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On the error term for the counting functions of finite Abelian groups. (English) Zbl 0763.11039

Let \(a(n)\) denote the number of non-isomorphic Abelian groups with \(n\) elements and let \[ \Delta(x):=\sum_{n\leq x} a(n)-\sum^ 9_{j=1}\underset{s=1/j}{\text{Res}}F(s) x^ s s^{-1}, \]
\[ \Delta_ 1(x):=\sum_{mn\leq x} a(m)a(n)-\sum^ 5_{j=1}\underset{s=1/j}{\text{Res}}F^ 2(s) x^ s s^{-1}, \] where \(F(s)=\zeta(s)\zeta(2s)\zeta(3s)\dots\). Thus \(\Delta_ 1(x)\) is the error term in the asymptotic formula for the sum \(\sum t(G)\), where \(t(G)\) denotes the number of direct factors of an Abelian group \(G\), and summation is extended over all Abelian groups whose orders do not exceed \(x\). By using the complex integration technique and power moments of \(\zeta(s)\) [see Chapter 8 of the author’s monograph, “The Riemann zeta- function” (1985; Zbl 0556.10026)] it is proved that \[ \int^ X_ 1\Delta(x)\;dx\ll_ \varepsilon\;X^{11/10+\varepsilon},\quad\int^ X_ 1\Delta_ 1(x)\;dx\ll_ \varepsilon\;X^{7/6+\varepsilon}, \]
\[ \int^ X_ 1\Delta^ 2_ 1(x)\;dx=\Omega(X^{3/2}\log^ 4 X),\quad\int^ X_ 1\Delta^ 2_ 1(x)\;dx\ll_ \varepsilon\;X^{8/5+\varepsilon}. \] Moreover, if \(\int^ T_ 1|\zeta(1/2+\text{it})|^ 8 dt\ll_ \varepsilon\;T^{1+\varepsilon}\) holds, then \(\int^ X_ 1\Delta^ 2_ 1(x)dx\ll_ \varepsilon\;X^{3/2+\varepsilon}\).
Reviewer: A.Ivić (Beograd)

MSC:

11N45 Asymptotic results on counting functions for algebraic and topological structures
11N37 Asymptotic results on arithmetic functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)

Citations:

Zbl 0556.10026
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References:

[1] Cohen, E.: On the average number of direct factors of a finite Abelian group. Acta Arith.6, 159–173 (1960). · Zbl 0113.25305
[2] Heath-Brown, D. R.: The number of Abelian groups of order at mostx. Proc. ”Journées Arithmétiques”, Luminy 1989. Astérisque198–200, 153–163 (1991).
[3] Ivić, A.: The Riemann Zeta-Function. New York: J. Wiley & Sons. 1985. · Zbl 0556.10026
[4] Ivić, A.: The number of finite non-isomorphic Abelian groups in mean square. Hardy-Ramanujan Journal9, 17–23 (1986). · Zbl 0666.10027
[5] Ivić, A.: The general divisor problem. J. Number Theory27, 73–91 (1987). · Zbl 0619.10040 · doi:10.1016/0022-314X(87)90053-9
[6] Ivić, A., Quellet, M.: Some new estimates in the Dirichlet divisor problem. Acta Arith.52, 241–253 (1989). · Zbl 0619.10041
[7] Krätzel, E.: On the average number of direct factors of a finite Abelian group. Acta Arith.11, 369–379 (1988). · Zbl 0633.10044
[8] Krätzel, E.: Lattice Points. Berlin: VEB Deutscher Verlag der Wissenschaften. 1988.
[9] Kuznetsov, N. V. Sums of Kloosterman sums and the eighth power moment of the Riemann zeta-function. In: Number Theory and Related Topics. pp. 57–117. Tata Institute (Bombay). Oxford University Press. 1989. · Zbl 0745.11040
[10] Liu, H.-Q.: On the number of Abelian groups of a given order. Acta Arith.56, 261–277 (1991). · Zbl 0737.11024
[11] Menzer, H.: Exponentialsummen und verallgemeinerte Teilerprobleme. Dissertation B. Jena: Friedrich - Schiller Universität. 1991.
[12] Menzer, H., Seibold, R.: On the average number of direct factors of a finite Abelian group. Monats. Math.110, 63–72 (1990). · Zbl 0731.11054 · doi:10.1007/BF01571277
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