Exposé Bourbaki 1161 : The Riemann zeta function in short intervals after Najnudel, and Arguin, Belius, Bourgade, Radziwiłł, and Soundararajan

Astérisque ◽  
2020 ◽  
Vol 422 ◽  
pp. 391-414
Author(s):  
Adam Harper
2021 ◽  
Vol 49 (6) ◽  
Author(s):  
Louis-Pierre Arguin ◽  
Frédéric Ouimet ◽  
Maksym Radziwiłł

Author(s):  
Antanas Laurincikas

We consider the approximation of analytic functions by shifts of the Riemann zeta-function ?(s+ikh) with fixed h > 0 when positive integers k run over the interval [N,N+M], where N1/3(logN)26=15 ? M ? N, and prove that those k have a positive lower density as N ? ?. The same is true for some compositions. Two types of h > 0 are discussed separately.


2009 ◽  
Vol 85 (99) ◽  
pp. 1-17 ◽  
Author(s):  
Aleksandar Ivic

It is proved that, for T? ? G = G(T)? ??T, ?T2T(I1(t+G,G)- I1(t,G))2 dt = TG ?aj logj (?T/G)+ O?(T1+? G1/2+ +T1/2?G? with some explicitly computable constants aj(a3>0)where, for fixed K ? N, Ik(t,G)= 1/?? ? ? -? |?(1/2 + it + iu)|2k e -(u/G)?du. The generalizations to the mean square of I1(t+U,G)-I1(t,G) over [T,T+H] and the estimation of the mean square of I2(t+ U,G) - I2(t,G) are also discussed.


Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1936
Author(s):  
Antanas Laurinčikas ◽  
Darius Šiaučiūnas

Let 0<γ1<γ2<⋯⩽γk⩽⋯ be the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function ζ(s). Using a certain estimate on the pair correlation of the sequence {γk} in the intervals [N,N+M] with N1/2+ε⩽M⩽N, we prove that the set of shifts ζ(s+ihγk), h>0, approximating any non-vanishing analytic function defined in the strip {s∈C:1/2<Res<1} with accuracy ε>0 has a positive lower density in [N,N+M] as N→∞. Moreover, this set has a positive density for all but at most countably ε>0. The above approximation property remains valid for certain compositions F(ζ(s)).


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