571p11 - =1 ( n ) q log(2 M + 1) + q q log q and deduce...

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Math 571 Analytic Number Theory I, Spring 2011, Problems 11 Due Tuesday 5th April 1. Let ϕ 2 ( q ) denote the number of primitive characters (mod q ). (a) Show that ϕ 2 ( q ) is a multiplicative function. (b) Show that ° d | q ϕ 2 ( d )= ϕ ( q ). (c) Show that ϕ 2 ( q )= q ± p ° q ² 1 2 p ³ ± p 2 | q ² 1 1 p ³ 2 . (d) Show that ϕ 2 ( q ) > 0 if and only if q °≡ 2 (mod 4). 2. (P´ olya 1919, I. M. Vinogradov 1919, Schur 1920.) Let χ be a non–principal character modulo q , and let N N with N<q . In parts (a)–(d) we assume that χ is primitive. (a) Prove that N ´ n =1 χ ( n )= 1 τ ( χ ) q ´ a =1 χ ( a ) N ´ n =1 e ( an/q ) . (b) Let M = µ q 1 2 , and θ =0when q is odd and θ =1when q is even. Prove that · · · · · N ´ n =1 χ ( n ) · · · · · 1 q q 1 ´ a =1 1 2 ± a/q ± M ´ a =1 q a + θ q . (c) Prove that for each a N we have 1 a log a + 1 2 a 1 2 and for each b N with b 2, log( b 1)+ 1 b log b . (d) Prove that · · · · · N ´ n
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Unformatted text preview: =1 ( n ) q log(2 M + 1) + q q log q and deduce that N n =1 ( n ) q log q holds for every N N . (e) Prove that for n N , d ( n ) 2 n . (f) Assume now that is a character modulo q induced by the primitive character modulo modulo r , so that r is the conductor of and r | q . Prove that N n =1 ( n ) = N n =1 ( n,q/r )=1 ( n ) = d | q/r ( d ) ( d ) m N/d ( m ) and that N n =1 ( n ) 2 q log q (the P olyaVinogradov inequality)....
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