Abstract
We writeandso that p(n) is the number of unrestricted partitions of n. Ramanujan [1] conjectured in 1919 that if q = 5, 7, or 11, and 24m ≡ 1 (mod qn), then p(m) ≡ 0 (mod qn). He proved his conecture for n = 1 and 2†, but it was not until 1938 that Watson [4] proved the conjecture for q = 5 and all n, and a suitably modified form for q = 7 and all n. (Chowla [5] had previously observed that the conjecture failed for q = 7 and n = 3.) Watson's method of modular equations, while theoretically available for the case q = 11, does not seem to be so in practice even with the help of present-day computers. Lehner [6, 7] has developed an essentially different method, which, while not as powerful as Watson's in the cases where Γ0(q) has genus zero, is applicable in principle to all primes q without prohibitive calculation. In particular he proved the conjecture for q = 11 and n = 3 in [7]. Here I shall prove the conjecture for q = 11 and all n, following Lehner's approach rather than Watson's. I also prove the analogous and essentially simpler result for c(m), the Fourier coefficient‡ of Klein's modular invariant j (τ) as
Publisher
Cambridge University Press (CUP)
Reference12 articles.
1. Proof of Ramanujan's partition congruence for the modulus IP;Lehner;Proc. Amer. Math. Soc.,1950
2. Ramanujans Vermutung über Zerfallungsanzahlen;Watson;J. Reine Angew. Math.,1938
3. Congruence properties of partitions
4. Some properties of p(n), the number ofpartitions of;Ramanujan;n Proc. Cambridge Phil. Soc.,1919
5. Divisibility Properties of the Fourier Coefficients of the Modular Invariant j(τ)
Cited by
93 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献