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SUMMARY:Maksym Radziwill (New York University)
DTSTART:20260714T183000Z
DTEND:20260714T185500Z
DTSTAMP:20260710T111708Z
UID:CANT2026/27
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/27/
 ">Exponential sums over primes</a>\nby Maksym Radziwill (New York Universi
 ty) as part of Combinatorial and additive number theory seminar (CANT 2026
 )\n\nLecture held in Science Center in the CUNY Graduate Center (4th floor
 ).\n\nAbstract\nA classical result of Vinogradov shows that\, for any $\\a
 lpha$ with $$\n\\Big | \\alpha - \\frac{a}{q} \\Big | \\leq \\frac{1}{q^2}
  \\ \, \\ q \\leq x^{1/2}\,\n$$ \nand for any $\\varepsilon > 0$\, we have
 \, $$\n\\Big | \\sum_{p \\leq x} e^{2\\pi i \\alpha p} \\Big | \\leq C(\\v
 arepsilon) x^{\\varepsilon} \\cdot \\Big ( \\frac{x}{\\sqrt{q}} + x^{4/5} 
 \\Big ).\n$$ \nwith $C(\\varepsilon) > 0$ a constant depending only on $\\
 varepsilon$.\nThis has resisted improvements for the past 80 years\, beyon
 d\nrefinements to the $x^{\\varepsilon}$ term. The $x / \\sqrt{q}$ term ca
 nnot be improved without eliminating the existence of a Siegel zero. I'll 
 discuss joint work with James Maynard and Mayank Pandey\, in which we redu
 ce the exponent $4/5$ appearing in $x^{4/5}$ to $19/24$\, which should hav
 e various applications to additive problems related to primes.\n
LOCATION:https://researchseminars.org/talk/CANT2026/27/
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