Related papers: On Ramanujan's formula for $\zeta(1/2)$ and $\zeta…
Example 7, after Entry 43, in Chapter XII of the first Notebook of Srinivasa Ramanujan is proved and, more generally, a summation theorem for $_3F_2(a,a,x;1+a,1+a+N;1)$, where $N$ is a non-negative integer, is derived.
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…
The cotangent zeta function is a very interesting object, which is related to partial zeta functions and Hecke $L$-functions of real quadratic fields. Its special values at odd integers greater than 1 are explicitly evaluated by Berndt in…
A two-term functional equation for an infinite series involving the digamma function and a logarithmic factor is derived. A modular relation on page 220 of Ramanujan's Lost Notebook as well as a corresponding recent result for the…
Through an application of a remarkable result due to Mishev in 2018 concerning the inverses for a class of transformations of sequences of complex numbers, we obtain a very simple proof for a famous series for $\frac{1}{\pi}$ due to…
Sander Zwegers showed that Ramanujan's mock theta functions are $q$-hypergeometric series, whose $q$-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form. George Andrews, Henri Cohen, Freeman Dyson,…
In this paper, we establish the irrationality of some open problems in mathematics based on using a recursive formula that generate the complete sequence of numbers. see [1] But before getting into that we begin with some Ramanujan notable…
The Riemann hypothesis, stating that the real part of all non-trivial zero points fo the zeta function must be $\frac{1}{2}$, is one of the most important unproven hypothesises in number theory. In this paper we will proof the Riemann…
The aim of this article is to generalize in several variables some formulae for Eisenstein series in one variable. For example the formula $2\zeta(2k) = (2\pi)^{2k} \frac{B_{2k}}{(2k)!} = Res_{z=0}(\frac{1}{z^{2k}(1-e^z)})$ for the values…
The secondary zeta function $Z(s)=\sum_{n=1}^\infty\alpha_n^{-s}$, where $\rho_n=\frac12+i\alpha_n$ are the zeros of zeta with $\Im(\rho)>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis…
Starting from the symmetrical reflection functional equation of the zeta function, we have found that the sigma values satisfying zeta(s) = 0 must also satisfy both |zeta(s)| = |zeta(1 - s)| and |gamma(s/2)zeta(s)| = |gamma((1 - s)/2)zeta(1…
For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the…
We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in Ramanujan's notebooks. The formula has a number of…
We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta…
A question proposed by Kimura and proved by Ruehr, Kimura and others in 1980 states that for any function $f$ continuous on $[-\frac{1}{2}, \frac{3}{2}]$ one has $$ \int_{-1/2}^{3/2} f(3x^2 - 2x^3) dx = 2 \int_0^1 f(3x^2 - 2x^3) dx. $$ In…
We improve the estimation of the distribution of the nontrivial zeros of Riemann zeta function $\zeta(\sigma+it)$ for sufficiently large $t$, which is based on an exact calculation of some special logarithmic integrals of nonvanishing…
We obtain an asymptotic series $\sum_{j=0}^\infty\frac{I_j}{n^j}$ for the integral $\int_0^1[x^n+(1-x)^n]^{\frac1{n}}dx$ as $n\to\infty$, and compute $I_j$ in terms of alternating (or "colored") multiple zeta value. We also show that $I_j$…
In this paper, by introducing a new operation in the vector space of analytic functions, the author presents a method for derivating the well-known formulas: $\zeta(1-k)=-\frac{B_k}{k}$ and $\zeta(1-n,a)=-\frac{B_n(a)}{n}$ , where $\zeta$,…
We have looked at the evaluation of the Riemann Zeta function at odd arguments and have provided a simple formula to approximate the value with exponential convergence. We have compared it with various other formulae present in literature.…
Explicit expressions for the hypergeometric series ${}_2F_1(-n, a; 2a\pm j;2)$ and ${}_2F_1(-n, a; -2n\pm j;2)$ for positive integer $n$ and arbitrary integer $j$ are obtained with the help of generalizations of Kummer's second and third…