nLab Gauss sum

Idea

A quadratic Gauss sum is a sum of square-powers of primitive roots of unity:

(1)$$\sum_{s = 0}^{k-1} e^{ \tfrac{2 \pi \mathrm{i}}{k} s^2 } \;\; \in \; \mathbb{C} \,, \;\;\;\;\;\; \text{for} \; k \in \mathbb{N}_{> 0} \,.$$

A generalized Gauss sum is a variant of this expression, admitting other prefectors or powers of $s$ in the exponent.

Properties

Ordinary quadratic Gauss sum

Proposition

(evaluation of the quadratic Gauss sum)
The basic quadratic Gauss sum (1) evaluates to:

(2)$$\sum_{s = 0}^{k-1} e^{ \tfrac{2 \pi \mathrm{i}}{k} s^2 } \;\; = \;\; \left\{ \begin{array}{ccl} (1 + \mathrm{i}) \sqrt{k} &\vert& k = 0 \;mod\; 4 \\ \sqrt{k} &\vert& k = 1 \;mod\; 4 \\ 0 &\vert& k = 2 \;mod\; 4 \\ \mathrm{i} \sqrt{k} &\vert& k = 3 \;mod\; 4 \mathrlap{\,.} \end{array} \right.$$

The original proof is due to Gauss 1811, an early alternative proof is due to Dirichlet 1835, reviewed in Dirichlet 1871. Several other proofs have been given. Modern review includes Lang 1970 p 87, Berndt & Evans 1981, Doyle 2016 (1.1), Ram Murty & Pathak 2017 Thm. 1.1, Taylor 2022 Thm. 1.16.

For odd $k$ we have more generally:

Proposition

(evaluation of the quadratic Gauss sum with multiple exponents)
For $k \in 2\mathbb{N} + 1$ and $q \in \mathbb{Z}$ we have

(3)$$\sum_{s = 0}^{k-1} e^{ \tfrac{2 \pi \mathrm{i}}{k} q s^2 } \;\;=\;\; (q \vert k) \sum_{s = 0}^{k-1} e^{ \tfrac{2 \pi \mathrm{i}}{k} s^2 } \;\; = \;\; \left\{ \begin{array}{lcl} \, (1 + \mathrm{i}) \sqrt{k} &\vert& k = 0 \;mod\; 4 \\ (q \vert k) \, \sqrt{k} &\vert& k = 1 \;mod\; 4 \\ \phantom{(q \vert k)} \; 0 &\vert& k = 2 \;mod\; 4 \\ (q \vert k) \, \mathrm{i} \sqrt{k} &\vert& k = 3 \;mod\; 4 \mathrlap{\,,} \end{array} \right.$$

where “$(a \vert b)$” is the Jacobi symbol.

(Lang 1970 “QS 4” (p 86), cf. also Doyle 2016 (1.1) but beware of typos there)

Qudratic Gauss sum with halved exponents

Proposition

(evaluation of quadratic Gauss sum with halved exponents)
For $k \in 2 \mathbb{N}_{\gt 0}$ the quadratic Gauss sum with halved exponents evaluates to:

$$\tfrac{1}{\sqrt{k}} \sum_{s = 0}^{k-1} e^{ \tfrac{\pi \mathrm{i}}{k} s^2 } \;\; = \;\; e^{\pi \mathrm{i}/4}$$

The following proofs are given in MO:a/4232289 (using Prop. ), in MO:a/489863 (using Prop. ), and in MO:a/489956 (using Prop. ).

Proof

Using the ordinary quadratic Gauss evaluation (Prop. ) we set $r \coloneqq k/2 \,\in\, \mathbb{N}$ and compute as follows:

$$\begin{array}{ccl} \sum_{s=0}^{k-1} \, e^{ \tfrac { \pi \mathrm{i}} { k } s^2 } &\equiv& \sum_{s=0}^{2r-1} \, e^{ \tfrac {2 \pi \mathrm{i}} { 4r } s^2 } \\ &=& \tfrac{1}{2} \Big( \sum_{s=0}^{2r-1} \,+\, \sum_{s=2r}^{4r-1} \Big) \, e^{ \tfrac {2 \pi \mathrm{i}} { 4r } s^2 } \\ &=& \tfrac{1}{2} \sum_{s=0}^{4r-1} \, e^{ \tfrac {2 \pi \mathrm{i}} { 4r } s^2 } \\ &=& \tfrac{1}{2} (1 + \mathrm{i}) \, \sqrt{4r} \\ &=& e^{\pi \mathrm{i}/4} \, \sqrt{2r} \\ &=& e^{\pi \mathrm{i}/4} \, \sqrt{k} \,, \end{array}$$

where the steps are, in order of appearance:

