On Hecke's Modular Functions, Zeta Functions, and some Other Analytic Functions in the Theory of Numbers

Prof. Siegel has been kind enough to point out to me the reason for the difference between my formula (6.23), p. 544, and the corresponding formula on p. 166 of Hecke's paper, my formula containing a constant C 11 which is absent from Hecke's formula. Hecka, in deducing his formula on p. 163, line 6 up, from the first formula on the same page (when k = 1) by deforming the path of integration, has overlooked the residue arising from the pole s = 0. The investigation of the simplest form and properties of C 11 would be interesting. I am also indebted to Prof. Siegel for the majority of the following corrections: P. 505, formula 2. 3. For μ c (mod a ) read μ ≡ c (mod a ). P. 534, l. 7 up. For 1, ε, ε 2 , ε 3 , … read ±1, ±ε, ±ε 2 , ±ε 3 , …. P. 536, l. 11. In the exponent, for λ′ e −t c . read λ′ e −t c ′. P. 537, l. 5 For a read a . P. 537, l. 10. For a = β = 1 read a = β = 0. P. 538, l. 11. For −i τ 2 πread −i τ 2 π Δ. P. 538, l. 2 up. For η + τ sgn η read η + τ sgn λ. P. 540, formula (6.10). For −2π i τ log ε read −2 i τ log ε. P. 541, foot. For | Δ | read| Δ |D. P. 542, l. 8. For S (λρ) ≡ S (σ′ρ) (mod aQD ) read S (λρ)≡ S (σ′ρ) (mod AQD ). P. 544, l. 4. For c τ read i τ. P. 544, l. 5. For D ( i η; ρ′, a ′) read D ( i η′, σ′, a ′). P. 548, 1. 4. In the exponent, for −τ(μ 2 t 2 + μ′ 2 ) read −2τ|μμ′|. P. 551, l. 1. For P read η. P. 551, l. 2. For ξ, η read ξ, P . P. 553, l. 8. For > read ⩾.