few years, this duality and identity (1) have
attracted considerable attention, and extended to the setting of algebraic
number fields by the use of the Chebotarev Density Theorem by several young researchers.
In the 1977 work, I established four general duality identities connecting the
smallest prime factor with the $k$-th largest prime factor, and the $k$-th
smallest prime factor with the largest prime factor, utilizing not just
$\mu(n)$, but also $\omega(n)$, the number the distinct prime factors of $n$, a function first systematically studied by Hardy and Ramanujan in 1917.
Recently, along with my PhD student Jason Johnson, I used the second order
duality together with the Prime Number Theorem for Arithmetic Progressions
to establish
$$
\sum_{n\ge 2, \, p(n)\equiv\ell(mod\,k)}\frac{\mu(n)\omega(n)}{n}=0.\tag2
$$
The proof involves a variety of elementary and analytic techniques, and a study
of the distribution of the second largest prime factor of $n$. Identity (2)
has more recently been extended to the setting of algebraic number fields
using the Chebotarev Density of Theorem by Sroyon Sengupta, another of my
PhD students. Starting from the 1977 results, I will
discuss all the recent developments pertaining to this problem.