1. definition of $r$,

2. observing that the summands are $2r$-periodic, because

   $$e^{ \tfrac{\pi \mathrm{i}}{2r} (n + 2r)^2 } \;=\; e^{ \tfrac{\pi \mathrm{i}}{2r} n^2 } e^{ 2\pi \mathrm{i} ( n + r ) } \;=\; e^{ \tfrac{\pi \mathrm{i}}{2r} n^2 } \,,$$

3. collecting summands,

4. the classical evaluation formula (3),

5. rearranging factors,

6. definition of $r$.

Alternatively, using reciprocity laws like the Landsberg-Schaar identity below:

The statement is the immediate special case of Prop. for $q \equiv r \coloneqq k/2$ and $p \equiv 1$.

Similarly, from Prop. we immediately get:

$$\begin{array}{ccl} \tfrac{1}{\sqrt{k}} \sum_{s = 0}^{k-1} e^{ \tfrac{\pi \mathrm{i}}{k} s^2 } &\equiv& G(k,0,1) \\ &=& e^{ \pi \mathrm{i}/4 } \, \overline{G(1,0,k)} \\ &\equiv& e^{ \pi \mathrm{i}/4 } \mathrlap{\,.} \end{array}$$

Landsberg-Schaar identity

The ordinary Gauss sums and those with halved exponents are related by:

Proposition

(Landsberg-Schaar identity) For $p, q \in \mathbb{N}_{\gt 0}$ we have

$$\tfrac{1}{\sqrt{p}} \sum_{n = 0}^{p-1} \, e^{ 2 \pi \mathrm{i} \tfrac{q}{p}n^2 } \;\;=\;\; \tfrac { e^{\pi \mathrm{i}/4} } { \sqrt{2q} } \sum_{n = 0}^{2q - 1} \, e^{ -\pi \mathrm{i} \tfrac{p}{2q} n^2 } \,.$$

(Schaar 1850, Dym & McKean 1972 §4.6, Armitage & Rogers 2000, Ustinov 2022)

Yet more generally:

Proposition

(reciprocity for generalized quadratic Gauss sum with halved exponents)
The expressions

(4)$$G(a,b,c) \;\coloneqq\; \tfrac{1}{\sqrt{c}} \sum_{n=0}^{c-1} \, e^{ \tfrac{\pi \mathrm{i}}{c} \big( a n^2 + 2 b n \big) } \,, \;\;\;\text{for}\;\; a, c \in \mathbb{N}_{\gt 0} ,\, b \in \mathbb{Z} ,\, a c \in 2 \mathbb{N} \,.$$

satisfy

$$G(a,b,c) \;=\; e^{ \big( \tfrac{\pi \mathrm{i}}{4} - \tfrac{b^2}{a c} \big) } \; \overline{G(c,b,a)}$$

(where $\overline{(-)}$ denotes complex conjugation).

This is proven by Harcos.

Related concepts

• quadratic reciprocity

References

The original proof of Prop. is due to

• Carl F. Gauss: Summatio Quarumdam Serierum Singularium Societas Regia Scientiarum Gottingensis (1811)

and an early alternative proof, using a variant of Poisson summation, is due to

• P. G. L. Dirichlet: Über eine neue Anwendung bestimmter Integrale auf die Summation endlicher oder unendlicher Reihen. Abhandlungen der Königlich Preussischen Akademie der Wissenschaften (1835)

reviewed in:

• P. G. L. Dirichlet: Vorlesungen über Zahlentheorie, Vieweg und Sohn (1871) [eudml:204463]

• Bill Casselman: Dirichlet’s calculation of Gauss sums, L’ Enseignement Mathématique 2 57 (2011) 281–301 [pdf, pdf]

