2. Let F(x) = a(n)z" for 2 <1 N=1 where d(n) denotes the number of divisors of n. Observe that the radius of con- vergen

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2 Let F X A N Z For 2 1 N 1 Where D N Denotes The Number Of Divisors Of N Observe That The Radius Of Con Vergen 1 (25.61 KiB) Viewed 18 times

2. Let F(x) = a(n)z" for 2 <1 N=1 where d(n) denotes the number of divisors of n. Observe that the radius of con- vergence of this series is 1. Verify the identity Σα(2" d(n)z" = n=1 n=1 1 1-r Using this identity, show that if z=r with 0 <r< 1, then |F(0)2 -log(1/(1-r)) as r → 1. Similarly, if 0 = 2p/q where p and q are positive integers and z = rete, then |F(re)1 26/07 -log(1/(1-r) as r - 1. Conclude that F cannot be continued analytically past the unit disc.