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SUMMARY:Harald Helfgott (Institut de Mathématiques de Jussieu-Paris Rive 
 Gauche and CNRS)
DTSTART:20260514T203000Z
DTEND:20260514T213000Z
DTSTAMP:20260526T085421Z
UID:MITNT/140
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MITNT/140/">
 Optimal bounds for sums of arithmetic functions</a>\nby Harald Helfgott (I
 nstitut de Mathématiques de Jussieu-Paris Rive Gauche and CNRS) as part o
 f MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons Bu
 ilding (building 2).\n\nAbstract\n(joint with Andrés Chirre)\n\nLet $A(s)
  = \\sum_n a_n n^{-s}$ be a Dirichlet series with meromorphic\ncontinuatio
 n. Say we are given information on the poles of $A(s)$ with\n$|\\Im s| \\l
 eq T$ for some large constant $T$. What is the best way to use\nsuch finit
 e spectral data to give explicit estimates for sums $\\sum_{n\\leq\nx} a_n
 $?\n\n  The problem of giving explicit bounds on the Mertens function $M(x
 ) =\n\\sum_{n\\leq x} \\mu(n)$ illustrates how open this basic question wa
 s.\nBounding $M(x)$ might seem equivalent to estimating $\\psi(x) = \\sum_
 {n\\leq\nx} \\Lambda(n)$ or the number of primes $\\leq x$. However\, we h
 ave long had\nfairly good explicit bounds on prime counts\, while bounding
  $M(x)$ remained\na notoriously stubborn problem.\n\n  We prove a sharp\, 
 general result on sums $\\sum_{n\\leq x} a_n n^{-\\sigma}$\nfor $a_n$ boun
 ded\, giving a  optimal way to use information on the poles of\n$A(s)$ wit
 h $|\\Im s|\\leq T$ and no data on the poles above.\n Our bounds on $M(x)$
  are stronger than previous ones by many orders of\nmagnitude. We also giv
 e a sharp result on such sums for a_n non-negative\nand not necessarily bo
 unded\, and apply it to obtain optimal bounds on\npsi(x)-x given finite ve
 rifications of RH.\n\n  Our proofs mix a Fourier-analytic approach in the 
 style of\nWiener--Ikehara with contour-shifting\, using optimal approximan
 ts of\nBeurling--Selberg type as in (Graham--Vaaler\, 1981) and\n(Carneiro
 --Littmann\, 2013). While our approach does not depend on existing\nexplic
 it work in number theory\, our method has an important step in common\nwit
 h work on another problem by (Ramana–Ramare\, 2020).\n
LOCATION:https://researchseminars.org/talk/MITNT/140/
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