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NMR implementations of Gauss sums  [PDF]
Jonathan A. Jones
Physics , 2008, DOI: 10.1016/j.physleta.2008.07.019
Abstract: I describe the use of NMR experiments which implement Gauss sums as a method for factoring numbers and discuss whether this approach can be computationally useful.
Note on the quadratic Gauss sums
George Danas
International Journal of Mathematics and Mathematical Sciences , 2001, DOI: 10.1155/s016117120100480x
Abstract: Let p be an odd prime and {χ(m)=(m/p)}, m=0,1,...,p−1 be a finite arithmetic sequence with elements the values of a Dirichlet character χ  modp which are defined in terms of the Legendre symbol (m/p), (m,p)=1. We study the relation between the Gauss and the quadratic Gauss sums. It is shown that the quadratic Gauss sums G(k;p) are equal to the Gauss sums G(k,χ) that correspond to this particular Dirichlet character χ. Finally, using the above result, we prove that the quadratic Gauss sums G(k;p), k=0,1,...,p−1are the eigenvalues of the circulant p×p matrix X with elements the terms of the sequence {χ(m)}.
Two truncated identities of Gauss  [PDF]
Victor J. W. Guo,Jiang Zeng
Mathematics , 2012, DOI: 10.1016/j.jcta.2012.12.004
Abstract: Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting the partitions of $n$ with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.
Gauss sums of cubic characters over $GF(p^r)$, $p$ odd  [PDF]
Michele Elia,Davide Schipani
Mathematics , 2011,
Abstract: An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown in terms of factorizations of the Gauss sums themselves, which are then rivisited in terms of prime ideal decompositions. Interestingly, one of these results gives a representation of primes $p$ of the form $6k+1$ by a binary quadratic form in integers of a subfield of the cyclotomic field of the $p$-th roots of unity.
Gauss Sums and Quantum Mechanics  [PDF]
Vernon Armitage,Alice Rogers
Physics , 2000, DOI: 10.1088/0305-4470/33/34/305
Abstract: By adapting Feynman's sum over paths method to a quantum mechanical system whose phase space is a torus, a new proof of the Landsberg-Schaar identity for quadratic Gauss sums is given. In contrast to existing non-elementary proofs, which use infinite sums and a limiting process or contour integration, only finite sums are involved. The toroidal nature of the classical phase space leads to discrete position and momentum, and hence discrete time. The corresponding `path integrals' are finite sums whose normalisations are derived and which are shown to intertwine cyclicity and discreteness to give a finite version of Kelvin's method of images.
NMR experiment factors numbers with Gauss sums  [PDF]
Michael Mehring,Klaus Mueller,Ilya Sh. Averbukh,Wolfgang Merkel,Wolfgang Schleich
Physics , 2006, DOI: 10.1103/PhysRevLett.98.120502
Abstract: We factor the number 157573 using an NMR implementation of Gauss sums.
On Quadratic Gauss Sums and Variations Thereof  [PDF]
Michael S. Milgram,Larry Glasser
Mathematics , 2014,
Abstract: A number of new terminating series involving $\sin(n^2/k)$ and $\cos(n^2/k)$ are presented and connected to Gauss quadratic sums. Several new closed forms of generic Gauss quadratic sums are obtained and previously known results are generalized.
Determinations of Analogues of Gauss Sums and Other Trigonometric Sums  [PDF]
Matthias Beck,Bruce C. Berndt,O-Yeat Chan,Alexandru Zaharescu
Mathematics , 2005,
Abstract: Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated with primitive odd characters. In this paper, we establish two general theorems involving both sines and cosines, with more attention given to cosine sums in the several examples that we provide.
The value distribution of incomplete Gauss sums  [PDF]
Emek Demirci Akarsu,Jens Marklof
Mathematics , 2012, DOI: 10.1112/S0025579312001179
Abstract: It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov's pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and the second author.
Gauss sum factorization with cold atoms  [PDF]
M. Gilowski,T. Wendrich,T. Müller,Ch. Jentsch,W. Ertmer,E. M. Rasel,W. P. Schleich
Physics , 2007, DOI: 10.1103/PhysRevLett.100.030201
Abstract: We report the first implementation of a Gauss sum factorization algorithm by an internal state Ramsey interferometer using cold atoms. A sequence of appropriately designed light pulses interacts with an ensemble of cold rubidium atoms. The final population in the involved atomic levels determines a Gauss sum. With this technique we factor the number N=263193.
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