In this paper we address some variants for the products of Hadamard and Patterson. We prove that all zeros of the Riemann $\Xi$--function are real. We also prove that the Riemann hypothesis is true. The equivalence theorems associated with the Riemann zeta--function are obtained in detail.

This paper attempts to extend the Riemann Zeta function of a complex number to a function of a matrix and/or a tensor $A$, namely $$\zeta (A)=\sum _{n=1}^{\infty} \frac{1}{n^{A}}= \sum _{n=1}^{\infty} \sum_{k=1}^n\lambda_k A^k$$ and inverse $$A=\sum _{n=1}^{\infty} \sum_{k=1}^n \mu(n)\lambda_k \zeta(A^k)$$ where $\mu(n)$ is the M\"obius function, $A$ is a complex matrix or tensor with any order, and $\lambda_k$ is eigenvalue of the matri/tensor $A$. This kind of calculations on the Riemann Zeta function has never been seen in the literature. Some examples are provided.

In this part of the series of two papers, we extend the theorems discussed in part I for infinite series. We then use these theorems to develop distinct novel results involving the Hurwitz zeta function, Riemann zeta function, Polylogarithm and Fibonacci numbers. In continuation, we obtain some zeros of the newly developed zeta functions and explain their behaviour using plots in complex plane. Furthermore, we provide particular cases for the theorems and corollaries which show that our results generalise the currently available functions and series such as the Riemann zeta function and the geometric series. Finally, we provide four miscellaneous examples to showcase the vast scope of the developed theorems.

In this article we propose the integral, series and product representations for the Ramanujan zeta-function. We suggest a variant for the Conrey-Ghosh product for the entire Ramanujan zeta-function. We present some variants for the product for the Ramanujan $\Xi$-function. We prove that all of its zeros are real. Along the way we obtain the truth of the Riemann-Hardy conjecture.

The basic idea is to expand the completed zeta function $\xi(s)$ in MacLaurin series (infinite polynomial), which can be further expressed as infinite product (Hadamard product) by conjugate complex roots. Finally, the functional equation $\xi(s)=\xi(1-s)$ leads to $(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$ with solution $\alpha_i= \frac{1}{2}, i \in \mathbb{N}$, where $\alpha_i$ are the real parts of the zeros of $\xi(s)$, i.e., $s_i =\alpha_i\pm j\beta_i, i\in \mathbb{N}$. Therefore, a proof of the Riemann Hypothesis is achieved.

The paper continues the study of efficient algorithms for the computation of zeta functions over the complex plane. We aim to apply the modifications of algorithms to the investigation of underlying fractal structures associated with the Riemann zeta function. We discuss the computational complexity and numerical aspects of the implemented algorithms based on series with binomial-like coefficients.

It is well known that the primes and prime powers have a deep relationship with the nontrivial zeros of Riemann’s zeta function. This is a reciprocal relationship. The zeros and the primes are encoded in each other and are reciprocally recoverable. Riemann’s zeta is an extended or continued version of Euler’s zeta function which in turn equates with Euler’s product formula over the primes. This paper shows that the zeros of the converging Dirichlet or Catalan beta function, which requires no continuation to be valid in the critical strip, can be easily determined. The imaginary parts of these zeros have a deep and reciprocal relationship with the odd primes and odd prime powers. This relationship separates the odd primes into those having either 1 or 3 as a remainder after division by 4. The vector pathway of the beta function is such that the real part of its zeros has to be a half.

It is due to Littlewood that the truth of the Riemann theorem implies that of the Lindel\"{o}f conjecture. This paper aims to use the idea of Littlewood to prove the Lindel\"{o}f conjecture for the Riemann zeta function. The Lindel\"{o}f $\mu $ function at the critical line is zero, with use of the Riemann theorem for the entire Riemann zeta function, proved based on the work of Heath-Brown. Our result is given to show that the Lindel\"{o}f conjecture, connected with the proof of the moment conjecture, is true.

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

In this paper, some convexity properties and some inequalities for the (p,k)-analogue of the Gamma function, Г_p,k(x) are given. In particular, a (p,k)-analogue of the celebrated Bohr-Mollerup theorem is given. Furthermore, a (p,k)-analogue of the Riemann zeta function, ζ_p,k(x) is introduced and some associated inequalities are derived. The established results provide the (p,k)-generalizations of some known results concerning the classical Gamma function.

