Z(1)Dci,(K,Rn/D) =

VOL. 402 1954 ERRATA: C. A. NICOL 363

6 R. Sager and S. Granick, Ann. New York Acad. Sci., 56, 831 (1953). 7Medium II (pH 6.2) contains, in gm/I: sodium acetate*3H20, 2 gm.; NaH2PO4H20, 3.67 gm.; K2HPO4, 1.15 gm.; NH4NO3, 0.3 gm.; MgSO47H20 0.3 gm.; CaCI2, 0.04 gm.; FeCl,3- 6H20, 0.01 gm.; sodium citrate-2H20, 0.5 gm.; trace-metal solution, 10 ml. Trace-metal stock solution contains, in mg/I: H3BO3, 100 mg.; ZnSO4 7H20, 100 mg.; MnSO44H20, 40 mg.; CoCI2 6H20, 20 mg.; NaMoO4-2H20, 20 mg.; CuSO4, 4 mg. 8 K. Mather and G. H. Beale, J. Genetics, 43, 1 (1942). 9 B. Ephrussi, H. Hottinguer, and J. Tavlitski, Ann. Inst. Pasteur, 76, 419 (1949). 10 M. B. Mitchell and H. K. Mitchell, these PROCEEDINGS, 38, 442 (1952).

ERRATA: On Restricted Partitions and a Generalization of the Euler so Number and the Moebius Function In the article of the foregoing title appearing in these PROCEEDINGS, 39, 963-968 (1953), the following corrections should be made: (1) Relation (8), page 965, should read, If n is odd, -)d¢b(k,1~ n/) din n°- if (k, n) =dn If n is even, n k n/ Z (1)dci,(k,rn/d) = d (n 0 otherwise (2) Relation (11), page 965, should read, E (d- 1)(_)da-1 (k, n/d) = 0 mod. (n), where a and ,B are the integral solutions of (n/d) a + g? = s such that 0 < a < d, 0 < ,BT. S., and Taussky, Olga, "Pairs of Matrices with Property L," Trans. Amer. Math. Soc., 73, 108-114, (1952).

ON RESTRICTED PARTITIONS AND A GENERALIZATION OF THE EULER p NUMBER AND THE MOEBIUS FUNCTION BY CHARLES A. NICOL UNIVERSITY OF TEXAS Communicated By H. S. Vandiver, June 23, 1953 Introduction.-In the present paper we shall treat the function n-I (z, x) III (z-x8) (1) s=1 mainly from an arithmetic standpoint. Functions of this type have been studied extensively in the theory of partitions of positive integers' where z is replaced by 1 or -1 and the range of the product is infinite. For example, a famous result due to Euler for lxi < 1 may be written as

(1 - X) (1 - X2) (1 - X3) . . . = 1 + E (_j)nX(1/i)n(3n+1) (2) n= -X The coefficients of the series admit the following combinatorial interpre- tation. If E(n) denotes the number of partitions of n into an even number of unequal parts and U(n) the number of partitions of n into an odd number of unequal parts then (2) may be stated in the following way.

= n = + - = E(n) U(n) except when 2 k(3k 1), when E(n) U(n) (-1)k The case where we have only a finite number of terms in (2) has received comparatively little attention. If in (1) we let z = 1 we have n-1 F,,- (1, x) = II (1 - Xe). (3) s=1 The coefficient of xk for k < n will be E(k) - U(k). But the coefficient of xk for k > n is the number of partitions of k into an even number of unequal parts, none of which is larger than n, minus the number of partitions of k 964 MA THEMA TICS: C. A. NICOL PROC. N. A. S. into an odd number of unequal parts, none of which is larger than n. This then is an extension of the analogous problem arising in the use of the infinite product. Also if z is replaced by -1 we consider the following function. n-i (-) n-lFn-_ (-I xX) = I(1 + x8). (4) s-i The coefficient of x" resulting from the expansion of this product is the number of partitions of k as a sum of distinct positive integers none of which is larger than n. This may also be stated as the number of solutions of the equation x1 + 2x2 + ... + (n -l)x-i= k, where for i = 1, 2, ..., (n - 1), xi is either zero or unity. The products (3) and (4) have been studied by Cauchy, T. Vahlen, von Sterneck, and others.2 In particular von Sterneck studied the case where the polynomial resulting from the expansion is reduced modulo a positive integer. Fundamental in this investigation will be the use of the number3

