scholarly journals Odd Ramanujan Sums of Complex Roots of Unity

2007 ◽  
Vol 14 (1) ◽  
pp. 20-23 ◽  
Author(s):  
Soo-Chang Pei ◽  
Kuo-Wei Chang
Keyword(s):  
Roots Of Unity ◽  
Complex Roots ◽  
Ramanujan Sums

scholarly journals Roots of Elliptic Scator Numbers

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 321
Author(s):  
Manuel Fernandez-Guasti
Keyword(s):  
Abelian Group ◽  
Geometric Interpretation ◽  
Roots Of Unity ◽  
Complex Roots ◽  
De Moivre’S Formula

The Victoria equation, a generalization of De Moivre’s formula in 1+n dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in S1+n of a real or a scator number, there are qn possible roots. For n=1, the usual q complex roots are obtained with their concomitant cyclotomic geometric interpretation. For n≥2, in addition to the previous roots, new families arise. These roots are grouped according to two criteria: sets satisfying Abelian group properties under multiplication and sets catalogued according to director conjugation. The geometric interpretation is illustrated with the roots of unity in S1+2.

Some Formulas Involving Ramanujan Sums

1962 ◽  
Vol 14 ◽  
pp. 284-286 ◽  
Author(s):  
C. A. Nicol
Keyword(s):  
Monic Polynomial ◽  
Cyclotomic Polynomial ◽  
Roots Of Unity ◽  
Ramanujan Sums ◽  
Image Position ◽  
Positive Integers

The purpose of this note is to establish an identity involving the cyclotomic polynomial and a function of the Ramanujan sums. Some consequences are then derived from this identity.For the reader desiring a background in cyclotomy, (2) is mentioned. Also, (4) is intimately connected with the following discussion and should be consulted.The cyclotomic polynomial Fn(x) is defined as the monic polynomial whose roots are the primitive nth roots of unity. It is well known that2.1For the proof of Corollary 3.2 it is mentioned that Fn(0) = 1 if n > 1 and that Fn(x) > 0 if |x| < 1 and 1 < n.The Ramanujan sums are defined by2.2where the sum is taken over all positive integers r less than or equal to n and relatively prime to n. It is also well known that2.3where the sum is taken over all positive divisors d common to n and k.

scholarly journals Generators and Relations for Cyclotomic Units

1966 ◽  
Vol 27 (2) ◽  
pp. 401-407 ◽  
Author(s):  
Hyman Bass
Keyword(s):  
Number Theory ◽  
Roots Of Unity ◽  
Generators And Relations ◽  
Complex Roots ◽  
Complete Set ◽  
Cyclotomic Units

We prove here an unpublished conjecture of Milnor which gives a complete set of multiplicative relations between the numbers e′(ζ) = 1−ζ,where ranges over complex roots of unity. Information of this type is useful in certain areas of topology as well as in number theory.

A note on B- and Br- incomplete topological Abelian groups

1969 ◽  
Vol 66 (2) ◽  
pp. 275-279 ◽  
Author(s):  
L. J. sulley
Keyword(s):  
Abelian Groups ◽  
Roots Of Unity ◽  
Complex Roots

From results of Baker (2) it appeared to be unlikely that, in Hausdorff topological Abelian groups, completeness would be implied by Br- or B-completeness (for definitions, see below). We show here (Corollary 1) that the group of all complex roots of unity, though not complete, is B-complete. Another example, for the suggestion of which we are indebted to Dr J. W. Baker, is used to show (Corollary 2) that B-completeness is not a consequence of Br-completeness. It is proved, however, that B- and Br-complete Hausdorff topological Abelian groups are embedded in their completions in special ways in relation to the closed subgroups of the completions. Also, the completions of B- and Br-complete Hausdorff topological Abelian groups are shown to be, respectively, B- and Br-complete.

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