1.4. Exercises 15 by L(s, χn) = ∞ a=1 χn(a) as . We are ready to write down the formula for S3(n), due to Gauss, Dirichlet and Shimura (there are also formulas for S5(n), due to Eisen- stein, Smith, Minkowski and Shimura, and a formula for S7(n), also due to Shimura). For simplicity, let us assume that n is odd and square free (for the utmost generality, please check [Shi02]): S3(n) = 0 if n ≡ 7 mod 8, 24 √ n π L(1,χn) otherwise. The reader is encouraged to investigate this problem further by at- tempting Exercises 1.4.6 and 1.4.7. 1.4. Exercises Exercise 1.4.1. Use the divisibility properties of integers to show that the only solutions to y2 = x(x + 1)(x + 2) with x, y ∈ Z are (0,0), (−1,0) and (−2,0). (Hint: If a and b are relatively prime and ab is a square, then a is a square and b is a square.) Exercise 1.4.2. Find all the Pythagorean triples (a, b, c), i.e., a, b, c ∈ Z and a2 + b2 = c2, such that b2 + c2 = d2 for some d ∈ Z. In other words, find all the integers a, b, c, d such that (a, b, c) and (b, c, d) are both Pythagorean triples. (Hint: You may assume that y2 = x(x + 1)(x + 2) has no rational points other than (0,0), (−1,0) and (−2,0).) Exercise 1.4.3. Prove Proposition 1.1.3 i.e., show that f((a, b, c)) is a point in En, that g((x, y)) is a triangle in Cn and that f(g((x, y))) = (x, y) and g(f((a, b, c))) = (a, b, c). Exercise 1.4.4. Calculate S4(n), for n = 1,3,5,6, by hand, using Jacobi’s formula and also by finding all possible ways of writing n as a sum of 4 squares. Exercise 1.4.5. The goal of this problem is to find the q-expansion of Θ6(q):

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