v = (n) (5) c1(k, n) (p(n/(k, n))_A(n/(k, n)), where k and n are positive integers and (k, n) denotes the greatest common divisor of k and n. If n is a positive integer so(n) denotes as usual the number of positive integers less than n and prime to it. (s(1) = 1.) Also, for n a positive integer, p(n) is zero if n contains a repeated prime factor. Otherwise u2(n) is equal to (-I)T where Y is the number of distinct prime factors of n. (,u(1)-1.) Note that (5) reduces to ,A(n) when (k, n) = 1 and p(n) when (k, n) n. Although (5) appears more compli- cated than its constituents it will be shown that many of the principal theorems concerning it are hardly more complex than those involving the (p or IA number alone. The properties of the coefficients in the development of (1) are extensively used to obtain properties of (5) and vice versa. We now state a number of theorems without proof. We hope to publish the proofs elsewhere. In the following paragraphs the symbol [x] will denote the largest integer contained in the real number x. Also the symbol $(a, b) will denote the number defined in (5) where a and b are positive integers. We have if k, r, and n are positive integers that4 F,E exp (2rirk/n) = c1(k, n), (6) (r,n) - 1 where the range of r is over all positive integers less than n and prime to it. (j2 = -1.) VOL. 39, 1953 MA THEMA TICS: C. A. NICOL 965

",We also have THIEOREM 1. -

d din O otherwise. Similarly (0 if (k, n) $ n. l)d D(k, n/d) = (-n if (k, n) = n, n even. (8) >Z(_din (Oif (k,n) = n,n odd. We may also prove THEOREM 2. Let a.(k) denote the sum of the divisors of k less than or equal to n. Then, n , [n/s]P(k, s) = oan(k). (9) s=1 In case k = 1 this becomes the well-known relation" n E [n/s]A(s) = 1. Also if k is replaced by n! we have another known result6

n sp(s) = n(n + 1)/2. s=lE [n/s] THEOREM 3. If 5 n, then (d/ ) -1 (k, b/d) 0 (mod. n). (10)

COROLLARY. If now p denotes an odd prime and a is a positive integer, then

pa) (pa/1) (mod. n). (lOa) THEOREM 4. , R,(d)cI(k, n/d) = 0 (mod. n), (11) dln where

Rs(d) = (d9 ) (_1)a

and this sum is over all integral solutions at of the equation (nld) a + , = s whereO < s< n, 0 . a

Consider now the function F,n-(z, x) defined in (1). If n > 1 and the product is expanded as a polynomial in x, we may write n FP-1(z, x) = E P,(z)x8, (12) s=O where ni = n(n - 1)/2 and P.(z) is a polynomial in z. Then we may define the polynomial B,(z) as Mg Bj(z) = E (13) k=O Pkn+t(Z), where n, = [(n - 1)/2 - t/n] and 1 < t < n. Then we may obtain THEOREM 5. Ifz is a number differentfrom unity, then

B,(z) = 1 (zn/d - (t, n/d), (14) n(z - 1) dln l)d-W where 1 < t < n and Bt(z) is defined in (13). Denote the polynomial defined by the function F.-1(1, x) in (3) by

E Aox, where nj = n(n - 1)/2. -(15) s-O Also define the number C, by the relation Mc C:=E Ak*+:, where n, = [(n - 1)/2 -t/n]. (15a) k=O Then we obtain the following THEOREM 6. Ct = n(t,n), (16) where C,is defined in (15a). THEOREM 7. in io(n) =- E C2, (17) n t=l where Cs is defined in (15a). In view of theorem 6 we may write theorem 7 as lin sp(n) = - Ej 12 (t, n). (18) n t=i An observation of possible interest may be made concerning theorem 7 if it is noted that C, (o(n). Then (17) becomes a quadratic relation in (p(n). Employing the quadratic formula we find VOL. 39, 1953 MA THEMA TICS: C. A. NICOL 967

(p(n) = (n ' V/n2 -4G(n))/2, (19) n-1 where G(n) = E C. Except in the case when so(n) = n/2 only one of the roots of (19) corre- sponds to sp(n). The significance of the remaining root has not been deter- mined and would seem to be of interest. If x is replaced by exp(iO), where i2 = -1, we obtain the following THEOREM 8.

