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SUMMARY:Krishnaswami Alladi (University of Florida)
DTSTART:20260714T190000Z
DTEND:20260714T195000Z
DTSTAMP:20260710T111542Z
UID:CANT2026/28
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/28/
 ">Duality between prime factors and prime numbers in arithmetic progressio
 ns</a>\nby Krishnaswami Alladi (University of Florida) as part of Combinat
 orial and additive number theory seminar (CANT 2026)\n\nLecture held in Sc
 ience Center in the CUNY Graduate Center (4th floor).\n\nAbstract\nIn 1977
 \, I noticed a duality between the largest and smallest\n 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\n for A
 rithmetic Progressions: \n If $k$ and $\\ell$ are positive\n integers\, wi
 th $1\\le \\ell\\le k$ and $(\\ell\, k)=1$\, then  $$ \n \\sum_{n\\ge 2\, 
 \\\, p(n)\\equiv\\ell(mod\\\,k)}\\frac{\\mu(n)}{n}=\\frac{-1}{\\phi(k)}\, 
 $$ where $\\mu(n)$ is the Mobius function\, $p(n)$ is the\n 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 algebr
 aic  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 ex
 ploit 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 $$ \n \\sum_{n\\ge 2\,\\\, p(n)\\equiv\\ell(mod\\\,k)}\\frac{\\mu(n
 )\\omega(n)}{n}=0. \n $$ 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  dist
 ribution of the largest prime factor. All results are established quantita
 tively. This is  joint work with my PhD student Jason Johnson. Recently\, 
 another PhD student of mine\,  Sroyon Sengupta\, has extended the Alladi-J
 ohnson results to algebraic number fields using the Chebotarev Density The
 orem. \nTowards the end of the talk\, we will briefly mention further join
 t work with Sengupta on consequences of such dualities involving the $k-th
 $ largest and smallest prime factors\, when $k\\ge 3$.\n
LOCATION:https://researchseminars.org/talk/CANT2026/28/
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