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37
Erd˝ os–Fuchs Theorem
As before, we will use Parseval on the first of these integrals, (4)
on the second, and Schwarz’s inequality together with Parseval on
the third.

n π iθ
So write S(z)A(z) c n z , and conclude that |S(re )
−π
iθ
2

2 2n
2
A(re )| dθ 2π |c n | r . Since the c n are integers, |c n |
2
2
2
c ≥ c n andsothisis,furthermore,≥ 2π c n r 2n 2πS(r )A(r ).
n
(The general fact then is that, if Fàz) has integral coefficients,
π iθ 2 2

−π |Fàre )| dθ ≥ 2πF(r ).)
Now we introduce a side condition on our parameters r and N
which we shall insist on henceforth namely that
1
≥ N. à 18)
1 − r 2
1 2
2
N
Thus, by (14), S(r )>NØ 2N ≥ N(1− 1 ) ≥ N(1− ) N ,
N 2 4
2 C
and by (13), A(r )> √ , and we conclude that
1−r 2
π C N

iθ iθ 2
|S(re )A(re )| dθ > √ , C > 0. à 19)
−π 1 − r 2
Next, (4) gives
π dθ e

2
CN 2 iθ ≤ MN log 2 (20)
−π |1 − re | 1 − r
and our last integral satisfies

π

iθ
S(re ) a n (re ) dθ
iθ n

−π

π π 2

2
iθ n

≤ S(re ) dθ a n (re ) dθ
iθ

−π −π
√

2α 2n
2 2n
2π r 2k |a n | r ≤ 2π NM n r .
k<N
Applying (13) and (14) again leads finally to
√

π M N

iθ iθ n
S re a n (re ) dθ ≤ . à 21)
2 α+1/2
−π (1 − r )