Abstract
Let τ(n) be Ramanujan's function,
In this paper it is shown that the Ramanujan congruence τ(n)=σd/nd11 mod 691 cannot be improved mod 6912. The following result is proved: for arbitrary r, s mod 691 the set of primes such that p ≡ r mod 691,τ (p) ≡ p11+1+691 · s mod 6912 has positive density.
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J.-P. Serre, A Course in Arithmetic [Russian translation], Moscow (1972).
S. Ramanujan, “On certain arithmetical functions,” Trans. Camb. Phil. Soc.,22, 159–184 (1916). (see also: S. Ramanujan, Collected Papers, Vol. 4 (1927), pp. 136–162).
P. Deligne, “Formes modulaires et représentationsl-adiques,” Séminaire Bourbaki,355, February (1969).
J.-P. Serre, “Une interprétation des congruences relatives a la fonction τ de Ramanujan,” Séminaire Delange—Pisot—Poitou,14, (1967–1968).
H. P. F. Swinnerton-Dyer, “Onl -adic representations and congruences for coefficients of modular forms,” in: Modular Functions of One Variable III, Lecture Notes in Mathematics, Vol. 350, Berlin, Heidelberg, and New York (1973).
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Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 255–264, February, 1975.
In conclusion, the author would like to thank Yu. I. Manin for posing this problem and for his continued interest.
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Panchishkin, A.A. There exist no Ramanujan congruences mod 6912 . Mathematical Notes of the Academy of Sciences of the USSR 17, 148–153 (1975). https://doi.org/10.1007/BF01161871
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DOI: https://doi.org/10.1007/BF01161871