The Riemann Zeta Function

In this discussion I give an explanation of the divergence and convergence of infinite series through the building blocks problem and at the same time I touch on the fact that mathematics is not just about manipulating complicated numerical formulas but also a field in which logical ways of thought are learnt. I emphasize that in order to overcome the aversion of university students to mathematics, teachers must pour their energies into developing study materials taken from topics relevant to the students.

Convergence of infinite series

By a series we mean a set of numbers a1, a2, a3… such that we have a rule for calculating a2, a3 etc. from the first number a1.Series occur in many problems in chemistry such as specific heats of solids, the theory of black-body radiation, solution of the Schrödinger equation, statistical thermodynamics and Fourier series in X-ray crystallography. In this chapter we consider the summation of series and the convergence of infinite series. We shall develop methods for the expansion of functions such as transcendental functions in terms of infinite series.

The classic problems of the two-state linear curve crossing both for the same and the opposite sign of slopes of the linear diabatic potentials are analyzed in a unified way by exactly dealing with the Stokes phenomenon for the four transition points of the associated second-order differential equations. First, distributions of the transition points and the Stokes lines are fully clarified for the whole range of the two parameters which effectively represent the coupling strength and the collision energy. Secondly, the so-called reduced scattering matrices are found to be expressed in terms of only one (complex) Stokes constant. This is made possible by finding the relations among the six Stokes constants. Finally, this one Stokes constant is given exactly and analytically by a convergent infinite series.

A one-parameter linear group in a vector space L is a continuous map \( g:t\mapsto {g^t}=g(t) \) on the space of parameters ℝ, taking values in a space of the linear operators on L and having the following properties: (1) g 0 = E and (2) the sum of parameter values corresponds to the composition \( {g^{s+t }}={g^s}\circ {g^t} \). Thus, one-parameter linear groups in ℝ are the continuous solutions of the functional equations g(0) = 1, g(t + s) = g(t)g(s). As is well known, exponential functions satisfy these equations. After solving the first problems in this chapter, readers will find that the continuous solutions are all exponential and, thus, will extract the usual properties of exponential functions from these functional equations (which corresponds to a real historical development).The second part of the chapter is devoted to proper generalizations for one-parameter matrix groups. Readers will find that all of them are given by the matrix exponential series, as in the one-dimensional case. However, a multidimensional picture looks much more complicated. Working on these problems will familiarize readers with matrix analysis. In addition, readers may need familiarity with some tools used by number theory and the theory of differential equations, such as Euclid’s algorithm, Chinese remainder theorem, and Poincaré’s recurrence theorem; we introduce and discuss these tools.In the third part of the chapter readers will find some elementary classic applications of one-parameter matrix groups in differential equation theory (Liouville formula), in complex analysis (e.g., complex exponential functions, Euler’s formula), for finite-dimensional functional spaces (spaces of quasipolynomials), and others. Readers will see how to deal with these problems using elementary tools of analysis and linear algebra.Further, part of this chapter contains problems for readers possessing more advanced preliminary experience (although we introduce and discuss the necessary tools). While working on these problems, readers will come across many interesting things regarding analysis and linear algebra and become acquainted with important concepts (symplectic forms and others).Readers will encounter far-reaching advances and applications of the subjects considered in the present chapters in powerful mathematical theories: of differential equations, Lie groups, and group representations (a guide to the literature is provided).KeywordsProblem GroupNumerical SeriesGreat Common DivisorExponential SeriesLess Common MultipleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

From their inception, Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. The author's aim is to present a straightforward and easily accessible survey of the main ideas of the theory at an elementary level, providing a sound basis from which the reader can study advanced works and undertake original research. This book is based on lectures given by the author for a number of years and is intended for a one-semester graduate course, though it can also be used profitably for self-study. The only prerequisites are a basic knowledge of algebra, number theory and complex analysis.

In this paper we present a simple method for finding the location of limit cycles of self excited oscillators whose damping functions can be represented by some infinite convergent series. We have used standard results of first-order perturbation theory to arrive at amplitude equations. The approach has been kept pedagogic by first working out the cases of finite polynomials using elementary algebra. Then the method has been extended to various infinite polynomials, where the fixed points of the corresponding amplitude equations cannot be found out. Hopf bifurcations for systems with nonlinear powers in velocities have also been discussed.

In these notes we try to demonstrate the utility of the theory of theta functions in combinatorial number theory and complex analysis. The main idea is to use identities among theta functions to deduce either useful number-theoretic information related to representations as sums of squares and triangular numbers, statements concerning congruences, or statements concerning partitions of sets of integers. In complex analysis the main utility is in the theory of compact Riemann surfaces, with which we do not deal. We do show how identities among theta functions yield proofs of Picard’s theorem and a conformal map of the rectangle onto the disk.

A new method called the modification of step variational iteration method (MoSVIM) is introduced and used to solve the fractional biochemical reaction model. The MoSVIM uses general Lagrange multipliers for construction of the correction functional for the problems, and it runs by step approach, which is to divide the interval into subintervals with time step, and the solutions are obtained at each subinterval as well adopting a nonzero auxiliary parameter ℏ to control the convergence region of series′ solutions. The MoSVIM yields an analytical solution of a rapidly convergent infinite power series with easily computable terms and produces a good approximate solution on enlarged intervals for solving the fractional biochemical reaction model. The accuracy of the results obtained is in a excellent agreement with the Adam Bashforth Moulton method (ABMM).

Accuracy of the Adomian decomposition method applied to the Lorenz system

We demonstrate that there exist globally convergent infinite tanh power series with which the spinless light-bullet solutions in the (3+1)D cubic-quintic complex Ginzburg-Landau equation can be exactly expressed. It is found that as these fixed-point bullet solutions exist, the series coefficients asymptotically approach certain nonzero constants; otherwise they oscillate and eventually decay to nil. In terms of the specific Pad\'e approximants, the analytical solutions obtained for either the conventional bullets or the composite ones are in excellent agreement with numerical simulations.

In this paper, the Homotopy analysis Method (HAM) is applied to the Maxwell system. The HAM yields an analytical solution in terms of a rapidly convergent infinite power series with easily computable terms.

1. IntroductionThe study of in¯nite series has been a scintillating sub-ject and its connection with the theory of numbers has avery long history. Inourcollege days, we are taughtvari-ous tests fordecidingwhetheragivenserieswillconvergeor not. When the series is convergent, one is interestedin ¯nding the sum ofthe series and the arithmetic natureof the sum. This still remains a challenging question formany well-known convergent series. In the present ar-ticle, we shall discuss the questions of irrationality andtranscendence of sums of some in¯nite series.2. De¯nitions and Some Basic ObservationsArealnumber® is called algebraic if there exists a poly-nomial P(x)=a

This paper describes a new method of obtaining an equivalent series from a given infinite convergent or divergent series. In many cases the new series is more convenient for summing than the original and, moreover, the same method may usually be repeated indefinitely to obtain an entire sequence of series each equivalent to the original and each better than its predecessor in summing properties. I h e new method differs from most summing methods heretofore employed in that terms of the transformed series are not linear functions of terms of the original series. The paper includes proofs of theorems indicating the scope of the new method and comparisons of results with various other methods for many specific examples.