The Riemann Hypothesis Essay Example for Free

The Riemann Hypothesis Essay The Riemann Zeta Function is defined by the following series: Here s is a complex number and the first obvious issue is to find the domain of this function, that is, the values of s where the function is actually defined. First of all, it is a well known result in calculus that, when s is real, the series is convergent for s1 (see [2]). For example, a simple application of the theory of Fourier series allows to prove that . For s=1, the series diverges. However, one can prove that the divergence is not too bad, in the sense that: In fact, we have the inequalities: Summing from 1 to , we find that and so which implies our claim. As a function of the real variable s,   is decreasing, as illustrated below.   for s real and 1 The situation is   more complicated when we consider the series as a function of a complex variable. Remember that a complex number is a sum , where   are real numbers (the real and the imaginary part of z, respectively) and , by definition. One usually writes   There is no ordering on the complex numbers, so the above arguments do not make sense in this setting.   We remind that the complex power   is defined by and Therefore, the power coincides with the usual function when s is real. It is not difficult to prove that the complex series is convergent if Re(s)1. In fact, it is absolutely convergent because where |z| denotes as usual, the absolute value: .   See [2] for the general criteria for convergence of   series of functions. Instead, it is a non-trivial task to prove that the Riemann Zeta Function can be extended far beyond on the complex plane: Theorem.   There exists a (unique) meromorphic function on the complex plane, that coincides with , when Re(s)1. We will denote this function again by We have to explain what ‘meromorphic’ means. This means that the function is defined, and holomorphic (i.e. it is differentiable as a complex function), on the complex plane, except for a countable set of isolated points, where the function has a ‘pole’. A complex function f(z) has a pole in w if the limit   exists and is finite for some integer m. For example,   has a pole in s=1. It is particularly interested to evaluate the Zeta Function at negative integers. One can prove the following: if k is a positive integer then where the Bernoulli numbers   are defined inductively by: Note that : the Bernoulli numbers with odd index greater than 1 are equal to zero. Moreover, the Bernoulli numbers are all rational. Of course, the number   is not obtained by replacing s=1-k in our original definition of the function, because the series would diverge; in fact, it would be more appropriate to write   where the superscript * denotes the meromorphic function whose values are defined, only when Re(s)1, by the series . There is a corresponding formula for the positive integers: 2 It is a remarkable fact that the values of the Riemann Zeta Function at negative integers are rational. Moreover, we have seen that   if n0 is even. The natural question arises: are there any other zeros of the Riemann Zeta Function? Riemann Hypothesis. Every zero of the Riemann Zeta Function must be either a negative even integer or a complex number of real part =  ½. It is hard to motivate this conjecture in an elementary setting, however the key point is that there exists a functional equation relating   and   (in fact, such a functional equation is exactly what is needed to extend   to the complex plane). The point   is the center of symmetry of the map It is also known that   has infinitely many zeros on the critical line Re(s)=1. Why is the Riemann Zeta function so important in mathematics? One reason is the strict connection with the distribution of prime numbers. For example, we have a celebrated product expansion: where the infinite product is extended to all the prime numbers and Re(s)1. So, in some sense, the Riemann Zeta function is an analytically defined object, encoding virtually all the information about the prime numbers. For example, the fact that   can be used to prove Dirichlet’s theorem on the existence of infinitely many prime numbers in arithmetic progression. The product expansion implies that   for every s such that Re(s)1. In fact, we have: and it is not difficult to check that this product cannot vanish. The following beautiful picture comes from Wikipedia. Bibliography [1] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 2000 [2] W. Rudin, Principles of Mathematical Analysis, McGraw Hill, 1976 [3] W. Rudin, Real and Complex Analysis, , McGraw Hill, 1986

