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References
Ahlgren, S.: The partition function modulo composite integers M. Math. Ann. 318, 795–803 (2000)
Ahlgren, S., Ono, K.: Congruence properties for the partition function. Proc. Natl. Acad. Sci. USA 98, 12882–12884 (2001)
Andrews, G.E.: The theory of partitions. Cambridge: Cambridge University Press 1998
Andrews, G.E., Garvan, F.: Dyson’s crank of a partition. Bull. Am. Math. Soc., New Ser. 18, 167–171 (1988)
Atkin, A.O.L.: Proof of a conjecture of Ramanujan. Glasg. Math. J. 8, 14–32 (1967)
Atkin, A.O.L.: Multiplicative congruence properties and density problems for p(n). Proc. Lond. Math. Soc., III. Ser. 18, 563–576 (1968)
Atkin, A.O.L., Lehner, J.: Hecke operators on Γ0(N). Math. Ann. 185, 134–160 (1970)
Atkin, A.O.L., O’Brien, J.N.: Some properties of p(n) and c(n) modulo powers of 13. Trans. Am. Math. Soc. 126, 442–459 (1967)
Atkin, A.O.L., Swinnerton-Dyer, H.P.F.: Some properties of partitions. Proc. Lond. Math. Soc., III. Ser. 4, 84–106 (1954)
Berndt, B.C., Ono, K.: Ramanujan’s unpublished manuscript on the partition and tau functions with commentary. Sémin. Lothar. Comb. 42, Art. B42 (1999)
Bruinier, J.H., Ono, K.: Coefficients of half-integral weight modular forms. J. Number Theory 99, 164–179 (2003)
Chan, H.H.: Private communication.
Dyson, F.J., Some guesses in the theory of partitions. Eureka (Cambridge) 8, 10–15 (1944)
Garvan, F., Kim, D., Stanton, D.: Cranks and t-cores. Invent. Math. 101, 1–17 (1990)
Gordon, B., Hughes, K.: Multiplicative properties of eta-products, II. Contemp. Math. 143, 415–430 (1993)
Kløve, T.: Recurrence formulae for the coefficients of modular forms and congruences for the partition function and for the coefficients of j(τ), (j(τ)-1728)1/2, and j(τ)1/3. Math. Scand. 23, 133–159 (1969)
Kiming, I., Olsson, J.: Congruences like Ramanujan’s for powers of the partition function. Arch. Math. 59, 348–360 (1992)
Kolberg, O.: Note on the parity of the partition function. Math. Scand. 7, 377–378 (1959)
Newman, M.: Periodicity modulo m and divisibility properties of the partition function. Trans. Am. Math. Soc. 97, 225–236 (1960)
Newman, M.: Construction and application of a certain class of modular functions. Proc. Lond. Math. Soc., III. Ser. 7, 334–350 (1957)
Ono, K.: Distribution of the partition function modulo m. Ann. Math. 151, 293–307 (2000)
Ramanujan, S.: On certain arithmetical functions. Trans. Cambridge Philos. Soc. 22, 159–184 (1916)
Ramanujan, S.: Some properties of p(n), the number of partitions of n. Proc. Cambridge Philos. Soc. 19, 207–210 (1919)
Ramanujan, S.: Congruence properties of partitions. Proc. Lond. Math. Soc., III. Ser. 18, xix (1920)
Ramanujan, S.: Congruence properties of partitions. Math. Z. 9, 147–153 (1921)
Serre, J.-P.: Formes modulaires et fonctions zêta p-adiques. Lect. Notes Math. 350, 191–268 (1973)
Swinnerton-Dyer, H.P.F.: On ℓ-adic representations and congruences for coefficients of modular forms. Lect. Notes Math. 350, 1–55 (1973)
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Ahlgren, S., Boylan, M. Arithmetic properties of the partition function. Invent. math. 153, 487–502 (2003). https://doi.org/10.1007/s00222-003-0295-6
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DOI: https://doi.org/10.1007/s00222-003-0295-6