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Dirichlet series associated with square of class numbers of binary quadratic forms

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You do site has therefore share! This download remark on the number of classes of binary quadratic forms of a given negative was normally numerous for the Agent-Based hundred citations. His first paper on the theory of numbers appeared in The results of ten years' study of everything published on the theory of numbers are contained in his Reports which appeared in the British Association volumes from to These reports are a model of clear and precise exposition and perfection of form.

They treat of linear indeterminate equations and congruences, and of the orders and genera of ternary quadratic forms. He contributed also two memoirs to the Proceedings of the Royal Society of and , in the second of which he remarks that the theorems of Jacobi, Eisenstein, and Liouville, relating to the representation of numbers by 4, 6, 8 squares, and other simple quadratic forms are deducible by a uniform method from the principles indicated in his paper.

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Theorems relating to the case of 5 squares were given by Eisenstein, but Smith completed the enunciation of them, and added the corresponding theorems for 7 squares. The solution of the cases of 2, 4, 6 squares may be obtained by elliptic functions, but when the number of squares is odd, it involves processes peculiar to the theory of numbers.

This class of theorems is limited to 8 squares, and Smith completed the group.

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In ignorance of Smith's investigations, the French Academy offered a prize for the demonstration and completion of Eisenstein's theorems for 5 squares. This Smith had accomplished fifteen years earlier. He sent in a dissertation in , and next year, a month after his death, the prize was awarded to him, another prize being also awarded to H. Minkowsky of Bonn. The theory of numbers led Smith to the study of elliptic functions. He wrote also on modern geometry. His successor at Oxford was J. Ernst Eduard Kummer — , professor in the University of Berlin, is closely identified with the theory of numbers.

In the effort to overcome this difficulty, Kummer was led to introduce the conception of "ideal numbers.

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Zolotareff of St Petersburg to the solution of a problem of the integral calculus, left unfinished by Abel Liouville's Journal , Second Series, , Vol. Dedekind has taken the roots of any irreducible equation with integral coefficients as the units for his complex numbers. Attracted by Kummer's investigations, his pupil, Leopold Kronecker — made researches which he applied to algebraic equations. On the other hand, efforts have been made to utilise in the theory of numbers the results of the modern higher algebra.

Seeber; and that of the arithmetical automorphics of such forms, by Eisenstein. Hermite showed that the number of non-equivalent classes of quadratic forms having integral coefficients and a given discriminant is finite, while Zolotareff and A. Korkine, both of St. Petersburg, investigated the minima of positive quadratic forms.

In connection with binary quadratic forms. Smith established the theorem that if the joint invariant of two properly primitive forms vanishes, the determinant of either of them is represented primitively by the duplicate of the other.

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The interchange of theorems between arithmetic and algebra is displayed in the recent researches of J. Glaisher of Trinity College born and Sylvester. Sylvester gave a Constructive Theory of Partitions, which received additions from his pupils, F. Franklin and G. The conception of "number" has been much extended in our time. With the Greeks it included only the ordinary positive whole numbers; Diophantus added rational fractions to the domain of numbers. Later negative numbers and imaginaries came gradually to be recognised.

Descartes fully grasped the notion of the negative; Gauss, that of the imaginary. With Euclid, a ratio, whether rational or irrational, was not a number. The recognition of ratios and irrationals as numbers took place in the sixteenth century, and found expression with Newton.

By the ratio method, the continuity of the real number system has been based on the continuity of space, but in recent time three theories of irrationals have been advanced by Weierstrass, J. Dedekind, G.

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  • Cantor, and Heine, which prove the continuity of numbers without borrowing it from space. They are based on the definition of numbers by regular sequences, the use of series and limits, and some new mathematical conceptions.

    Continued fractions, binary quadratic forms and Markov spectrum

    The immediate question that we ask is: can we use the values of a lax superbase to find values elsewhere? The answer is yes. We can find the form that it represents. So given a superbase, we can determine a binary quadratic form that represents it. We can find other outputs on the topograph by forming arithemtic progressions elsewhere. This is best show by example. You can think of the topograph as a tree that branches out infinitely. This tree, or Range Topograph as we call it, represents all the values of quadratic forms at primitive lax vectors. In fact, each binary qua- dratic form that the topograph represents is equivalent to every other quadratic binary quadratic form represented by the topograph.

    For different discriminants, the topograph has different features. Since we are only concerned with indefinite binary quadratic forms we can omit certain features. There are few things that all topographs share in common. One of them is what we call The Climbing Lemma: Lemma 3. Suppose a, b, c, d are positive integers that label a cell in the Range Topograph represented below. All topographs that represent any indefinite integral binary quadratic form have a feature called the river.

    Definition 3. The river is the set of all edges on a range topograph that have both a positive and negative value adjacent to it. There are infinitely many such edges on the topograph, and the river is an infinite line composed of edges in a topograph.


    A very important fact about the river: the river is periodic, meaning that values on the river occur in a repeating pattern. Next we introduce the idea of a river bend: Definition 3. We will call these conditions on u, v, e, f the river-bend condition. Since the river is periodic, there are a finite number of river bends on a given topograph. This concludes our introduction to topographs.

    We will now present a few particularly interesting theorems and proofs that are results of the topograph. To restate the theorem: Theorem 4. We observe that 1 is on the river, and since the river is periodic by the definition, 1 occurs an infinite amount of times on the topograph. We conclude that there are infinitely many inputs that yield the output 1. Definition 4.