Representing Primes as x^2 + 5y^2: An Inductive Proof that Euler Missed

Instant Evaluation and Demystification of zeta(n),L(n,chi) that Euler,Ramanujan Missed - II

For Hurwitz zeta function, we obtain power series expression in second variable for its higher order derivatives (with respect to first variable) at non-positive integer arguments and consequently obtain rapidly decreasing series expression for Riemann zeta fuction at positive odd integer arguments. Further, we obtain corresponding results for Dirichlet L-series. We also a unified proof of various classical identities involving Riemann zeta values.

Instant Evaluation and Demystification of zeta(n),L(n,chi) that Euler,Ramanujan Missed III

We show that for a non-positive value of the first variable,Hurwitz zeta function becomes a polynomial in the second variable. We show this, using 'integration approach', instead of 'power series approach', which we had resorted to, in our earlier paper with the same title. This, in particular, explains why Riemann zeta function at positive even integer arguments, can be evaluated and why it cannot be evaluated explicitly at positive odd integer arguments.

Instant Evaluation and Demystification of $ζ(n),L(n,χ)$ that Euler,Ramanujan Missed - I

For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive integral argument in first variable.On using functionalequations this results in instant evaluation of Riemann Zeta function at positive even integral values of its argument and of Dirichlet L series at positive integral values of its argument,when the argument and the corresponding Dirichlet character are both even or both odd.We also obtain finite sum expressio…

特殊函数 $e^{x}E_{1}(x)$ 的有理函数逼近及其在 Euler-Gompertz 常数 $δ$ 的无理性中的应用

Rational function approximations of the special function $e^{x}E_{1}(x)$ and applications to irrationality of Euler-Gompertz constant $δ$

本文提出一种构造特殊函数 $e^{x}E_{1}(x)$ 连分数逼近的方法，证明了其收敛性，并以此构造逼近 Euler-Gompertz 常数 $δ$ 的有理数列，给出了证明其无理性的充分条件。