Geodesic
Asymptotic Expansion of Certain Power Series with Multiplicative Coefficients near the Unit Circle
Matematičeskie zametki, Tome 100 (2016) no. 6, pp. 887-899.

Voir la notice de l'article provenant de la source Math-Net.Ru

An asymptotic theorem important for the study of many power series with multiplicative coefficients is proved. Examples of concrete series to which the theorem can be applied are given. It is shown that power series of many classical arithmetic sequences can be expanded in asymptotic series as the variable tends to the roots of unity along the radii of the unit circle.
Keywords: power series, multiplicative function, classical arithmetic function, asymptotics, sum of divisors. 
@article{MZM_2016_100_6_a10,
     author = {O. A. Petruschov},
     title = {Asymptotic {Expansion} of {Certain} {Power} {Series} with {Multiplicative} {Coefficients} near the {Unit} {Circle}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {887--899},
     publisher = {mathdoc},
     volume = {100},
     number = {6},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2016_100_6_a10/}
}
TY  - JOUR
AU  - O. A. Petruschov
TI  - Asymptotic Expansion of Certain Power Series with Multiplicative Coefficients near the Unit Circle
JO  - Matematičeskie zametki
PY  - 2016
SP  - 887
EP  - 899
VL  - 100
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2016_100_6_a10/
LA  - ru
ID  - MZM_2016_100_6_a10
ER  - 
%0 Journal Article
%A O. A. Petruschov
%T Asymptotic Expansion of Certain Power Series with Multiplicative Coefficients near the Unit Circle
%J Matematičeskie zametki
%D 2016
%P 887-899
%V 100
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2016_100_6_a10/
%G ru
%F MZM_2016_100_6_a10
O. A. Petruschov. Asymptotic Expansion of Certain Power Series with Multiplicative Coefficients near the Unit Circle. Matematičeskie zametki, Tome 100 (2016) no. 6, pp. 887-899. http://geodesic.mathdoc.fr/item/MZM_2016_100_6_a10/

[1] G. H. Hardy, J. E. Littlewood, “Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes”, Acta Math., 41:1 (1916), 119–196 | DOI | MR

[2] I. Katai, “On oscillations of number-theoretic functions”, Acta Arith., 13 (1967), 107–122 | MR | Zbl

[3] S. Gerhold, “Asymptotic estimates for some number-theoretic power series”, Acta Arith., 142:2 (2010), 187–196 | DOI | MR | Zbl

[4] W. D. Banks, F. Luca, I. E. Shparlinski, “Irrationality of power series for various number theoretic functions”, Manuscripta Math., 117:2 (2005), 183–197 | DOI | MR | Zbl

[5] S. Wigert, “Sur la série de Lambert et son application à la théorie des nombres”, Acta Math., 41:1 (1916), 197–218 | DOI | MR

[6] O. Petrushov, “On the behavior close to the unit circle of the power series with Möbius function coefficients”, Acta Arith., 164:2 (2014), 119–136 | DOI | MR | Zbl

[7] E. Knepp, Ellipticheskie krivye, Faktorial Press, M., 2004 | Zbl

[8] P. Flajolet, X. Gourdon, P. Dumas, “Mellin transforms and asymptotics: harmonic sums”, Theoret. Comput. Sci., 144:1-2 (1995), 3–58 | DOI | MR | Zbl

[9] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Oxford Univ. Press, Oxford, 1979 | MR | Zbl

[10] H. L. Montgomery, R. C. Vaughan, Multiplicative Number Theory. I. Classical Theory, Cambridge Stud. in Adv. Math., 97, Cambridge Univ. press, Cambridge, 2006 | MR