Abstract
In his 1916 memoir entitled “On certain arithmetic function,” Ramanujan considered the three fundamental Eisenstein series P,Q, and R. In that paper, he derived a system of nonlinear differential equations satisfied by them. These equations played a fundamental role in the 1996 work of Nesterenko who calculated the transcendence degree of the field generated by the special values of these Eisenstein series. In this chapter, we discuss the significance of this work in transcendental number theory.
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References
B.C. Berndt, Ramanujan’s Notebooks, Part II (Springer, Berlin, 1989)
B.C. Berndt, Ramanujan’s Notebooks, Part V (Springer, Berlin, 1998)
B.C. Berndt, Ramanujan reaches his hand from his grave to snatch your theorems from you. Asia Pac. Math. Newsl. 1(2), 8–13 (2011)
S. Chowla, A. Selberg, On Epstein’s zeta function. J. Reine Angew. Math. 227, 86–110 (1967)
W. Duke, Continued fractions and modular functions. Bull. Am. Math. Soc. 42(2), 137–162 (2005)
D. Duverney, K. Nishioka, K. Nishioka, I. Shiokawa, Transcendence of Jacobi’s theta series and related results, in Number Theory, Eger, 1996 (de Gruyter, Berlin, 1998), pp. 157–168
B. Gross, On an identity of Chowla and Selberg. J. Number Theory 11(3), 344–348 (1979). S. Chowla Anniversary issue
E. Grosswald, Die Werte der Riemannschen Zetafunktion an ungeranden Argumentstellen. Nachr. Akad. Wiss., Gottinger Math.-Phys. Kl. 2, 9–13 (1970)
S. Gun, M.R. Murty, P. Rath, Algebraic independence of values of modular forms. Int. J. Number Theory 7, 1065–1074 (2011)
S. Gun, M.R. Murty, P. Rath, Transcendental values of certain Eichler integrals. Bull. Lond. Math. Soc. 43(5), 939–952 (2011)
G.H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, 3rd edn. (Chelsea, New York, 1978)
C. Krattenthaler, T. Rivoal, W. Zudilin, Séries hypergéométriques basiques, q analogues des valeurs de la fonction zeta et séries d’Eisenstein. J. Inst. Math. Juissieu 5(1), 53–79 (2006)
S. Lang, Elliptic Functions, 2nd edn. (Springer, New York, 1987)
D.W. Masser, Elliptic Functions and Transcendence. Lecture Notes in Mathematics, vol. 437 (Springer, Berlin, 1975)
M.R. Murty, P. Rath, Introduction to transcendental number theory (in press)
M.R. Murty, C. Smyth, R. Wang, Zeros of Ramanujan polynomials. J. Ramanujan Math. Soc. 26, 107–125 (2011)
M.R. Murty, C. Weatherby, Special values of the gamma function at CM points. Int. J. Number Theory (in press)
Yu.V. Nesterenko, Algebraic independence for values of Ramanujan functions, in Introduction to Algebraic Independence Theory. Lecture Notes in Math., vol. 1752 (Springer, Berlin, 2001), pp. 27–46
S. Ramanujan, On the product \(\prod_{n=0}^{\infty}[1 + (\frac{x }{a+nd})^{3}]\). J. Indian Math. Soc. 7, 209–211 (1915)
S. Ramanujan, Some definite integrals. Messenger of Math. 44, 10–18 (1915)
S. Ramanujan, Proof of certain identities in combinatory analysis. Proc. Camb. Philos. Soc. 19, 214–216 (1919)
T. Rivoal, La fonction zeta de Riemann prend une infinité de valeurs irrationelles aux entiers impairs. C. R. Acad. Sci. Paris Sér. I Math. 331(4), 267–270 (2000)
L.J. Rogers, Second memoir on the expansion of certain infinite products. Proc. Lond. Math. Soc. 25(1), 318–343 (1894)
I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbruche. Berliner Sitzungsber. 23, 301–321 (1917)
A. Selberg, Über einge arithmetische identitäten. Avh. Norske Vidensk. Akad. Oslo 1(8), (1936), 23s
D. Zagier, Elliptic Modular Forms and Their Applications, in The 1–2–3 of Modular Forms. Universitext (Springer, Berlin, 2008), pp. 1–103
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Murty, M.R., Murty, V.K. (2013). Ramanujan and Transcendence. In: The Mathematical Legacy of Srinivasa Ramanujan. Springer, India. https://doi.org/10.1007/978-81-322-0770-2_6
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