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448 HISTORY OF THE THEORY OP NUMBERS. [CHAP, xix R(i), — 1 if a product of an odd number, 0 if # is divisible by the square of a prime of R(i). Let {m} denote a complete set of residues 5^0 of complex integers modulo m. Then the sum of the values of f(x) for all complex integers x relatively prime to a given one n, which are in {m}, equals S/i(d)S/(cfo), where d ranges over all divisors of n in {m}, and x ranges over {m/d}. This is due, for the case of real numbers, to Nazimov167 of Chapter V. Again, 2/*(d!) = 1 or 0 according as norm n is 1 or >1. Also 2/(d) =F(n) implies ^(d)F(n/d) =/(n). J. C. Kluyver33 employed Kronecker's30 identity for special functions / and obtained known results like Scos—=/i(n), II2sin— = ey(n)> f\i 71 where v ranges over the integers <n and prime to n, while 7(71) is Bouga-ief's16 function v(n). P. Fatou34 noted that Merten's cr(ri) does not oscillate between finite limits. E. Landau35 proved that it is at most of the order of ne', where t = — aVlogn. Landau36 noted that Furlan37 made a false use of analysis and ideal theory to obtain a result of Landau's on Merten's25 <r(ri). 0. Meissner37" employed primes pi} qt. For n=TLpiei set Z(n) =![<?/» and Z2(n)=Z{Z(n)J. Then Z(n)=n only if n is Ilpfi or 16 or TLp'<q'*. Next, Z2(n)=n in these three cases and when the exponents e< in n are distinct primes; otherwise, Z2(n)<n. We have [l/Z(n)]=/>i2(n). R. Hackel38 extended the method of von Sterneck28 and obtained various closer approximations, one39 being where o = l, 6, 10, 14, 105; 6 = 2, 3, 5, 7, 11, 13, 385, 1001. W. Kusnetzov40 gave an analytic expression for //(n). K. Knopp160 of Ch. X gave many formulas involving A. Fleck400 generalized At(ra)=Mi(ra) by setting Using the zeta function (12) of Ch. X, and <j>k of Fleck125 of Ch. V, we have 83Verslag. Wiss. Ak. Wetenschappen, Amsterdam, 15, 1906, 423-9. Proc. Sect. Sc. Ak. Wet., 9, 1906, 408-14. 34Acta Math., 30, 1906, 392. 3BRend. Circ. Mat. Palermo, 26, 1908, 250. 36Rend. Circ. Mat. Palermo, 23, 1907,367-373- "Monatshefte Math. Phys., 18, 1907, 235-240. 370Math. Naturw. Blatter, 4, 1907, 85-6. 38Sitzungsber. Ak. Wiss. Wien (Math.), 118,1909, II a, 1019-34. "Sylvester, Messenger Math., (2), 21, 1891-2, 113-120. 40Mat. Sbornik (Math. Soc. Moscow), 27, 1910, 335-9. 400Sitzungsber. Berlin Math. Gesell., 15, 1915, 3-8.