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SUMMARY:Harald Helfgott (CNRS/Institut de Math\\' ematiques de Jussieu)
DTSTART:20250520T113000Z
DTEND:20250520T115500Z
DTSTAMP:20260423T005752Z
UID:CANT2025/2
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/2/"
 >Explicit estimates for sums of arithmetic functions\, or the optimal use 
 of finite information on Dirichlet series</a>\nby Harald Helfgott (CNRS/In
 stitut de Math\\' ematiques de Jussieu) as part of Combinatorial and addit
 ive number theory (CANT 2025)\n\nLecture held in CUNY Graduate Center - Sc
 ience Center (4th floor).\n\nAbstract\nLet $F(s) = \\sum_n a_n n^{-s}$ be 
 a Dirichlet series. Say we have an analytic continuation of $F(s)$\, and i
 nformation on the poles of $F(s)$ with $|\\Im s|\\leq T$ for some large co
 nstant $T$. \n What is the best way to use this information to give explic
 it estimates on sums $\\sum_{n\\leq x} a_n$? \n\n  The problem of giving e
 xplicit bounds on the Mertens function $M(x) = \\sum_{n\\leq x} \\mu(n)$ i
 llustrates how open this basic question was.\n  One might think that bound
 ing $M(x)$ is essentially equivalent to estimating $\\psi(x) = \\sum_{n\\l
 eq x} \\Lambda(n)$ or the number of primes $\\leq x$.\n  However\, we have
  long had fairly satisfactory explicit bounds on $\\psi(x)-x$\, whereas bo
 unding $M(x)$ well was a notoriously recalcitrant problem.\n\nWe give an o
 ptimal way to use information on the poles of $F(s)$ with $|\\Im s|\\leq T
 $. In particular\, we give bounds on the Mertens function much stronger th
 an those in the literature\, while also substantially improving on estimat
 es on $\\psi(x)$.\n\n We use functions of "Beurling-Selberg" type -- namel
 y\, an optimal approximant due to Carneiro-Littmann and an optional majora
 nt/minorant due to Graham-Vaaler. Our procedure has points of contact \n w
 ith Wiener-Ikehara and also with work of Ramana and Ramaré\, but does not
  rely on results in the explicit analytic-number-theory literature. \n\n(j
 oint work with Andrés Chirre)\n
LOCATION:https://researchseminars.org/talk/CANT2025/2/
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