D O C U M E N T 1 0 3 N O V E M B E R 1 9 2 5 1 2 5 . (This would actually already have to be taken into account in deriving (XX), but it probably cannot yield anything there yet.) It is only within the range of complete degeneracy that the behavior is easily indicated again, because there the sum of the states is limited to the first term it becomes . What must be emphasized is that all assertions about only the observable thermo- dynamic behavior of do not depend on the absolute quantity of the gas, which could perhaps not have been foreseen with certainty by the method of quantizing the gaseous body as a whole. In order to follow the degeneration itself, the series must be approximated more precisely than by the integral. I haven’t quite finished this up yet, but I do believe it will work by means of the Riemannian ζ-function. For, the beginning of the se- ries is very precisely approximated by . The bracket for is simply , for this series (naturally not the summed state!) is divergent. In any case, the ζ-series offers the possibility to develop the approximation quite far at low temperatures and perhaps find a perfect connection.—[8] I think this really is the clear implementation of Planck’s argument now, because every other assumption about the phase region in the vicinity of leads to impossibilities. What do you think about publication? The basic idea is yours, I only calculated it consequently, you must decide on the further fate of your child, even if it is unpleasant for you because you prefer your first-born degeneracy theory.[9] — I do not even need to point out that it would be a special honor for me to be permitted to publish together with you. My most cordial regards to you, esteemed professor, and thank you once again for the pleasure that the pursuit of your thought afforded me. Yours sincerely, E. Schrödinger β l ∼ β l) » ( Ψ kβ klg(N!) +–= E β∂ ∂T - –kT2----- R 3h2N3 2 -- - 4πe5 /3mk ----------------------- - 1- V3 2 -- - ----- ⋅ ⋅ = = p β∂ ∂T - –kT2----- R h2N3 2 -- - 2πe5 /3mk ----------------------- - 1 V3 5 -- - ------ ⋅ ⋅ = = p 2 3 --EV - = N V --- - e–β§ 1 1- 2α ----- 1- 3α ----- 1- 4α ----- ....· + + + + © ¹ α β l -- -= α 1 ζ α) ( α 1 ≤ E 0=