Survey and review:

• Bruce C. Berndt, Ronald J. Evans: The determination of Gauss sums, Bull. Amer. Math. Soc. 5 (1981) 107-129 [ams:bull/1981-05-02/S0273-0979-1981-14930-2, euclid:1183548292]

• Bruce C. Berndt, Ronald J. Evans, Kenneth S. Williams: Gauss and Jacobi Sums, John Wiley & Sons (1998) [ISBN:978-0-471-12807-6]

• Laurence R. Taylor: Gauss Sums in Algebra and Topology [arXiv:2208.06319]

Further discussion:

• Serge Lang, §IV.3 in: Algebraic number theory, Graduate Texts in Mathematics 110, Springer (1970, 1986, 1994, 2000) [doi:10.1007/978-1-4612-0853-2]

• Kenneth Ireland, Michael Rosen: Quadratic Gauss Sums, chapter 6 in: A Classical Introduction to Modern Number Theory, Graduate Texts in Mathematics 84, Springer (1990) [doi:10.1007/978-1-4757-1779-2_6]

• Lisa Jeffrey, Props. 2.3 and 4.3 in: Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun.Math. Phys. 147 (1992) 563–604 [doi:10.1007/BF02097243, spire:342446]

  (reciprocity formulas in the context of Chern-Simons theory)

• George Danas: Note on the quadratic Gauss sums, Int. J. Math and Math. Sciences (2001) [doi:10.1155/S016117120100480X]

• Kh. M. Saliba & V. N. Chubarikov: A generalization of the Gauss sum, Moscow Univ. Math. Bull. 64 (2009) 92–94 [doi:10.3103/S0027132209020132]

• Greg Doyle: Quadratic Form Gauss Sums, PhD thesis, Ottawa (2016) [doi:10.22215/etd/2016-11457, pdf]

• M. Ram Murty, Siddhi Pathak: Evaluation of the quadratic Gauss sum, The Mathematics Student 86 1-2 (2017) 139-150 [pdf, pdf, pdf]

• Ramin Takloo-Bighash: Gauss Sums, Quadratic Reciprocity, and the Jacobi Symbol, in: A Pythagorean Introduction to Number Theory, Undergraduate Texts in Mathematics, Springer (2018) [doi:10.1007/978-3-030-02604-2_7]

• Frederik Broucke, Jasson Vindas, section 2 of: The pointwise behavior of Riemann’s function, J. Fractal Geom. 10 3/4 (2023) 333-349 [arXiv:2109.08499, doi:10.4171/jfg/137]

• Nilanjan Bag, Antonio Rojas-León, Zhang Wenpeng: On some conjectures on generalized quadratic Gauss sums and related problems, Finite Fields and Their Applications 86 (2023) 102131 [doi:10.1016/j.ffa.2022.102131]

• Alexander P. Mangerel: On a rigidity property for quadratic Gauss sums [arXiv:2502.16014]

See also:

• Wikipedia: Quadratic Gauss sum

• Wikipedia: Gauss sum

On the Landsberg-Schaar identity:

• M. Schaar: Mémoire sur la théorie des résidus biquadratiques, Mémoires de l’Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique 24 (1850) [biodiversitylibrary:20728]

• Harry Dym, Henry P. McKean, section 4.6 in: Fourier series and integrals, Probability and Mathematical Statistics 14, Academic Press (1972)

• Vernon Armitage, Alice Rogers: Gauss Sums and Quantum Mechanics, J. Phys. A: Math. Gen. 33 (2000) 5993 [doi:10.1088/0305-4470/33/34/305, arXiv:quant-ph/0003107]

  (via tools from quantum mechanics)

• Alexey Ustinov: A Short Proof of the Landsberg–Schaar Identity, Mathematical Notes 112 (2022) 488–490 [doi:10.1134/S0001434622090188]. Russian original: А. В. Устинов, Короткое доказательство тождества Ландсберга–Шаара, Математические заметки, 2022, том 112, выпуск 3, страницы 478–480, doi.

• Wikipedia: Landsberg-Schaar relation

Further generalization:

• Gergely Harcos: The reciprocity of Gauss sums via the residue theorem [pdf, pdf]