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22 SECTIONAL ADDRESSES. 



More accurately, the number of primes less than ./■ is, to a first 

 approximation, 



X 



log X 

 The chance that a large number n, selected at random, should be 



prime is, we may say, about Still more precisely, the ' logarithm- 

 log n 

 integral ' 



clt 



logt 



gives a Aery good approximation to the numljer of primes. This 

 number differs from Li x by a function of x which oscillates continually, 

 as Mr. Littlewood, in defiance of all empirical evidence to the contrary, 

 has shown, between positive and negative values, and is sometimes 

 large, of the order of magnitude V.r or thereabouts, but always small 

 in comparison with the logarithm-integral itself. 



Except for one lacuna, which I must pass over in silence now, this 

 problem of the general distribution of primes, the first and central 

 problem of the theory, is in all essentials solved. But a variety of most 

 exciting problems remain as to the distribution of primes among numbers 

 of special forms. The first and simplest of these is that of the arith- 

 metical progressions : How are the primes distributed among all possible 

 arithmetical progressions an+b ? We may leave out of account the case 

 in which a and b have a common factor; this case is trivial, since an + b 

 is then obviously not prime. 



The first step towards a solution was made by Dirichlet, who proved 

 for the first time, in 1837, that any such arithmetical progression contains 

 an infinity of primes. It has since been shown that the primes are, 

 to a fhst approximation at any rate, distributed evenly among all the 

 arithmetical progressions. When we pursue the analysis further 

 differences appear; there are on the average, for example, more primes 

 4n + 3 than primes 4n-f 1, though it is not true, as the evidence of 

 statistics has led some mathematicians to conclude too hastily, that 

 there is always an excess to whatever point the enumeration is carried. 



The problem of the arithmetical progressions, then, inay also be 

 regarded as solved; and the same is true of the problem of the primes 

 of a given quadratic form, say am^ + 2bmn + en-, homogeneous in the 

 two variables m and n. To take, for instance, the simplest and most 

 striliing case, there is the natural and obvious number of primes 

 irr + n-. A prime is of this form, as I have mentioned already, if and 

 only if it is of the form 4fc + 1. The quadratic problem reduces here to a 

 particular case of the problem of the arithmetical progression. 



When we pass to cubic forms, or forms of higher degree, we come 

 to the region of the unlinown. This, however, is not the field of inquiry 

 which I wish now to commend to your attention. The quadratic forms 

 of which I have spoken are forms in two independent variables m and n ; 

 tlie form n'- + 1 of my question is a non-homogeneous form in a single 

 variable n, the simplest case of the general form an--\-2bn + c. It is 

 clear that one may ask the same question for forms of any degree : 



