Ramanujan Series Upside-Down

Abstract

We prove that there is a correspondence between Ramanujan-type formulas for 1/π, and formulas for Dirichlet L-values. If we have an identity of the formwhere (s) n = Γ(s + n)/Γ(s), then under certain conditions we prove thatreduces to Dirichlet L-values evaluated at 2. The two sums rarely converge at the same time, however divergent formulas make sense when they are interpreted as values of analytically continued hypergeometric functions. The same method also allows us to resolve certain values of the Epstein zeta function in terms of rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker in [7].

A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds

“…Those fit a general picture highlighted in the observations above, except that the modular form f (τ ) is replaced by a quadratic character so that a critical L-value L(f, m) is replaced by the critical value of the corresponding Dirichlet L-series. This is transparent from supercongruence observations in [37] and, in addition, from a noncongruence (bilateral) counterpart experimentally discovered by J. Guillera in [12] (see also the related prequel [13]).…”

“…We also stress on the fact that L(E z , 1), therefore L(z, 1) in (13), vanishes when the (analytic) rank of the elliptic curve E z is positive. In such situations, numerics suggests no relation between the hypergeometric functions F 0 (z), F 1 (z) in question and the first nonzero derivative of L(E z , s) (or of L(z, s)) at s = 1.…”

Ramanujan Series With a Shift

“…These substitutions are justified because they preserve formally the recurrence equation Γ(x + 1) = x Γ(x); see the duality property [9,Ch. 7] and the application shown in [7,Section 4], and see [8] for the analytic interpretation. If |z| > 0, we understand the 'divergent' series (1.1) as its analytic continuation, and, if |z| < 0, we interpret the 'divergent' series (1.2) in the same way.…”

“…If |z| > 0, we understand the 'divergent' series (1.1) as its analytic continuation, and, if |z| < 0, we interpret the 'divergent' series (1.2) in the same way. While in [8] we have studied the 'upside-down' transformation, in this paper we consider the transformation with a shift. In [8] we prove that the upside-down transformation modifies the value of the modular variable q.…”

“…While in [8] we have studied the 'upside-down' transformation, in this paper we consider the transformation with a shift. In [8] we prove that the upside-down transformation modifies the value of the modular variable q. Here we will see that a shift does not modify it.…”