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SUMMARY:Marco Cantarini (University of Perugia\, Italy)
DTSTART:20260718T153000Z
DTEND:20260718T155500Z
DTSTAMP:20260710T111640Z
UID:CANT2026/83
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2026/83/
 ">Averages of the diagonal Elliott-Halberstam problem twisted by the Möbi
 us function with Sobolev and Hölder-Zygmund weights</a>\nby Marco Cantari
 ni (University of Perugia\, Italy) as part of Combinatorial and additive n
 umber theory seminar (CANT 2026)\n\nLecture held in Science Center in the 
 CUNY Graduate Center (4th floor).\n\nAbstract\nRecalling that the so-calle
 d Elliott-Halberstam conjecture twisted\nby the Möbius function $\\mu(n)$
  claims that $$\\sum_{q\\leq N^{\\theta}}\\max_{y\\leq N}\\max_{(a\,q)=1}\
 \left|\\sum_{\\underset{{\\scriptstyle n\\equiv a\\\,\\mod\\\,q}}{n\\leq y
 }}\\Lambda(n)\\mu\\left(N-n\\right)-\\frac{1}{\\varphi\\left(q\\right)}\\s
 um_{n\\leq y}\\Lambda(n)\\mu\\left(N-n\\right)\\right|\\ll\\frac{N}{\\log\
 \left(N\\right)^{A}} \n$$\nfor every $A>0$\, where $0<\\theta<1$ is fixed\
 , and also recalling\nthat the validity of this conjecture\, in combinatio
 n with the validity\nof the classical Elliott-Halberstam for suitable $\\t
 heta$\, proves\nthe binary Goldbach conjecture\, in this talk we analyze w
 eighted average\nvariants of this problem. We will show that\, under Gener
 alized Riemann\nHypothesis\, a weak version of the Gonek-Hejhal conjecture
  and working\nwith weights belonging to the Sobolev space $W^{2\,1}$ or in
  the Hölder-Zygmund\nspaces $\\mathcal{C}^{\\delta}$ for suitable range o
 f $\\delta$\, the\nbound of the average is consistent with the bound of th
 e ``diagonal\nversions'' of this conjecture (that is\, taking $y=N$ and ta
 king\n$n\\equiv N\\mod q)$. In particular\, in the case of weights in Sobo
 lev\nspace\, the consistent upper bound holds for the whole $0<\\theta<1$\
 nand\, in the case of weights in the Hölder-Zygmund class $\\mathcal{C}^{
 \\delta}$\,\nfor $\\theta$ that depends on the choice of $\\delta$ but sti
 ll not\nbelow the $1/2-2\\varepsilon$ threshold.\n
LOCATION:https://researchseminars.org/talk/CANT2026/83/
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