Let be the number of Schur's partitions of n, i.e. the number of partitions of n into distinct parts congruent to 1, . We prove
Soit le nombre de partitions de Schur de n, c'est-à-dire le nombre de partitions de n en parts distinctes congrues à . Nous montrons que :
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Shi-Chao Chen 1
@article{CRMATH_2019__357_5_418_0, author = {Shi-Chao Chen}, title = {Parity of {Schur's} partition function}, journal = {Comptes Rendus. Math\'ematique}, pages = {418--423}, publisher = {Elsevier}, volume = {357}, number = {5}, year = {2019}, doi = {10.1016/j.crma.2019.05.006}, language = {en}, }
Shi-Chao Chen. Parity of Schur's partition function. Comptes Rendus. Mathématique, Volume 357 (2019) no. 5, pp. 418-423. doi : 10.1016/j.crma.2019.05.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.05.006/
[1] Generalizations of Schur's partition theorem, Manuscr. Math., Volume 79 (1993), pp. 113-126
[2] On Schur's second partition theorem, Glasg. Math. J., Volume 9 (1967), pp. 127-132
[3] The Theory of Partitions, Encyclopedia of Mathematics, vol. 2, Addison Wesley, 1976
[4] Schur's theorem, Capparelli's conjecture, and the q-trinomial coefficients, Contemp. Math., Volume 166 (1994), pp. 141-154
[5] Schur's theorem, partitions with odd parts and the Al-Salam–Carlitz polynomials (M.E.H. Ismail; D. Stanton, eds.), q-Series from a Contemporary Perspective, Contemporary Mathematics, vol. 254, 2000, pp. 45-56
[6] A refinement of Alladi–Schur theorem (G.E. Andrews; C. Krattenthaler; A. Krinik, eds.), Lattice Path Combinatorics and Applications, Developments in Mathematics, vol. 58, Springer, Cham, 2019, pp. 71-77
[7] Nonzero coefficients of half-integral weight modular forms mod ℓ, Res. Math. Sci. (2018), pp. 5-6
[8] Parité des coefficients de formes modulaires, Ramanujan J., Volume 40 (2016), pp. 1-44
[9] Über die Darstellung von positiven, ganzen Zahlen durch die primitiven, binären quadratischen Formen einer nicht-quadratischen Diskriminante, 1912 (Dissertation, Göttingen, Germany)
[10] A combinatorial proof of Schur's 1926 partition theorem, Proc. Amer. Math. Soc., Volume 79 (1980), pp. 338-340
[11] On generalized Schur's partitions, Int. J. Number Theory, Volume 13 (2017), pp. 1381-1391
[12] Odd values of the Rogers–Ramanujan functions, C. R. Acad. Sci. Paris, Ser. I, Volume 356 (2018), pp. 1081-1084
[13] Primes of the Form : Fermat, Class Field Theory and Complex Multiplication, Wiley, New York, 1989
[14] Roots of unity and nullity modulo n, Proc. Amer. Math. Soc., Volume 138 (2010), pp. 2729-2743
[15] Uber einen Satz van Herrn I. Schur, Math. Z., Volume 28 (1928), pp. 372-382
[16] The number of representations function for binary quadratic forms, Amer. J. Math., Volume 62 (1940), pp. 589-598
[17] Elementary proofs of various facts about 3-cores, Bull. Aust. Math. Soc., Volume 79 (2009), pp. 507-512
[18] On norms of integers in a full module of an algebraic number field and the distribution of values of binary integral quadratic forms, Mathematika, Volume 22 (1975), pp. 108-111
[19] Nonvanishing of quadratic twists of modular L-functions with applications for elliptic curves, J. Reine Angew. Math., Volume 533 (2001), pp. 81-97
[20] The parity of the partition function, Adv. Math., Volume 225 (2010), pp. 349-366
[21] The structure of the number of representations function in a positive binary quadratic form, Math. Z., Volume 36 (1933), pp. 321-343
[22] On the distribution of parity in the partition function, Math. Comput., Volume 21 (1967), pp. 466-480
[23]
, S.–B. Akad. Wiss. Phys.–Math. KL, Berlin (1926), pp. 488-495 (Reprinted in Gesammelte Abhandlungen, Vol. 2, 1973, Springer Verlag, Berlin, pp. 43-50)[24] et al. http://www.sagemath.org/ SageMath Software (Version 8 (1)
[25] Das asymptotische verhalten von summenüber multiplikative funktionen, Math. Ann., Volume 143 (1961), pp. 75-102
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