Search results
Results from the WOW.Com Content Network
Most of Ramanujan's mistakes arise from his claims in analytic number theory, where his unrigorous methods led him astray. In particular, Ramanujan thought his approximations and asymptotic expansions were considerably more accurate than warranted. In [12], these shortcomings are discussed in detail. [12] is Berndt, Ramanujan's Notebooks, Part IV.
Bruce Berndt has claimed that all the claims in Ramanujan's "Lost Notebook" have been proved, with a solution to the the final problem being published by Berndt, Li, and Zaharescu in J. London Math. Soc. in 2019. However, I am not sure that this means that all the formulas in Ramanujan's other writings have been proved.
Ramanujan's approach therefore also involves a great deal of labour involving highly efficient manipulation. A person who used similar approach was Jacobi. It's rather unfortunate that the techniques of Jacobi and Ramanujan have been ditched in modern times. $\endgroup$
Riemann Hypothesis and Ramanujan’s Sum Explanation. RH: All non-trivial zeros of the Riemannian zeta-function lie on the critical line. ERH: All zeros of L-functions to complex Dirichlet characters of finite cyclic groups within the critical strip lie on the critical line. Related Article: The History and Importance of the Riemann Hypothesis
I noticed an apparent conflict in the definition in literature about what is a "Ramanujan graph, which I was wondering if someone could kindly clarify. (1) The Hoory-Linial-Wigderson review on expanders in its definition 5.11 calls a d-regular graph to be Ramanujan if the second highest (adjacency?) eigenvalue is bounded above by $2\sqrt{d-1}$.
This is one of those precious cases when Ramanujan himself provided (a sketch of) a proof. The identity was published in his paper "Some definite integrals" (Mess. Math. 44 (1915), pp. 10-18) together with several related formulae.
$\begingroup$ A lot of nonsense around Ramanujan's master theorem. In your linked article they only define $\Phi(x)$ by its power series for $|x|$ small but they don't mention we are looking at its analytic continuation (has to be proven it exists) in $\int_0^\infty \Phi(x)x^{s-1}dx$. $\endgroup$
Ramanujan graphs are defined as $(q+1)$-regular graphs for which all nontrivial eigenvalues of the ...
Here are some additions to your list of explicit and inexplicit Ramanujan graphs, all from the recent decade. Ballantine and Ciubotaru constructed inexplicit infinite families of $(q+1,q^3+1)$-bigregular Ramanujan graphs here. See also this followup. I gave inexplicit "new" infinite families of regular Ramanujan graphs (and complexes) in my paper.
The Ramanujan tau function $\tau(n)$ is given by the expansion of the Dedekind eta function $\eta(z)$'s ...