Abstract:
Due to strong absorption of the incident light, the media with high refractive index are considered restrictive for applications in photonic crystals (PhCs). The possibility to resolve this problem by optical saturation effectively minimizing the absorption of the PhC medium is discussed. Such approach might be promising for the significant broadening of the photonic band-gap.

Abstract:
We derived the sum identities for generalized harmonic and corresponding oscillatory numbers for which a sieve procedure can be applied. The obtained results enable us to understand better the properties of these numbers and their asymptotic behavior. On the basis of these identities a simple proof of the Prime Number Theorem is represented.

Abstract:
Using a sieve procedure akin to the sieve of Eratosthenes we show how for each prime $p$ to build the corresponding M\"obius prime-function, which in the limit of infinitely large primes becomes identical to the original M\"obius function. Discussing this limit we present two simple proofs of the Prime Number Theorem. In the framework of this approach we give several proofs of the infinitude of primes.

Abstract:
The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At least one of these identities may be applied to prove the Riemann Hypothesis by induction. Additionally using this approach, the new series for Euler's constant gamma has been found.

Abstract:
Dirac delta function (delta-distribution) approach can be used as efficient method to derive identities for number series and their reciprocals. Applying this method, a simple proof for identity relating prime counting function (pi-function) and logarithmic integral (Li-function) can be obtained.

Abstract:
We obtained the probabilities for the values of the M\"obius function for arbitrary numbers and found that the asymptotic densities of the squarefree integers among the odd and even numbers are $8/\pi^2$ and $4/\pi^2$, respectively. It is determined that statistics of successive outcomes of the M\"obius function for very large squarefree odd and even numbers behaves similar to statistics of heads and tails of two flipping coins. These preliminary results are giving arguments supporting the Riemann Hypothesis. Its plausibility is based on statistical phenomena for integers.

Abstract:
We obtained the formulas for the quantities of positive, negative and zero values of the Mobius function for any real x in terms of the Mobius function values for square root of x - similar to the identities we found earlier for the Mertens function [1]. Using the Dirac delta function approach [3, 2] we propose the equations showing how the Mertens and related functions can generate the output values.

Abstract:
The prime detecting function (PDF) approach can be effective instrument in the investigation of numbers. The PDF is constructed by recurrence sequence - each successive prime adds a sieving factor in the form of PDF. With built-in prime sieving features and properties such as simplicity, integro-differentiability and configurable capability for a wide variety of problems, the application of PDF leads to new interesting results. As an example, in this exposition we present proofs of the infinitude of twin primes and the first Hardy-Littlewood conjecture for prime pairs (the twin prime number theorem). On this example one can see that application of PDF is especially effective in investigation of asymptotic problems in combination with the proposed method of test and probe functions.

Abstract:
Using the theorem of residues Chiarella and Reichel derived a series that can be represented in terms of the complex error function (CEF). Here we show a simple derivation of this CEF series by Fourier expansion of the exponential function $\exp ({- {\tau ^2}/4})$. Such approach explains the existence of the lower bound for the input parameter $y = \operatorname{Im} [z]$ restricting the application of the CEF approximation. An algorithm resolving this problem for accelerated computation of the CEF with sustained high accuracy is proposed.