A missing sequence from group theory?
N. J. A. Sloane
njas at research.att.com
Wed Nov 17 22:23:32 CET 2004
To MathFun and SeqFans:
A possibly new sequence? Could someone calculate it?
Let t(G) = no. of unitary factors of the abelian group G, and let
T(n) = Sum t(G)
taken over all abelian groups of order <= n.
There are several papers giving estimates for T(n), e.g.
T(x) = c_1 x (log x + 2 gamma - 1) + c_2 x + ...
My question is, how does the sequence T(n) begin?
References and definitions follow, from MSN.
NJAS
MR1850168 Zhai, Wen Guang On the average number of unitary factors of
finite abelian groups. II [MR1813450 (2001j:11092)]. (Chinese) Acta
Math. Sinica 44 (2001), no. 4, 667--672. 11N45
MR1813450 (2001j:11092) Zhai, Wenguang On the average number of unitary
factors of finite abelian groups. II. Acta Math. Sin. (Engl. Ser.) 16
(2000), no. 4, 549--554. (Reviewer: R. C. Baker) 11N45 (11L07 20K01)
MR1641042 (99e:11124) Zhai, Wenguang; Cao, Xiaodong On the average
number of unitary factors of finite abelian groups. Acta Arith. 85
(1998), no. 4, 293--300. (Reviewer: R. C. Baker) 11N45
MR1613290 (99f:11124) Wu, J. On the average number of unitary factors
of finite abelian groups. Acta Arith. 84 (1998), no. 1, 17--29.
(Reviewer: R. C. Baker) 11N45 (11L07 20K01)
1225427 (94k:11108) Schmidt, Peter Georg(D-MRBG) Zur Anzahl unitrer
Faktoren abelscher Gruppen. (German) [On the number of unitary factors
in abelian groups] Acta Arith. 64 (1993), no. 3, 237--248.
Part of this review follows:
Let $\scr A$ denote the set of all abelian groups. Under the operation
of direct product, $\scr A$ is a semigroup with identity element $E$,
the group with one element. $G_1$ and $G_2$ are relatively prime
(German: teilerfremd) if the only common direct factor of $G_1$ and
$G_2$ is $E$. We say that $G_1$ and $G_2$ are unitary factors of $G$ if
$G=G_1\times G_2$ and $G_1,G_2$ are relatively prime. Let $t(G)$ denote
the number of unitary factors of $G$, and let $T(x)=\sum_{G\in {\scr
A}, |G|\le x} t(G).$ There are constants $A,B,$ and $C$ such that
$\Delta(x)\colon =H(x)-Ax\log x - Bx -C\sqrt x=o(\sqrt x).$ In this
paper, the author shows that $\Delta(x)\ll x^{3/8} \log^{7/2}x.$ The
previous best estimate for $\Delta(x)$ had been $\Delta(x)\ll x^{31/82}
\log^2 x$; this was given by H. Menzer \ref["Exponentialsummen und
verallgemeinerte Teilerprobleme", Habilitationsschrift,
Friedrich-Schiller-Univ., Jena, 1992; per bibl.].
Further references from the Wu (1998) paper:
Many accented letters have been dropped!
# R. C. Baker and G. Harman, Numbers with a large prime factor, Acta
Arith. 73 (1995), 119--145. MR1358192 (97a:11138)
# E. Cohen, On the
average number of direct factors of a finite abelian group, ibid. 6
(1960), 159--173. MR0118764 (22 #9535)
# E. Fouvry and H. Iwaniec,
Exponential sums with monomials, J. Number Theory 33 (1989), 311--333.
MR1027058 (91b:11097)
# S. W. Graham and G. Kolesnik, Van der Corput's
Method of Exponential Sums, Cambridge Univ. Press, Cambridge, 1991.
MR1145488 (92k:11082)
# C. H. Jia, The distribution of square-free
numbers, Sci. China Ser. A 36 (1993), 154--169. MR1223084 (94f:11095)
# G. Kolesnik, On the number of abelian groups of a given order, J. Reine
Angew. Math. 329 (1981), 164--175. MR0636451 (83b:10055)
# E. Krtzel,
On the average number of direct factors of a finite abelian group, Acta
Arith. 51 (1988), 369--379. MR0971087 (89m:11091)
# H.-Q. Liu, The
greatest prime factor of the integers in an interval, ibid. 65 (1993),
302--328. MR1259341 (95d:11117)
# H.-Q. Liu, On some divisor problems,
ibid. 68 (1994), 193--200. MR1305200 (96a:11089)
# H.-Q. Liu, Divisor
problems of 4 and 3 dimensions, ibid. 73 (1995), 249--269. MR1364462
(96k:11118)
# P. G. Schmidt, Zur Anzahl unit\"arer Faktoren abelscher
Gruppen, ibid. 64 (1993), 237--248. MR1225427 (94k:11108)
# P. Sargos
and J. Wu, Multiple exponential sums with monomials and their
applications in number theory, Prpublications 97/ n37 de l'Institut lie
Cartan, Universit Henri Poincar (Nancy 1).
# J. Wu, On the
distribution of square-full and cube-full integers, Monatsh. Math., to
appear. cf. MR1657818 (2000a:11125)
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