This paper attempts to propose a Riemann spring model via an analogy between the Riemann Zeta function of a complex number and the elastic spring in series.

In the paper, the authors discover several series identities involving the Catalan numbers, the Catalan function, the Riemanian zeta function, and the alternative Hurwitz zeta function.

Euler’s product formula over the primes and Euler’s zeta function equate to enshrine the Fundamental Theorem of Arithmetic that every integer > 1 is the product of a unique set of primes. The product formula has no zero, and with a domain ≤1 Euler’s zeta diverges. Dirichlet’s eta function η(s), negates alternate terms of zeta, permitting convergence when s∈C and Re(s) < 1, and its non-trivial zeros {ρ}, have a deep relationship with the distribution of the primes. The Riemann Hypothesis is that all the non-trivial zeros have Re(ρ) = 1/2. This work examines the symmetries in a partial Euler’s zeta series with a complex domain equating it to the difference between two finite vector series whose matched terms have mirror-image arguments, but whose magnitudes differ when Re(s) ≠ 1/2. Analytical continuation generates a modified eta series η_l(s), in which every l^th term is multiplied by (1-l). If the integer l is appropriately determined by the Im(s), similar paired finite vector series have a difference that closely follows η_l (s) and their terminal vectors intersect in a unique way permitting zeros only when Re(s) = 1/2. Furthermore, those vectors tracking the derivatives of the series, have a special relationship permitting zeros of the differential only when Re(s) > 1/2.

Proof of the Riemann hypothesis is obtained using the properties of the Gamma function, the Bernoulli function, and limitations imposed by the rational relations of the complex roots of the Riemann zeta function, and the rational relations of the complex Gamma functions.

The interaction of solution of poly (diallyldimethylammonium) chloride, PDADMAc and sodium dodecylsulfate, SDS has been studied using conductometric, densimeter and zeta potential methods. the critical micellization concentration (cmc) was determinated with conductance results and increased with the addition of PDADMAc comparing with solutions containing pure SDS. Further the degree of dissociation,  and thermodynamic parameters viz. free energy of micellization, and entropy of micellization, has been evaluated. there results demonstrated that surfactant/polymer complex is governed by both electrostatic and hydrophobic interactions. The apparent molar volume, Vϕ and adiabatic compressibility, Kϕ;s have been determined for the SDS solution from density and speed sound data. the negative charge of the SDS solutions with polymer shows that these complexes probably do not contain surfactant in the form of micelles.

The aim of this work was to prepare bioplastics from renewable and biodegradable molecules. In particular, the bioplastics were produced by using as biopolymer source the grass pea (Lathyrus sativus L.) flour, the proteins of which were structurally modified by means of microbial transglutaminase, an enzyme able to catalyze isopeptide bonds between glutamines and lysines. We analyzed, by means of Zeta-potential, the flour suspension with the aim to choose which pH is more stable for the production of film-forming solutions. The bioplastics were produced by casting and they were characterized according to several technological properties. Optical analysis demonstrated that films cast in the presence of the microbial enzyme are more transparent compared to the untreated ones. Moreover, the visualization by Scanning Electron Microscopy demonstrated that the enzyme-modified films possessed a more compact and homogeneous structure. Furthermore, the presence of microbial transglutaminase allowed to obtain film more mechanically resistant. Finally, digestion experiments under physiological conditions performed in order to obtain information useful for applying these novel biomaterials as carriers in the industrial field, indicated that the enzyme-treated coatings might allow the delivery of bioactive molecules in the gastro-intestinal tract.

The case of a virtual bosonic scalar field phi(x) gravitationally coupled in a conformally flat spacetime is investigated. The action S is known to be Weyl invariant only for specific expressions of the potential V(phi). In the conformally flat FLRW case the harmonic angular frequency omega_k(eta) becomes uniquely temporally independent. Therefore any inertial observers embedded in a conformally flat FLRW spacetime all agree on the choice of virtual vacuum states. We postulate that the wavenumber k must be quantized in order to be able to regulate the vev by zeta regularization. Some fundamental implications of this result is derived, and likely conclusions drawn.

