By Marco Brunella

The textual content provides the birational category of holomorphic foliations of surfaces. It discusses at size the idea built by way of L.G. Mendes, M. McQuillan and the writer to review foliations of surfaces within the spirit of the class of advanced algebraic surfaces.

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In summary, they are as follows: - - (a) Is Riemann’s estimate of the number of roots p on the line segment from to 3 iT correct as T 03 ? 15 valid? ) (c) Is the product formula for &(s) valid? ) Im (d) Is Riemann’s estimate of the number of roots p in the strip (0 I p T)correct? ) (e) Is the prime number theorem true? 1 (f) Is the Riemann hypothesis true? 1 INTRODUCTION In 1893 Hadamard published a paper [Hl] in which he studied entire functions (functions of a complex variable which are defined and analytic at all points of the complex plane) and their representations as infinite products.

It was proved in 1914 that <(J it) has infinitely many real roots (Hardy [H3]), in 1921 that the number of real roots between 0 and T is at least KT for some positive constant K and all sufficiently large T (Hardy and Littlewood [H6]), in 1942 that this number is in fact at least KT log T for some positive K and all large T (Selberg, [Sl]), and in 1914 that the number of complex roots t of it) = 0 in the range (0 I Re 1 I T, --E I Im t I r } is equal, for any E > 0, to (1) with a relative error which approaches zero as T 00 (Bohr and Landau, [B8]).

14 The Principal Term of J(x) 27 that is, it is the Cauchy principal value of the divergent integral r(dt/log t). His argument is as follows: Fix x 1 and consider the function of B defined by so that the desired number is F(1). j and defining log[(s/B) - 11 to be log@ - B) - log 8, where, as usual, log z is defined for all z other than real z 5 0 by the condition that it be real for real z > 0. The integral F(B) converges absolutely because is integrable while xsoscillates on the line of integration.