Weighted value distributions of the Riemann zeta function on the critical line
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Abstract We prove a central limit theorem for log | ζ ( 1 2 + i t ) | {\log\lvert\zeta(\frac{1}{2}+it)\rvert} with respect to the measure | ζ ( m ) ( 1 2 + i t ) | 2 k d t {\lvert\zeta^{(m)}(\frac{1}{2}+it)\rvert^{2k}\,dt} ( k , m ∈ ℕ {k,m\in\mathbb{N}} ), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure | ζ ( 1 2 + i t + i α ) | 2 k d t {\lvert\zeta(\frac{1}{2}+it+i\alpha)\rvert^{2k}\,dt} , with α ∈ ( - 1 , 1 ) {\alpha\in(-1,1)} . Finally, we prove unconditionally the analogue result in the random matrix theory context.
2004 ◽
Vol 109
(2)
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pp. 240-265
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Keyword(s):
Generalized Wirtinger inequalities, random matrix theory, and the zeros of the Riemann zeta-function
2002 ◽
Vol 97
(2)
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pp. 397-409
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2008 ◽
Vol 128
(10)
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pp. 2836-2851
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2010 ◽
Vol 467
(2128)
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pp. 1073-1100
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2000 ◽
Vol 456
(2003)
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pp. 2611-2627
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2016 ◽
Vol 472
(2194)
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pp. 20160548
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2003 ◽
Vol 36
(12)
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pp. 2907-2917
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2015 ◽
Vol 11
(07)
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pp. 2087-2107
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