Highly colloidal Eu-doped Gd2O3 nanoparticles(core-NPs) were synthesized by thermal decomposition via weak base at low temperature (150oC), subsequently, silica layers were deposited to increased colloidal stability, solubility, biocompatibility and no-toxicity at the environmental condition. XRD results indicate the highly purified, crystalline, single phase cubic phase Gd2O3 nanocrystals. TEM image shows the mesoporous thick silica layer was effectively coated over the core nanocrystals, which have irregular size with nearly spherical shape and a mean grain size is about 10-30 nm. Absorption spectra and zeta potential results in aqueous media revealed that solubility, colloidal stability, and biocompatibility character was enhanced from core to core-shell structure because of silica layer surface encapsulation. The samples demonstrate excellent photoluminescence properties (dominant emission 5D0→7F2 transition in red region at 610 nm) indicated the advantage to use in optical bio-detection and bio-labeling etc. The photoluminescence intensity of the silica shell modified core/shell nanoparticles were suppressed relatively core-nanoparticles, it indicates the multi-photon relaxation pathways arising from the surface coated high vibrational energy molecules of the silanol groups. The core/nSiO2/mSiO2 nanocrystals display strong emission (5D0→7F2) transition along with excellent solubility and biocompatibility, which may find promising applications in photonic based biomedical applications.

Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function

Low temperature cofired ceramics (LTCC) have conquered a niche segment in biological microelectromechanical systems (BioMEMS) during the last two decades. Since 3-dimensional assembly and rapid prototyping capability are outstanding features of the technology, bioreactors with complex geometry can easily be produced. Particularly needed functions are working electrodes, e.g. for impedance measurements and reference electrodes consisting in platinum or silver-silver chloride. Although the distinct grainy surface of thick film materials influences the double layer properties, data describing the bioelectronic interface and biocompatibility of common thick film materials are rarely published up to now. This works aims to fill this gap by studying the surface properties, composition, electrochemical properties and biocompatibility of commercially available thick film materials. It was found that thick film gold is suitable as electrode material in direct cell contact since an appropriate proliferation of the cell culture was observed. Poly(3,4-ethylenedioxythiophene) coating decreases the absolute value of the zeta potential of thick film gold from 16.55 V to 11.78 mV without any change in vitality of the cells. Thick film platinum layers have a porous structure entailing an enlargement of the effective surface by a factor of 21.6 and can be used as reference instead of silver/silver chloride, which was identified to be incompatible with cell culture. The investigations show that some commercially available cost-effective thick film materials are compatible with cell culture and the here presented data give an orientation for the use of the same in LTCC bioreactors.

Protective coatings, in recent years also from nanocomposite formulations, are commonly applied onto architectural stone and stone artefacts, mainly to prevent absorption into the porous stone structure of condensed water and dissolved atmospheric pollutants. While standard protocols are available to assess a coating’s performance, understanding the response of the coating-stone system is a complex task, due to the interplay of various factors determining the overall behaviour. Characterization techniques allowing to correlate the extent and nature of surface modification upon treatment with the most relevant physical properties (i.e water absorption and surface wettability) are thus of great interest. Electrokinetic analysis based on streaming current measurements, thanks to its sensitivity towards even minor changes in the surface chemical composition, may fulfil such requirement. Indeed, by involving the interaction with a testing aqueous electrolyte solution, this technique allows to probe not only the outer surface but also the outermost layer of the pore network, which plays a crucial role in the interaction of the stone with condensed atmospheric water. In this work a correlation was found between the extent of surface modification, as determined by streaming current measurements, surface wettability and capillary water absorption of 6 coating-lithotype combinations (3 lithotypes and 2 nanocomposites).

In the paper, by virtue of some properties for the Riemann zeta function, the author finds a double inequality for the ratio of two consecutive Bernoulli numbers with even indexes and analyzes the approximating accuracy of the double inequality.

