There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

https://chaos.if.uj.edu.pl/~karol/erratum10.pdf

Geometry of Quantum States

Geometry of quantum states: an introduction to quantum entanglement by Ingemar Bengtsson and Karol Zyczkowski

Theory of Equations. By J. V. Uspensky. Pp. vii, 353. 27s. 1948. (McGraw-Hill)

Quantum wave packet revivals

Gauss Sums Jacobi Sums and Cyclotomic Numbers Evaluation of Jacobi Sums Over Fp Determination of Gauss Sums Over Fp Difference Sets Jacobsthal Sums Over Fp Residuacity Reciprocity Laws Congruences for Binomial Coefficients Diagonal Equations over Finite Fields Gauss Sums over Fq Eisenstein Sums Brewer Sums A General Eisenstein Reciprocity Law Research Problems Bibliography Notation Indexes

/pdf/gauss-and-jacobi-sums-24cgbz8sph.pdf

Gauss and Jacobi sums

Algebraic number theory is a subject which came into being through the attempts of mathematicians to try to prove Fermat's last theorem and which now has a wealth of applications to diophantine equations, cryptography, factoring, primality testing and public-key cryptosystems. This book provides an introduction to the subject suitable for senior undergraduates and beginning graduate students in mathematics. The material is presented in a straightforward, clear and elementary fashion, and the approach is hands on, with an explicit computational flavour. Prerequisites are kept to a minimum, and numerous examples illustrating the material occur throughout the text. References to suggested reading and to the biographies of mathematicians who have contributed to the development of algebraic number theory are given at the end of each chapter. There are over 320 exercises, an extensive index, and helpful location guides to theorems and lemmas in the text.

/pdf/introductory-algebraic-number-theory-mb2542qwe0.pdf

Introductory Algebraic Number Theory

A Simple Norm Inequality

Joseph Liouville is recognised as one of the great mathematicians of the nineteenth century, and one of his greatest achievements was the introduction of a powerful new method into elementary number theory. This book provides a gentle introduction to this method, explaining it in a clear and straightforward manner. The many applications provided include applications to sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums involving the divisor functions, and many others. All of the topics discussed have a rich history dating back to Euler, Jacobi, Dirichlet, Ramanujan and others, and they continue to be the subject of current mathematical research. Williams places the results in their historical and contemporary contexts, making the connection between Liouville's ideas and modern theory. This is the only book in English entirely devoted to the subject and is thus an extremely valuable resource for both students and researchers alike.

/pdf/number-theory-in-the-spirit-of-liouville-461qwaryds.pdf

Number Theory in the Spirit of Liouville

arises in the study of the gamma function (see, for example, [3, p. 47]). The series T (r, s, t) is finite if and only if (1.1) r + t > 1, s+ t > 1, and r + s+ t > 2. The basic properties of T (r, s, t) (assuming (1.1)) are the following T (r, s, t) = T (s, r, t), (1.2) T (r, s, 0) = ζ(r)ζ(s), (1.3) T (r, 0, t) + T (t, 0, r) = ζ(r)ζ(t)− ζ(r + t), r ≥ 2, (1.4) T (r, s− 1, t+ 1) + T (r − 1, s, t+ 1) = T (r, s, t), r ≥ 1, s ≥ 1, (1.5) where ζ denotes the Riemann zeta function.

/pdf/on-tornheim-s-double-series-38yqas8voq.pdf

Kenneth S. Williams

Papers

Gauss and Jacobi sums

Introductory Algebraic Number Theory

A Simple Norm Inequality

Number Theory in the Spirit of Liouville

On Tornheim's double series