The Euler characteristic of the moduli space of curves.
Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that , where P(y) is a cubic polynomial in y and , with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), . In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce . In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove .
Let k ≥ 1 denote any positive rational integer. We give formulae for the sums (where χ ranges over the ϕ(f)/2 odd Dirichlet characters modulo f > 2) whenever k ≥ 1 is odd, and for the sums (where χ ranges over the ϕ(f)/2 even Dirichlet characters modulo f>2) whenever k ≥ 1 is even.
It is shown that the multiplicative independence of Dedekind zeta functions of abelian fields is equivalent to their functional independence. We also give all the possible multiplicative dependence relations for any set of Dedekind zeta functions of abelian fields.