A series of LaF₃:Ce³⁺ phosphors for the application in photodynamic therapy is synthesized using a one-stage solvothermal synthesis. The conditions providing the maximum intensity of UV- and X-ray-excited luminescence, lowest size and highest colloidal stability of the phosphor nanoparticles are found to respectively include cerium content 5% mol., use of ethanol as the reaction medium for the solvothermal synthesis and addition of polyvinylpyrrolidone as a stabilizer in an optimized amount.

A concise proof of the Riemann Hypothesis is presented by clarifying the Hadamard product expansion over the zeta zeros, demonstrating that the Riemann Hypothesis is true.

This article discusses a transformer-free, high-efficiency DC-DC converter besides renewable energy applications. The traditional buck-boost, classic Zeta, Sepic, and Cuk converter does have the benefits of a simple design, low cost, as well as the capacity to execute voltage step-up and step-down. Conversely, because of the detrimental consequences of the parasitic constraints of the device, the voltage conversion gain of the traditional DC-DC converter is much more restricted and the efficiency is also significantly smaller, whereas this proposed converter does have a higher voltage gain and efficiency because it is used in a single power switch, resulting in reduced switching losses and voltage stress. The said converter's design is very simple, which simplifies the operation control and reduces switching and conduction losses, leading to an efficiency of 97.4 percent. This converter seems to have the same capabilities as the Zeta converter, including continuous desired output current and desired buck-boost operation. Such an article offers the operation principle and steady evaluation, as well as a comparison with other existing high step-up configurations. The proposed converter employs a fuzzy logic controller, which improves the voltage level as well as reduces the time taken to set the voltage output of a conventional PI and ANN controller, especially in comparison to the FLC controller. For deployment, Experimental Result and MATLAB/Simulink has been used, and the modeling results indicate that the proposed controller performance has improved

This paper addresses a variant of the product for the Dirichlet $L$--functions. We propose a completely detailed proof for the truth of the generalized Riemann conjecture for the Dirichlet $L$--functions, which states that the real part of the nontrivial zeros is $1/2$. The Wang and Hardy--Littlewood theorems are also discussed with removing the need for it. The results are applicable to show the truth of the Goldbach's conjecture.

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

In this article, author performs computational analysis for the generalized zeta functions by using computational software Mathematica. To achieve the purpose recently obtained difference equations are used. These difference equations have a computational power to compute these functions accurately while they can not be computed by using their known integral represenations. Several authors investigated such functions and their analytic properties, but no work has been reported to study the graphical representations and zeors of these functions. Author performs numerical computations to evaluate these functions for different values of the involved parameters. Taylor series expansions are also presented in this research.

This paper presents Mathematical Formulation for Neutrino-Fission of a heavy nuclei in Neutrino-environments. It has been proposed that neutrino fission is a dynamic two-process reaction, where more than one neutrino can be captured by target nucleus. A quantity named Neutrino-Zeta with units of eV^-1 has been defined to represent Neutrino-environments. Neutrino-zeta with assumed excited energy states for heavy nuclei A∼230-270 is estimated to be 0.0212 MeV^-1, with the probability of fission Barrier energy state P(E_x,f)= 0.169064014 with E_x,f =6MeV. P(E_x,f)=0.222664 is highest at zeta=0.1MeV^-1 . It has been shown that probabilities of certain excitation energy states and their corresponding decay reactions at those excitation energies can be equal for two neutrino-zetas. Neutron induced fission for different neutrinos hypothesize that probability of symmetric fission channels is higher at low neutrino-zetas, which can be tested experimentally.

The renewable energy source plays a major role in the grid side power production. The stability analysis is very essential in the renewable energy converters. In this paper the bifurcation is analyzed in ZETA converter and Continuous input and output(CIO) power Buck Boost converter. The ZETA converter gives positive step down and step up output voltage and the CIO power converter gives the negative step up and step down output voltage. These converters are used in the DC micro grid with renewable energy as the source. The current mode control technique is applied to analyze the bifurcation behavior and the reference current is taken as the bifurcation parameter. When the reference current is varied, both the converters loses its stability and it enters into chaotic region through period doubling bifurcation. The simulation results are presented to study the performance behavior of both the converters. The stability region of both the converters are determined by deriving the Monodromy matrix approach.

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.