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By Euler L.

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13) - Re £ ' ^f^ < l£2 _ _ L _ log(1/5) + 0ttK{l) p xo(£,K), u < x, and \t\ < C2. 7) will then follow by choosing 8 small enough in terms of e and K. 13). We may therefore suppose that x6 < u < x. We consider first the case when \t\ < 1/(8C). 3). Since ip(p) is either 0 or a root of unity of order tp(s) < s < K, it follows that ( \ — y by our assumption K > 10. 3), #{1 < a the prime < s , a = '• V ( ) number theorem ? 5 for arithmetic progressions, and since ~~1} ^ V ( )/4 for a non-real character -0.

6) is satisfied with a constant cs(e) > ci(8). 11) holds. In this case we fix a constant D = D(e) > 1 that will be chosen later and evaluate M(g-p;x,a) by the Dirichlet hyperbola method, using the representation g-p = ( 1 / i d ) * l^c * fi. We have M(gv;x,a) =^ I — */xj (d) Y lTc(n)e(adn) d say. Now, for any given d < D, Dirichlet's theorem guarantees the existence of coprime integers ad and qd such that \ad — ad/qd\ < Dci/(qdx) and 1 < qd < x/(Dci), where c\ — ci(8) is given as before.

Wefixa parameter 7 £ (0,1/2) and set k(v) = fc7r_7jW+7(v) and XQ = exp ( £ 7 ) , x1 = max(xo, e 1'*), so that 2 < XQ < x\. , ^ p . R e V ' - ^ ^ p -it zZz-x* E V + w p

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