More than five-twelfths of the zeros of $$\zeta $$ are on the critical line

Abstract

Dedicated to Brian Conrey on the occasion of the 30th anniversary of his 'Two-fifths' paper.Abstract. The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form (µ Λ k 1

Zeros of $\mathrm{GL}_2$ $L$-functions on the critical line

“…Conrey [10] refined Levinson's ideas to prove that κ > 2 5 . Most recently, Pratt, Robles, Zaharescu, and Zeindler [19] proved that κ > 0.41729, and this seems to be close to the limit of Levinson's method. The Riemann hypothesis asserts that all of the nontrivial zeros lie on the line Re(s) = 1 2 , which would imply that κ = 1.…”

Maxima of log-correlated fields: some recent developments

“…Conjecture 1.1 (Riemann hypothesis). Re(ρ n ) = 1 2 ∀ n. The proportion of non-trivial Riemann zeros that lie on the critical line is currently known to be at least 0.41729 ('more than 5/12') [126]. This improves a result of Conrey [49], which gave 'more than 2/5' of zeros on the critical line, and the subsequent improvement of Bui et al [36] to 'more than 41%'.…”

“…In [87], under the Riemann hypothesis, Harper proves that there exists a set H of measure at least 0.99 with H ⊆ [T, T + 1] such that (126) log |ζ(…”

Riemann Hypothesis Proof using Balazard, Saias and Yor integral

“…Shaoji Feng [7], in 2012 proved that atleast 41.28 % of the zeros of Riemann zeta function are on the critical line. Pratt et al [8] in 2020 proved that more than five-twelfths of the zeros are on the critical line.…”