Elliptic Modular Forms and Their Applications - Up To

14 D. Zagierfor the Eisenstein series (the factor 1 2arises because (c d) and (−c − d) givethe same element of Γ 1 \Γ 1 ). It is easy to see that this sum is absolutelyconvergent for k>2 (the number of pairs (c, d) with N ≤|cz + d| 2, z∈ H) , (10)where the sum is again absolutely and locally uniformly convergent for k>2,guaranteeing that G k ∈ M k (Γ 1 ). The modularity can also be seen directly bynoting that (G k | k γ)(z) = ∑ m,n (m′ z + n ′ ) −k where (m ′ ,n ′ )=(m, n)γ runsover the non-zero vectors of Z 2 {(0, 0)} as (m, n) does.In fact, the two functions (9) and (10) are proportional, as is easily seen:any non-zero vector (m, n) ∈ Z 2 can be written uniquely as r(c, d) with r (thegreatest common divisor of m and n) a positive integer and c and d coprimeintegers, soG k (z) = ζ(k) E k (z) , (11)where ζ(k) = ∑ r≥1 1/rk is the value at k of the Riemann zeta function. Itmay therefore seem pointless to have introduced both definitions. But in fact,this is not the case. First of all, each definition gives a distinct point of viewand has advantages in certain settings which are encountered at later pointsin the theory: the E k definition is better in contexts like the famous Rankin-Selberg method where one integrates the product of the Eisenstein series withanother modular form over a fundamental domain, while the G k definition isbetter for analytic calculations and for the Fourier development given in §2.2.