Averaging Effects on Irregularities in the Distribution of Primes in Arithmetic Progressions
نویسندگان
چکیده
Let t be an integer taking on values between 1 and x (x real), let irhc(t) denote the number of positive primes < t which are s c (mod b), and let li t denote the usual integral logarithm of /. Further, let the ratio of quadratic nonresidues of b > 2 to quadratic residues of bbey(b) to 1, and let l/ * * , Ah(x) = (i/y(b))-\ £ n.c(0-r(») L "i,X>)\ M 1 as x -» oo, and Schinzel has provided a heuristic argument that no amount of averaging of A6(x) will provide an asymptotic relationship of this sort. However, let h(,)(x) = h{x), A£\x) = A6(x), and for k > 1 let ''u +V) = \ Í *a,0). 4k+l)(x) = \ i ^(O7=1 7=1 Assuming the truth of the generalized Riemann hypothesis for L(s, x), X 'he nonprincipal character mod 6, we prove ¿6k)(x) ¿6k)(x) lim lim -= 1 = lim lim :_.,» h^(x) *-. x— A<*>(X) A The behavior of Ab(x) is a special case of a far more general phenomenon. In Section 3, reasons are given why Ah(x) can be expected to oscillate more or less symmetrically about h(x) for every modulus b > 2.
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