Fourier Expansion of the Riemann zeta function and applications

We study the distribution of values of the Riemann zeta function $\zeta(s)$ on vertical lines $\Re s + i \mathbb{R}$, by using the theory of Hilbert space. We show among other things, that, $\zeta(s)$ has a Fourier expansion in the half-plane $\Re s \geq 1/2$ and its Fourier coefficients are the binomial transform involving the Stieltjes constants. As an application, we show explicit computation of the Poisson integral associated with the logarithm of $\zeta(s) - s/(s-1)$. Moreover, we discuss our results with respect to the Riemann and Lindel\"{o}f hypotheses on the growth of the Fourier coefficients.Comment: 21 page

Log-tangent integrals and the Riemann zeta function

We show that integrals involving the log-tangent function, with respect to any square-integrable function on  , can be evaluated by the harmonic series. Consequently, several formulas and algebraic properties of the Riemann zeta function at odd positive integers are discussed. Furthermore, we show among other things, that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and its values depend on the Dirichlet series , where . &nbsp

A Pseudo-Random Number Generator Using Double Pendulum

Chaos in the double pendulum motion has been proved in several studies. Despite this useful cryptographic propriety, this system has not been applied to cryptography yet. This paper presents a new pseudo random number generator based on a double pendulum. Randomness of the numbers generated by the proposed generator is successfully tested by NIST and DIEHARDER tests. The results of the security analysis asserted the appropriateness of the new generator for cryptographic applications

Optimization of Surplus Reinsurance Treaty using the Conditional Tail Expectation

In this work, we propose a new optimization strategy for reinsurance using the genetic algorithms. This approach is to determine an optimal structure of a "surplus" reinsurance contract by finding the optimal cession rates through an optimization model which is based on the minimization of the Conditional Tail Expectation (CTE) risk measure under the constraint of technical benefit. This approach can be seen as a decision support tool that can be used by managers to minimize the actuarial risk and maximize the technical benefit in the insurance company