相关论文: Ramanujan's Most Singular Modulus

By using the theory of vertex operator algebras, we gave a new proof of the famous Ramanujan's modulus 5 modular equation from his "Lost Notebook" (p.139 in \cite{R}). Furthermore, we obtained an infinite list of $q$-identities for all odd…

Inspired by the work of S. Ramanujan, many people have studied generalized modular equations and the numerous identities found by Ramanujan. These identities known as modular equations can be transformed into polynomial equations. There is…

Ramanujan made many beautiful and elegant discoveries in his short life of 32 years, and one of them that has attracted the attention of several mathematicians over the years is his intriguing formula for $\zeta(2n+1)$. To be sure,…

At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli $\alpha_n$. All those results were proved by Berndt et. al by employing Weber-Ramanujan's class invariants. In this paper, we initiate to derive the…

This is an elementary explanation of a cubic composition formula due to Ramanujan.

Ramanujan in his notebook recorded two modular equations involving multiplier with moduli of degrees (1,7) and (1,23). In this paper, we find some new Ramanujan's modular equations involving multiplier with moduli of degrees (3,5) and…

S. Ramanujan introduced a technique in 1913 for providing analytic expressions for certain Mellin-type integrals which is now known as Ramanujan's Master Theorem. This technique was communicated through his "Quarterly Reports" and has a…

When Mike Hirschhorn showed us his lovely gem, that gives the simplest-to-date proof of Ramanujan's famous result that p(11n+6) is divisible by 11, we realized that his amazing method can be extended, and taught to a computer, and can prove…

Ramanujan Master Theorem is a technique developed by the indian mathematician S. Ramanujan to evaluate a class of definite integrals. This technique is used here to calculate the values of integrals associated with specific Feynman…

In this paper we present a probabilistic algorithm to compute the coefficients of modular forms of level one. Focus on the Ramanujan's tau function, we give out the explicit complexity of the algorithm. From a practical viewpoint, the…

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number of solutions of the linear congruence $a_1x_1+\cdots +a_kx_k\equiv b \pmod{n}$, with…

Ramanujan's Master Theorem is a decades-old theorem in the theory of Mellin transforms which has wide applications in both mathematics and high energy physics. The unconventional method of Ramanujan in his proof of the theorem left…

Towards the end of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently. Nevertheless, this manuscript gave rise to a lot of subsequent work. In…

The modular transformations of Ramanujan's tenth order mock theta functions are computed, beginning from Choi's Hecke-type identites and using Zwegers' results on indefinite theta series. Explicit completions and shadows are found as an…

In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series…

We investigate Ramanujan congruences for the function which counts the overpartitions of n with restricted odd differences. In particular, we show that only one such congruence exists. Our method involves using the theory of modular forms…

Let $\Lambda$ be a ring and $\mathcal N$ a class of $\Lambda$-modules. A $\Lambda$-module is said to be generated by $\mathcal N$ provided that it is a factor module of a direct sum of modules in $\mathcal N$. The semi-simple…

Ramanujan's last letter to Hardy explored the asymptotic properties of modular forms, as well as those of certain interesting $q$-series which he called \emph{mock theta functions}. For his mock theta function $f(q)$, he claimed that as $q$…

In this paper, we initiate a generous amount of new-found general theorems for explicit evaluations of product of the theta functions $b_{m, n}$ using Kronecker's limit formula and other various novel explicit evaluations that were…

We prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the largest modulus is at least 108. We also prove that if the smallest modulus of a covering system with distinct moduli is $5$, then the least…