By Henry B. Mann.

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**Sample text**

M(a;) are all prime. W e now work modulo a prime # . ,fm(x) ~ 0 (m odp). This means that x must be confined to certain residue classes m od#. W e therefore divide the residue classes m od # into a set H (#) o f /(# ) forbidden classes and a set K (p ) o f g(p) = # —/ ( # ) permitted classes; h m od # is forbidden if and only if one o f the p o ly n o m ia ls /^ ) is a multiple o f # . I f x falls into a forbidden class for any prime # smaller than each o f the fi(x), then one at least o f t h e / i(x) cannot be prime.

W e shall show in Chapter 8 that, i f J/' satisfies the above condition, the number o f members o f J f in any interval o f N consecutive integers is N v ... 3) g(i) = ? IT { l - f i v ) h ? } v\a (6-14) W e shall work out examples o f this upper bound in Chapter 8 ; in each case the leading term is a multiple o f the leading term in the conjectured formula. Upper bounds o f the right order o f magnitude were first found by Viggo Brun using combinatorial arguments. 12), which was found in a different way b y Selberg.

W e shall prove later that the hypothesis is true for one linear polynom ial/(a;) = qx-\-a\ this is the prime-number theorem for arithmetical progressions, but the error term in the asym ptotic formula will only be shown to be slightly smaller than the leading term. The next simplest case concerns two linear polynomials, f ^ x ) = x, f 2(x) = x — 2. 6 W e shall now describe how to write down the conjectured asymptotic formulae. Let a/ \ S (a )= v 2, p^N n. m o\ (6>3) such an expression is called an exponential sum or a trigonometric sum.