More than five-twelfths of the zeros of ζ documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$zeta $$end{document} are on the critical line

摘要

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form ( μ ? Λ 1 ? k 1 ? Λ 2 ? k 2 ? ? ? Λ d ? k d ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$(mu star Lambda _1^{star k_1} star Lambda _2^{star k_2} star cdots star Lambda _d^{star k_d})$$end{document} is computed unconditionally by means of the autocorrelation of ratios of ζ documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$zeta $$end{document} techniques from Conrey et al. (Proc Lond Math Soc (3) 91:33–104, 2005), Conrey et al. (Commun Number Theory Phys 2:593–636, 2008) as well as Conrey and Snaith (Proc Lond Math Soc 3(94):594–646, 2007). This in turn allows us to describe the combinatorial process behind the mollification of ζ ( s ) + λ 1 ζ ′ ( s ) log T + λ 2 ζ ′ ′ ( s ) log 2 T + ? + λ d ζ ( d ) ( s ) log d T , documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$egin{aligned} zeta (s) + lambda _1 rac{zeta '(s)}{log T} + lambda _2 rac{zeta ''(s)}{log ^2 T} + cdots + lambda _d rac{zeta ^{(d)}(s)}{log ^d T}, end{aligned}$$end{document} where ζ ( k ) documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$zeta ^{(k)}$$end{document} stands for the k th derivative of the Riemann zeta-function and { λ k } k = 1 d documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${lambda _k}_{k=1}^d$$end{document} are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (Res Number Theory 4:9, 2018), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.