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1. This paper contains a short account of results whose detailed proofs will be published later.We define the function Z(s) by Z(s) = 13'(am2 + bmn + cnl2)8(1) where s = a + it(a and t, real), o > 1, and the summation is for all integers m, n (each going fromto + co), while the dash indicates that m = n = 0 is excluded from the summation; further a and c are positive numbers while b is real and subject to 4ac -b = A > 0. It is well known that the function Z(s), defined for a > 1 by (1), can be continued analytically over the whole s-plane, and satisfies a functional equation similar to the one satisfied by the Riemann Zeta Function. The function Z(s), thus defined, is a meromorphic function with a simple pole at s = 1. Deuring (Math. Ztschr., 37, 403-413 (1933)) obtained an important formula for Z(s). Deuring's work led Heilbronn (Quart. J. Maths., Oxford, 5, 150 (1934)) to the proof of the following famous conjecture of Gauss on the class-number of binary quadratic forms with a negative fundamental discriminant: let h (-A) denote the number of classes of binary quadratic forms of negative fundamental discriminant -iA = --4ac, then h(-A)-co as A -*c (2) Again using the ideas of Heilbronn and Deuring, Siegel proved that h(-A.) > A'12e [A > A(e)] (3) which is a great advance on (2). Our starting point is the formula: Z(s) = 2¢(2s)a-S + 2 -a /2 V (2s -l(s -'/2) + Q(s) (4) r(s) As where Q(s) = r S-2S +3/2 c o W (2 (n7rb) f -8/2 Q(S) a T() As/_ -/4 f al-2ln) cos fo -7rnA&/" exp( 2a (cj$+ c-1jdo (4) VOL. 35, 1949 371 372 MATHEMATICS: CHOWLA AND SELBERG PROC. N. A. S.Here ak(n) denotes the sum of the kth powers of the divisors of n, and D(s) is Riemann's Zeta Function. The series for Q(s) is highly convergent.Taking a crude estimate of the series for Q(s) we obtain the formula of Deuring referred to above.2. The formula (4) can be applied to the proof of the positiveness of certain Dirichlet L-functions at s = l/2. In fact we define for s > 0, L4(s) = n nS where (n/p) is Legendre's symbol defined as follows:If n 0 (mod p), then (n/p) = + 1 if the congruence X2 = n (mod p)The positiveness of Lp(s) for 0 < s _ 1 was proved by S. 89, 97. But no information was obtained in the cases p = 43, 67, 163 (here the class number h(-p) is small). Heilbronn (Acta Arithmetica, Band 2, 212 (1937) proved that there are infinitely many primes p for which the method of Chowla gives no information. Curiously enough, the present method is more successful with precisely those cases like p = 43, 67, 163 (class number h(-p) = 1) where the previous method failed. In these three cases we obtain L4(1/2) > 0 (Rosser has recently, in an unpublished paper, settled the cases p = 43 and p = 67 by an entirely different method).That L,(1/2) > 0 in these cases, is not surprising, for if there is a prime p such that LX(/2) < 0 then the extended Riemann hypothesis is false! These results are deduced from the following THEOREM: If p is an odd prime >7 and if h(-p) = 1, then (c = r/2) WI/2)Lp(1/2) = Y + log (P) + 80. e cPwhere 'y is Euler's constant and 0 ...

Let it be required to arrange v elements into v sets such that every set contains exactly k distinct elements and such that every pair of sets has exactly elements in common . This combinatorial problem is studied in conjunction with several similar problems, and these problems are proved impossible for an infinitude of v and k. An incidence matrix is associated with each of the combinatorial problems, and the problems are then studied almost entirely in terms of their incidence matrices. The techniques used are similar to those developed by Bruck and Ryser for finite projective planes [3]. The results obtained are of significance in the study of Hadamard matrices [6;8], finite projective planes [9], symmetrical balanced incomplete block designs [2; 5], and difference sets [7].

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