16 1. Introduction (1) Find by hand the values of S6(n), for n = 0,1,2 i.e., find all possible ways to write n = 0,1,2 as a sum of 6 squares. (2) Using Sage, calculate the dimension of M k 2 (Γ1(4)) (see Ap- pendix A.2) and a basis of modular forms for k = 6. (3) Write Θ6 as a linear combination of the basis elements found in part 2. (4) Use part 3 to write the q-expansion of Θ6 up to O(q20). (5) Use the expansion of Θ6 to verify that S6(4) = 252. Also, calculate S6(19) using Jacobi’s formula and verify that it coincides with the coeﬃcient of Θ6 in front of the q19 term. Exercise 1.4.6. Show that any integer n ≡ 7 mod 8 cannot be rep- resented as a sum of three integer squares. Exercise 1.4.7. Find the number of representations of n = 3 as a sum of 3 squares. Then compare your result with the value of the formula given in Example 1.3.3 i.e., use a computer to approximate S3(3) = 24 √ 3 π L(1,χ3) = 24 √ 3 π ∞ a=1 ( −3 a ) a by adding the first 10,000 terms of L(1,χ3). Do the same for n = 5 and n = 11. Does the formula seem to work for n = 2? (Note: the command kronecker(-n,m) calculates the Kronecker symbol ( −n m ) in Sage.) Exercise 1.4.8. Prove that the Riemann zeta function ζ(s) = ∑ ∞ n=1 1 ns has an Euler product i.e., prove the following formal equality of series ∞ n=1 1 ns = p prime 1 1 − p−s . (Hint: There are two possible approaches: Hint (a). Expand the right-hand side using the Fundamental Theorem of Arithmetic and the algebraic equality 1 1+x = ∑ ∞ k=0 xk. [This approach helps build an intuition about what is going on, but may be hard to write into a rigorous proof] Hint (b). Calculate (1 − 1/2s)ζ(s) and (1 − 1/3s)(1 − 1/2s)ζ(s), etc.)

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