A Closure Problem Related to the Riem Ann Zeta-Function

J. M. Cummings, L. Goldstein, and A. F. Blakeslee, these PROCEEDINGS,in press. R. S. Caldecott, E. F. Frolik, and R. Morris, these PROCEEDINGS,38, 804-809, 1952. J. S. Kirby-Smith and C. P. Swanson, Science, 119, 42-45, 1954. A. D. Conger, Science, 119, 36-42, 1954. E. F. Frolik, TID-5098, United States Atomic Energy Commission, pp. 81-87, 1953. CLOSURE PROBLEM RELATED TO THE RIEMANN ZETA-FUNCTION BY ARNE BEURLING INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW JERSEY Communicated by Herman Weyl, March 14, 1955. It is rather obvious that any property of the Riemann zeta-function may be Dressed in terms of some other property of the function p(x) defined as the frac- nal part of the real number x, i.e., x = p(x) mod 1. This note will deal with a ality of the indicated kind which may be of some interest due to its simplicity statement and proof. In the sequel, C will denote the linear manifold of func- Yns f(x) -= zECIp 0 < i 1, n = 1, 2,..., n Lerethe c, are constants such that f(i + 0) = cO,,c= 0. THEOREM. The Riemann zeta-function is free from zeros in the half-plane a > 1/p, ' p < oo, if and only if C is dense in the space LP(O, 1). Let CP denote the closure of C in the space L" = L"(0, 1), and let Ta, 0 < a < 1, the operator which takes a function f(x) defined over (0, 1) into the function lich is equal to f(x/a) for 0 < x < a and equal to 0 for a < x < 1. This semi- up of operators has the following properties which will be important for our prob- n: Each Ta carries C into itself and is norm-diminishing in each space LP. From is we easily conclude that C is dense in LP if and only if C" contains the function k ich is equal to 1 over the unit interval. For, if k belongs to CP, the same must be true the characteristic function of any subinterval (a, b) of (0, 1), this function being ual to Tlk - Tak. We next point out that, for a > 0, fo p -)x dxc --O - ---.s) (1) \X/ S-1 S ,r f e C we will have n c(s) Zcr6A" = fol f(x)xs-l dx ----0. > (2) ;sume first that CP = L'. We can then find an f e C such that Ill + fi p < e, lere e is a given positive number. By equation (2), o' (I + f(x)) Xs-' dx = ( - (s) ZCvs) a > 0. (3) r > l/p, Holder's inequality yields the following majorant of equation (3) ' 11 + fllll xllq'< q(- l/p) nsequently, n c - s q's Il - -sD(s)^c^i E < ( o >l/p. (4) I c q(a - l/p)' is proves that v 7 0 in the region Dp, where the right-hand member of relation is < 1, i.e., for 1 eq a> - + Islq p q e 0, Dp,, increases and exhausts the half-plane c > 1/p. We should also ob- ve at this instance that Dp, e is convex and that its boundary intersects the line = 1 at the two points where Is = 1/e. Phe proof of the necessity of the condition CP = LP is less trivial. If C is not ise in LP, we know by a classical theorem of F. Riesz and Banach that the dual .ce Lq must contain a nontrivial element g which is orthogonal to C in the sense It 0 = fol g(x)f(x) dx, feC. (5) 0ceTa takes C into itself, it follows that 0 = folg(x)Taf(x) dx = afol g(ax)f(x) dx, feC, 0 < a < 1. (6) Es, 1 < r < q, be the closed linear subset of LI spanned in the topology of btspace by the set g(ax), 0 < a 1 }. From equation (6) and the fact that C isists of bounded functions, we conclude that each f e C is orthogonal to each Lctionbelonging to any of the sets Eg. We now recall that the positive reals < 1 m a semigroup S under multiplication and that each continuous (and normal- i) character of S has the form p = xX, where X is an arbitrary complex or real mber. Clearly s e Lq if and only if (R(X) > - 1/q. The problem of whether or not et of the kind EP contains a character is of considerable complexity. It has mnstudied earlier by the author', 3 and by Nyman.2 However, the following re- t3 can be proved by elementary function theoretic means: Let g(x) belong to a ice LQ(O, 1), 1 < q < o, and have the property fS Ig(x) dx > O, x > O. (7) en there exists at least one character xX, (Re(X) > - l/q, which is contained in each Egfor 1 < r < q (but not necessarily in E'). Cnorder to apply this theorem, we have first to show that condition (7) is satisfied our function g. For this pupose assume that g vanishes a.e. on some interval a), a > 0. Choose b such that a < b < min (1, 2a), and f(x) = bp(a/x) - (b/x). Thisf belongs to C; it vanishes for x > b and takes the value a for a < x < Therefore, 0 = fo'f(x)g(x) dx = afa g(x) dx, a < b < min (1, 2a), (8) d it follows that g = 0 a.e. for x < min (1, 2a). On repeating the same argument nite number of times, we find that g = 0 a.e. over (0, 1), which is a contradiction. Lecited theorem may thus be applied to g and yields the existence of a number R,e(X)> - l/q, such that xXis orthogonal to C. On defining So = 1 + X, we will ve, in particular == ) 0 o p()- (p(0)) xS?-x ( Osi (so), 0 < 6 < 1. (9) )viously, so 7 1. Since 0 can be chosen such that, at any given point # 1, 06-1 - 7 0, it follows that so is a zero of P'. We also have 6e, (So) > 1 - q = l/p, and this concludes the proof. We finally point out that the problem of how well k = 1 can be approached by ictions e C has a direct bearing on the distribution of the primes even in case toes have zeros arbitrary close to the line a = 1. A. Beurling, "On Two Problems concerning Linear Transformations in Hilbert Space," Acta ath., Vol. 81, 1949. B. Nyman, "On Some Groups and Semi-groups of Translations" (thesis, Uppsala, 1950). A. Beurling, "A Theorem on Functions Defined on a Semi-group," Math. Scand., Vol. 1, 1953. rTEGRABLE AND SQUARE-INTEGRABLE REPRESENTATIONS OF A SEMISIMPLE LIE GROUP BY HARISH-CHANDRA DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY Communicated by Paul A. Smith, March 11, 1955 Let G be a connected semisimple Lie group. We shall suppose for simplicity at the center of G is finite. Let w7be an irreducible unitary representation of G a Hilbert space ,5. We say that ir is integrable (square-integrable) if there ists an element V # 0 in ~ such that the function (A, r(x) /) (x e G) is integrable quare-integrable) on G, with respect to the Haar measure. Assuming that the aar measure dx has been normalized in some way once for all and that 7ris square- legrable, we denote by d, the positive constant given by the relation' f n nt ) ,tr n dx iere V is any unit vector in ,. Let C, (G) denote the set of all complex-valued nctions on G which are everywhere indefinitely differentiable and which vanish Itside a compact set. Then the following result is an easy consequence of the ;hurorthogonality' relations. .

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