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DTSTART:20240310T070000
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DTSTART;TZID=America/Toronto:20250128T100000
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URL:https://uwaterloo.ca/pure-mathematics/events/number-theory-seminar-136
SUMMARY:Number Theory Seminar
CLASS:PUBLIC
DESCRIPTION:AKASH SINGHA ROY\, UNIVERSITY OF GEORGIA\n\nResidue-class dist
 ribution and mean values of multiplicative functions\n\nThe distribution o
 f values of arithmetic functions in residue classes\nhas been a problem of
  great interest in elementary and analytic number\ntheory. The analogous q
 uestion commonly studied for multiplicative\nfunctions is the distribution
  of their values in coprime residue\nclasses. In work studying this proble
 m for large classes of\nmultiplicative functions\, Narkiewicz obtained cri
 teria deciding when a\nfamily of such functions is jointly uniformly distr
 ibuted among the\ncoprime residue classes to a fixed modulus. In the first
  part of this\ntalk\, we shall extend Narkiewicz's criteria to moduli that
  are allowed\nto vary in a wide range. Our results are essentially the bes
 t possible\nanalogues of the Siegel-Walfisz theorem in this setting. One o
 f the\nprimary themes behind our arguments is the quantitative detection o
 f a\ncertain \"mixing\" (or ergodicity) phenomenon in multiplicative group
 s\nvia methods belonging to the \"anatomy of integers\"\, but we also rely
 \nheavily on more classical analytic arguments\, tools from arithmetic\nan
 d algebraic geometry\, and from linear algebra over rings.\n\nIn the secon
 d part of this talk\, we shall gain a finer understanding\nof these distri
 butions\, such as the second-order behavior. This shall\nrely on extending
  some of the most powerful known estimates on mean\nvalues of multiplicati
 ve functions (precisely\, the\nLandau-Selberg-Delange method) to a result 
 that is much more uniform\nin certain important parameters. We will see se
 veral applications of\nthis extended result in other interesting settings 
 as well.\n\nThis talk is partially based on joint work with Prof. Paul Pol
 lack.\n\nJoin on Zoom\n[https://uwaterloo.zoom.us/j/98942212227?pwd=huSbGS
 NTP1ODaePFVsXb4FJy6Deite.1MeetingID:98942212227Passcode:112827]
DTSTAMP:20260408T022809Z
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