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International symposium "Arithmetic days in Moscow"
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Some identities of Ramanujan type Yu. Nesterenko Moscow State University |
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Abstract: We will discuss some algebraically independent functions connected to Eisenstein series and having degeneracy of the transcendence degree of the field generated by their values. This degeneracy is a consequence of some quasi- modular identities that generalize the classical identities of S. Ramanujan and E. Grosswald. Values of these functions at the point i are connected to Riemann zeta-values. For example the following classical equality of M. Lerch holds $$ \zeta(3)=\frac{7\pi^3}{180}-2\sum_{n=1}^\infty\sigma_{-3}(n)e^{-2\pi n}, \qquad \sigma_{-3}(n)=\sum_{d|n}d^{-3}. $$ Language: English |
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