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Full text of "History Of The Theory Of Numbers - I"

446                    HISTORY OF THE THEORY OP NTJMBERS.           [CHAP, xix
of ni,..., n,.   Then, if a ranges over those divisors d which are divisible by no one of v\,..., *v, chosen from ni}..., na,
The left member equals F(mt N, c) constructed for the numbers other than PI,.. ., v,' of the set wi,..., n.   For $'=s, we have
/(, AT, ) =F(m, AT, e)-^([f]- f' ) +   - 
The latter becomes the series in Bachmann20 when w=, 2V=0, =1, while nu n2,... are primes.
F. Mertens25 considered <r(n) =/i(l)+jLt(2) +. -  +M(W) and proved that
- 2 /
r, ==!
By means of a table (pp. 781-830) of the values of v(n) and /i(n) for ?i< 10000, it is verified that \<r(n) \< Vn for Kn< 10000.
D. von Sterneck26 verified the last result up to 500 000, and for 16 larger values under 5 million.
A. Berger27 noted that, if g(m)g(ri) =g(mri), g(I) = l,
where d ranges over all divisors of n, p over the prime divisors of n.   li 2g(m) is absolutely convergent,
where p ranges over all primes.
D. von Sterneck28 noted that, if 6(x) ^ 1 for every x and if
then
In particular, |S/x(A;)|<8+n/9.
D. F. Seliwanov29 gave Dedekind's formula with application to <(n). H. von Koch29a defined /*(&) by use of infinite determinants.
^Sitzungsber. Ak. Wiss. Wien (Math.), 106, Ila, 1897, 761-830.                                         ""
M/6w?., 835-1024; 110, Ila, 1901, 1053-1102; 121, Ila, 1912, 1083-96; Proc. Fifth Intern. Congress Math., 1912, I, 341-3.
"Ofversigt Vetenskaps-Akad. Forhand., Stockholm, 55, 1898, 579-618. "Monatshefte Math. Phys., 9, 1898, 43-5. "Math. Soc. St. P^tersbourg, 1899, 120. 29aOfversigt K. Vetensk.-Akad. Forhand., Stockholm, 57, 1900, 659-68.