Essay --

1. On April 22, 2013 a jury trial was planned for commence as the result of Timelines, Inc. suing Facebook, Inc. for trademark infringement. Timelines, Inc. filed with the courts after noticing that Facebook, Inc. added the new â€Å"Timeline† feature to its member profiles. Facebook motioned for a summary judgment, but the U.S. District Court for Northern District of Illinois turned down Facebook’s motion for a summary judgment. The reason they denied this motion is because Facebook failed to prove that the word â€Å"timeline† was of fair use, and that the term â€Å"timelines† was just a generic term. A party that moves for a summary judgment is trying to save the time and expense of a full trial when the outcome is usually obvious. Since Timelines, Inc. has a registered trademark of the â€Å"timelines† phrase; the court affirmed that they were entitled to a presumption of validity. If I were a member of the jury, I would rule in favor of Timelines, Inc. because Facebook went against the trademark laws, and was not able to prove that the phrase â€Å"timeline† was of fair use. Fair use can be proved only if the defense can prove that such use was for a purpose other than that normally made of a trademark. The party that is being charged with trademark infringement may defeat proceedings ONLY if it can prove a valid exception of fair use. If companies did not trademark their mark, multiple companies would use similar marks causing consumer confusion about the origin of the defendant’s goods/services. As much as I love Facebook, I feel as though they took someone else’s creation, and used their power to make the concept legendary. Trademark infringement is a violation of the exclusive rights attached to a trademark without the authorization of the tra... ... the duplication of the copyrighted work must be outside the exceptions. The combination, the lyrics and associated melodies, intonations, and usage are too similar to the original. This violates the copyright trademark laws, and it is protectable material. The puzzle pieces match knowing that the composer’s advisors, John and Charles, listened to her song in 2008, critiqued it, and told her it was best as a duet. Copyright infringement is the act of violating any of a copyright owner’s exclusive rights that were granted by the Federal Copyright Act. The only exceptions to the copyright law are fair use, face-to-face instruction, and virtual instruction. These exceptions allow someone to use work without permission from the copyright holder, and potentially paying fees. If Bowen wins the case, she could potentially receive the actual dollar amount of the profits. Essay -- 1. On April 22, 2013 a jury trial was planned for commence as the result of Timelines, Inc. suing Facebook, Inc. for trademark infringement. Timelines, Inc. filed with the courts after noticing that Facebook, Inc. added the new â€Å"Timeline† feature to its member profiles. Facebook motioned for a summary judgment, but the U.S. District Court for Northern District of Illinois turned down Facebook’s motion for a summary judgment. The reason they denied this motion is because Facebook failed to prove that the word â€Å"timeline† was of fair use, and that the term â€Å"timelines† was just a generic term. A party that moves for a summary judgment is trying to save the time and expense of a full trial when the outcome is usually obvious. Since Timelines, Inc. has a registered trademark of the â€Å"timelines† phrase; the court affirmed that they were entitled to a presumption of validity. If I were a member of the jury, I would rule in favor of Timelines, Inc. because Facebook went against the trademark laws, and was not able to prove that the phrase â€Å"timeline† was of fair use. Fair use can be proved only if the defense can prove that such use was for a purpose other than that normally made of a trademark. The party that is being charged with trademark infringement may defeat proceedings ONLY if it can prove a valid exception of fair use. If companies did not trademark their mark, multiple companies would use similar marks causing consumer confusion about the origin of the defendant’s goods/services. As much as I love Facebook, I feel as though they took someone else’s creation, and used their power to make the concept legendary. Trademark infringement is a violation of the exclusive rights attached to a trademark without the authorization of the tra... ... the duplication of the copyrighted work must be outside the exceptions. The combination, the lyrics and associated melodies, intonations, and usage are too similar to the original. This violates the copyright trademark laws, and it is protectable material. The puzzle pieces match knowing that the composer’s advisors, John and Charles, listened to her song in 2008, critiqued it, and told her it was best as a duet. Copyright infringement is the act of violating any of a copyright owner’s exclusive rights that were granted by the Federal Copyright Act. The only exceptions to the copyright law are fair use, face-to-face instruction, and virtual instruction. These exceptions allow someone to use work without permission from the copyright holder, and potentially paying fees. If Bowen wins the case, she could potentially receive the actual dollar amount of the profits.

