The weak Gram law for Hecke $$ L $$ -functionsThe Ramanujan Journal - Tập 60 - Trang 981-997 - 2022
Sebastian Weishäupl
We generalize a theorem by Titchmarsh about the mean value of Hardy’s
$$Z$$
-function at the Gram points to the Hecke
$$L$$
-functions, which in turn implies the weak Gram law for them. Instead of p...... hiện toàn bộ
On an extension of a van Hamme supercongruenceThe Ramanujan Journal - Tập 42 - Trang 713-723 - 2016
Dian-Wang Hu, Guo-Shuai Mao
We prove a conjectural extension of a van Hamme supercongruence due to Sun. The proof is inspired by recent techniques due to Osburn and Zudilin, in particular the use of a certain WZ pair and congruences for quotients of Pochhammer symbols.
On modular equations of degree 25The Ramanujan Journal - Tập 56 - Trang 743-752 - 2020
K. R. Vasuki, M. V. Yathirajsharma
On page 237–238 of his second notebook, Ramanujan recorded five modular equations of composite degree 25. Berndt proved all these using the method of parametrization. He also expressed that his proofs undoubtedly often stray from the path followed by Ramanujan. The purpose of this paper is to give direct proofs to four of the five modular equations using the identities known to Ramanujan.
Progress on Olga Taussky-Todd’s circulant problemThe Ramanujan Journal - Tập 28 - Trang 45-60 - 2012
Norbert Kaiblinger
Determining the possible values of integer circulant determinants is an open problem proposed by Taussky-Todd. Recent interest in this question comes from studying the Lehmer constant of finite cyclic groups. By refining the approach by Laquer and Newman we contribute to the circulant determinant problem in the case that the order is a power of two.
Dynamics of a Ramanujan-type continued fraction with cyclic coefficientsThe Ramanujan Journal - Tập 16 - Trang 285-304 - 2008
Jonathan M. Borwein, D. Russell Luke
We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the continued fraction were determined for certain complex parameters (Borwein et al., Exp. Math. 13, 275–286, 2004, Ramanujan J., in press, 2004; Borwein and Crandall, Exp. Math. 12, 287–296, 2004). A study o...... hiện toàn bộ