History Of The Theory Of Numbers - I

CHAP.X] SUM AND NUMBER OF DlYISORS. 283
account of the order of the factors. The number of pairs of relatively prime integers £, t\ "for which fr^n is therefore
For the preceding C and T(ri), it is proved that
where m is of the order of magnitude of ns, 6>y/2, while 7 is determined by Ss^=l (s = 2 to co). Moreover, 5P(w) is the number of pairs of integers x, y for which xy^n. He noted that
and that the difference between this sum and ?r2n2/12 is of an order of magnitude not exceeding n loge n.
G. H. Burhenne18 proved by use of infinite series that
"
and then expressed the result as a trigonometric series.
V. Bouniakowsky19 changed x into x8 in (6), multiplied the result by and obtained
Thus every number 8m+4 is a sum of four odd squares in cr(2m+l) ways. By comparing coefficients hi the logarithmic derivative, we get
(8) (l2-2m+l)(r(2m+l)4-(32--2w--l)(r(2w-l) + (52--2m-~5)(r(2m---5)
+ ...=o,
in which the successive differences of the arguments of a are 2, 4, 6, 8, .... For any integer N,
(9) (l2-AT)<r(#) + (32-]^^
+ ...=0,
where cr(0) , if it occurs, means ]V/6. It is proved (p. 269) by use of Jacobi's10 result for s3 that
"Archiv Math. Phys., 19, 1852, 442-9.
"M&n. Ac. Sc, St. P^tersbourg (Sc. Math. Phya.), (Q), 4, 1850, 25^-295 (presented, 1848). Extract in Bulletin, 7, 170 and 15, 1857, 267-9.