Native Americans in the United States and Hardy Individualism Essay

Prompt: Although the development of the Trans-Mississippi West is popularly associated with hardy individualism, it was in fact largely dependent on the federal government. Assess the validity of this statement with specific reference to western economic activities in the 19th century. In the late 1800s and early 1900s, the idea of the far west captivated many. The chance to begin life anew attracted thousands of individuals and families alike to move out west and escape their current life, which was usually full of poverty and for some, full of discrimination. As the west expanded and grew into an important part of the United States, westerners found it somewhat difficult to survive with important resources going scarce. Although the development of the Trans-Mississippi west is mainly associated with hardy individualism, the west? s development as a whole was largely the result of the aid of the federal government by constructing railroads, promoting and protecting the land, and removing the Indian tribes. Railroads were an integral part of the west; without them the West would not be successful. The distance of the west from the rest of the country was large and the only way to reach the west was through a long, tiresome journey by wagon. The Pacific Railroad Act of 1862 paved the way for the expansion of the railroads. The Act gave companies land to build railroads. The faster the company built the railroad, the faster they could get more land, which they usually sold for profit later on. The construction of the railroad benefitted many who were not living in the west, namely Chinese immigrants. With thousands of workers, railroad companies had to ensure their safety to prevent being sued and frowned upon by the public. To prevent that, railroad companies provided many necessities for their workers like shelter.

Cyber Threat, Its Scope And Its Impact On National Security

Cyberspace which is an offshoot for the development of computer and digital communication technologies, has in recent decades become part and parcel of our daily lives. It has become an ideal source for cyber criminals to remain active while preying on victims. Higher the number of cyberspace users’, higher is the opportunities for exploitation. So it is the need of the hour to protect our computers, networks, digital applications and our data from unintended or unauthorized source, change or destruction. This paper proposes a policy directed examination on cyber threat, its scope and its impact on national security. It includes a profile of cooperation among criminals, terrorists and hostile organizations and discusses future of cybercrime in 2020. In recent years, â€Å"Information warfare† a new way of terrorism has become the major concern for information security specialists; terrorists might tamper with computers and electronic devices to commit secured information based threats to nations, to businesses and to individuals. INTRODUCTION In a world where information and communications technology (ICT) that provides the mode so people can work with each other electronically in a digital form over large distances, cyber threats and crimes are of great concern. Computerized technologies are implemented to improve and enhance the efficiency of creative and working processes in every facet of life and the world of cybercrime is no exception. Cybercrime is an illegal activityShow MoreRelatedCyber Threat, Its Scope And Its Impact On National Security2043 Words   |  9 Pageslives. It has become an ideal source for cyber criminals to remain active while preying on victims. 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For the military, an overhaul of the whole concept of Defence has gradually changed as cyber security is streamlined into defence policy coupled with economic influence and globalisation. ‘In the military, information and intelligence operations, routine administrative functions, and a wide array of everyday jobs have been increasingly developed and transformed with the support of interconnected electro-electronic devices’ . Defence policy and military strategy go hand in hand and DefenceRead MoreInternational Terrorism And The Security Of The United Kingdom1665 Words   |  7 Pagesinternational terrorism represents the greatest threat to the security of the United Kingdom. It will begin by defining, within the context of this essay, what is meant by the concepts of national security, terrorism and international terrorism, and how international terrorism threatens our nation through both direct and indirect means. Throughout, it will identify a snapshot of the current gamut of security threats to the United Kingdom and analyse, through these threats, how international terrorism poses theRead MoreCyber Crime Case1458 Words   |  6 Pagesquickly. Cyber crime which are increasing at faster rate should be controlled to protect the organization, CSIRT is the pillar for any organization should act effectively in responding. As many other developing countries, the internet usage in Colombia has also increas ed rapidly. The Government of Colombia want to develop CSIRT called ColCERT to secure their organizations. In Colombia use communication technologies has increased significantly, this raised the countrys level of exposure to cyber threatsRead MoreThe Department Of Health And Human Services1414 Words   |  6 Pages(DHHS) security classifications of its system. †¢ This policy has the purpose to ensure that the Department of Health of Human Services (DHHS) is in compliance with the security standards established by NIST SP 800-53. †¢ The policy is also schemed to identify the five core framework functions: identify, protect, detect, respond, and recover. This will help the Department of Health and Human Services (DHHS) to manage cybersecurity risks in both internal and external manners. Scope: †¢ The scope of theRead MoreGovernment Enforced Cyber Security, a Public Good?3640 Words   |  15 PagesGovernment enforced Cyber Security; A public good? 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The United States Executive Office of the President stated, â€Å"The President identified cybersecurity as one of the top priorities of his administration in doing so, directed a 60-day review to assess polices.† (United States Executive Office of the President, 2009, p.2). Furthermore, critical infrastructure, our network, and internet alike are identified as national assets upon which the administration will orchestrate integrated