Geodesic
Note on the mean absolute value theorem for the Dirichlet's $L$-function in the critical stripe
Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 67-75

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In the paper we are continued investigations on a generalization and a improvement of the R. T. Turganaliev's result by the deduction of the asymptotic formula for the mean-value of the Rieman's zeta-function in the critical stripe with the rest term, having the power in the reduction. We are found the asymptotics of Dirichlet's $L$-function in the critical stripe, which improves the R. T. Turganaliev's theorem on the zeta-function for all values of the real part ($1/2\mathrm{Re}\, s\leq 1$). This result are got for the account of the different using of estimations of trigonometric sums on the base of the second derivative in the exponent.
Keywords: Dirichlet's characters, Dirichlet's functions, the zeta-sum twisted together with the Dirichlet's character. 
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     author = {L. G. Arkhipova and V. N. Chubarikov},
     title = {Note on the mean absolute value theorem for the {Dirichlet's} $L$-function in the critical stripe},
     journal = {\v{C}eby\v{s}evskij sbornik},
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     publisher = {mathdoc},
     volume = {22},
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     year = {2021},
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     url = {http://geodesic.mathdoc.fr/item/CHEB_2021_22_1_a4/}
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L. G. Arkhipova; V. N. Chubarikov. Note on the mean absolute value theorem for the Dirichlet's $L$-function in the critical stripe. Čebyševskij sbornik, Tome 22 (2021) no. 1, pp. 67-75. http://geodesic.mathdoc.fr/item/CHEB_2021_22_1_a4/