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On some problems involving Hardy’s function

100%
Ivić A.
Open Mathematics
|
2010
|
tom 8
|
nr 6
1029-1040
EN
Some problems involving the classical Hardy function $$ Z\left( t \right) = \zeta \left( {\frac{1} {2} + it} \right)\left( {\chi \left( {\frac{1} {2} + it} \right)} \right)^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} , \zeta \left( s \right) = \chi \left( s \right) \zeta \left( {1 - s} \right) $$, are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.
2
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Upper bounds for the moments of derivatives of Dirichlet L-functions

100%
Sono K.
Open Mathematics
|
2014
|
tom 12
|
nr 6
848-860
EN
In this paper, we give certain upper bounds for the 2k-th moments, k ≥ 1/2, of derivatives of Dirichlet L-functions at s = 1/2 under the assumption of the Generalized Riemann Hypothesis.
3
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Basel Problem – Preliminaries

81%
Korniłowicz A., Pąk K.
Formalized Mathematics
|
2017
|
tom 25
|
nr 2
141-147
EN
In the article we formalize in the Mizar system [4] preliminary facts needed to prove the Basel problem [7, 1]. Facts that are independent from the notion of structure are included here.
4
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Basel Problem

81%
Pąk K., Korniłowicz A.
Formalized Mathematics
|
2017
|
tom 25
|
nr 2
149-155
EN
A rigorous elementary proof of the Basel problem [6, 1] ∑n=1∞1n2=π26 $$\sum\nolimits_{n = 1}^\infty {{1 \over {n^2 }} = {{\pi ^2 } \over 6}} $$ is formalized in the Mizar system [3]. This theorem is item #14 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.
5
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An explicit formula of Atkinson type for the product of the Riemann zeta-function and a Dirichlet polynomial

81%
Ishikawa H., Matsumoto K.
Open Mathematics
|
2011
|
tom 9
|
nr 1
102-126
EN
We prove an explicit formula of Atkinson type for the error term in the asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial. To deal with the case when coefficients of the Dirichlet polynomial are complex, we apply the idea of the first author in his study on mean values of Dirichlet L-functions.
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