4(t, n) = 2 f {Fn_- (exp(i6)) E exp(-(kn + t)iO)} dO, (20) n-I where Fn_i(exp(iO)) = I (1 - exp(siO)) andlnf = [(n - 1)/2 - t/n]. $=1 Similarlyfor the numbers Ag defined in (15) we have 1 (2w At = Jr 1Fn-l (exp(iO)) (exp(-tiO))) dO. (21) Furthermore the numbers A, defined in (15) have the following properties: n > n, (22) s=O jA1A where ni = n (n - 1)/2. A rather unusual property of these numbers is: THEOREM 9. Ifdi (n - 1), then A = 0, (23) (s, n-1) =d where 0 < s < n(n - 1)/2. A by-product of these investigations is the following result: If p is an odd prime, the integral roots of the congruence p-1 1 + E cb(s,p - 1)xSO0 (mod. P) (24) s= 1 are the incongruent primitive roots modulo p. If we consider the function defined in (4) we may obtain the following: If n is an odd positive integer and t is an integer such that 1 < t < n, then

= - E 2d 4(t, (25) Bt(-1) 2n din nld), where B,(z) is defined-in (13). 968 9.68MA THEMA TICS: C. A. NICOL PROC. N. A. S.

Since B,(z) is a polynomial with integral coefficients the number B,( 1) is an integer. In particular, if n is odd and t is less than n and prime to it, thern

Bt(-1) = -E 2d ,(n/d). (26) 2ndl-n This number is a generalization of the Fermat Quotient, (2P-1 - 1)/p, where p denotes an odd prime. Acknowledgment.-The author is indebted to H. S. Vandiver for his many helpful suggestions and encouragement. I Dickson, L. E., History of the Theory of Numbers, Vol. 2, chapter 3, Carnegie Inst. of Washington, Publication No. 256 (1920). Bachmann, P., "Niedere Zahlentheorie," Zweiter Teil, chapters 3-6, B. G. Teubner, Leipzig, 1910. 2 Vahlen, T., Bachmann, P., Ibid., 116, 167, 273. Von Sterneck, Sitzungsber. d. Wiener Akad., 111, 1567 (1902); 113, 326 (1904); 114,711(1905). Cauchy, Oeuvres D'Augustin Cauchy, 5 (series 1), 81-85, 135-152. Paris, Gauthier- Villars (1885). 3 Von Sterneck (P. Bachmann, Ibid.) introduced a function equivalent to cb(k, n). He used it to obtain results concerning partitions modulo a positive integer. Employing this function he obtained a special case of theorem 1. The number '(k, n) was used by R. Moller in the following result (Math. Monthly, 59, No. 4, 228 (April 1952)). If the numbers gd are all of the incongruent integers belonging to d modulo p, p being an odd prime and d a divisor of p - 1, then for any r, E gd =(r, d) (mod. p) 4 The sum E exp(2xrirk/n) is known as Ramanujan's sum (cf., Hardy, G. H. (r, n) = 1 and Wright, E. M., Introduction to the Number Theory, Oxford, 1938. pp. 55,237). Another closed form for this sum was found previously by T. M. Apostal and D. R. Anderson and stated by them in an abstract in Bull. Am. Math. Soc., 58, No. 5, 559 (1952). The form they found in our notation is cJ(b) 3(a)/op(c) where a = (n, k): b = n/a, and c = (a, b). If ,u(b) . Owe have the relation p(n)/,o(b) = c,(a)/,i(c). Nagell, T., Introduction to Number Theory, Uppsala, 1951, p. 43. Review of Perez-Cacho, "The Function E(x) in the Theory of Numbers," Math. Reviews, 18, 913, (1952).