On Quadratic Gauss Sums and Variations Thereof

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.


Introduction
In a recent work (Glasser & Milgram, 2014), a number of new integrals were evaluated analytically, and in the process, we noted that the limiting case of some of those integrals reduced to simpler known forms that involved trigonometric series with quadratic dummy indices of summation. A reasonably thorough search of the literature indicates that such series are not very well tabulated-in fact only two were found in tables (Hansen, 1975)-those two corresponding to the real and imaginary part of Gauss quadratic sums, the series associated with classical number theory (Apostol, 1998), and, recently applied to quantum mechanics (Armitage & Rogers, 2000;Gheorghiu & Looi, 2010). Hardy and Littlewood (1914), in their original work considered integrals that are similar to those considered here (see (2.1) below) but not in closed form; a comprehensive historical summary can be found in Berndt and Evans (1981, Section 2.3).
In Section 2, we summarize the integrals upon which our results are based; all eight variations of alternating quadratic sums analogous to classical quadratic Gauss sums are developed in Section 3.1. These new sums are then connected to the classical sums (Section 3.2), and that relationship is used to find new closed forms for odd-indexed variants of the classical sums (Section 3.3), all of which are believed to be new. The upper summation limit of an interesting subset of these results is extended by the addition of a new parameter p (Section 4.1), and a template proof is provided in the Appendix.
Miscellaneous results arising during our investigation are also listed (Section 4.2). The trigonometric

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Scientists frequently need to add numbers up, and this paper presents a new way of determining the sum of a general collection of complicated things without doing any addition. Mathematicians always like to know that such things will work for any case that they may try, so when a new result is presented, it must be proven valid in a relatively rigorous manner. That is why this paper is somewhat longer than it needs to be-the new results must be proven, rather than simply stated.
forms of the results presented in the first four sections are finally rewritten in the canonical form of a generalized Gauss quadratic sum involving the roots of unity, again leading to new expressions, best described as alternating and extended Gauss quadratic sums (Section 5). It is recognized that many of the results derived here can be obtained by clever-and lengthy-manipulation (Chapman, 2014) of known results from number theory starting from the classical Gauss sum. This work provides a simple method of obtaining sums that are not listed in any of the tables or reference works usually consulted (e.g. Hansen, 1975), particularly motivated by the fact that number theoretic methods are not well known in other fields where such sums arise (Armitage & Rogers, 2000;Gheorghiu & Looi, 2010).

The basic integrals
In the previous work (Glasser & Milgram, 2014, Equation 3.33), with k ∈ ℕ, {a, b, s} ∈ ℜ, we evaluated a family of related integrals of the form where 4a ≤ (2k − 1) , and, for particular choices of s, this family of integrals (Glasser & Milgram, 2014, Equations 3.28, 3.34, 3.46 with s = 0 and b → i a and 3.37 and 3.41 combined) gives rise (respectively) to the following particular results: where and (2.1)

Alternating Gauss quadratic sums
All the main results in this section depend upon the following well-known result (Gradshteyn & Ryzhik, 1980, Equation 3.691) (see cover image) In (2.2), we evaluate the limit a = k ∕2, and, after setting k → 2k, we eventually arrive at after setting k → 2k − 1, we find Both of these results may be characterized as Alternating Gauss Quadratic Sums for which we can find no references in the literature (e.g. Apostol, 1998). In Equations 2.3 and Equations 2.4 let a = (2k − 1) and a = 2k , respectively, and, by comparing real and imaginary parts, eventually arrive at and All of these are believed to be new.
If a = (2k − 1) in Equation 2.3, after setting k → 2k we find and, after setting k → 2k − 1 we obtain Adding and subtracting Equations 3.6 and 3.8 gives and while performing the same operations on Equations 3.7 and 3.9 yields and all of which, together with Equations 3.2-3.5 are believed to be new; note the fourfold modularity of these results, and the fact that Equations 3.10-3.13 can be also written, respectively, (and more conveniently)

Variations
An interesting finite sum identity/equality arises from Equation 2.5. Set b = 0, a = (2k − 1) in the real and imaginary parts to discover By removing odd multiples of ∕2 from the arguments of each of the above trigonometric functions, .18 can alternatively be written in the form of odd-indexed quadratic Gauss sums The fact that the arguments of both the sine and cosine terms are fractional multiples of ∕4 may explain the intriguing equality of these sine and cosine sums of equal argument, which also hints at the existence of some more general property. The only previously tabulated sums of this genre, corresponding to Gauss' classical result for quadratic sums (Apostol, 1998, Equation 9.10(30); Berndt & Evans' 1981) are to be found in the following tabular listing (Hansen, 1975 cos n 2 + 2 n z + i sin n 2 + 2 n z also recognized as a generalized Gauss quadratic sum (Apostol, 1998, Theorem 9.16).

Odd indexed quadratic sums
Break Equations 3.2 and 3.3 into their even and odd parts to find and Employing the method outlined in the Appendix, and using Equation 4.13 (see next section below) we find and From Equations 3.31 and 3.32, the odd indexed quadratic sums appearing in Equations 3.29 and 3.30 eventually give a generalization of Equation 3.19. This is discussed further in Section 5.

Generalizations
Along the foregoing lines, we also note that some of the above may be generalized by extending the upper limit of summation. If p is a positive integer, it may be shown (by induction and removing all integral multiples of from the arguments of the corresponding trigonometric functions-see the Appendix for a sample proof) that Equation 3.2 generalizes to Equation 3.5 generalizes to Equation 3.15 generalizes to

Other miscellaneous results
Ancillary to proofs of the above, we obtained a number of miscellaneous results, all of which can be proven by using the methods outlined in the previous section with the help of the Appendix.