Rational trigonometric sums

Rational trigonometric sums are complex sums of a special kind that can be used to prove theorems in analytic number theory

Content

1 Definition
2 Some ratings
- 2.1 Special cases
  - 2.1.1 Linear Sums
  - 2.1.2 Gauss sums (quadratic)
- 2.2 General estimates
- 2.3 Partial linear sums
- 2.4 Impossibility of some nontrivial estimates
3 Application
4 History
5 See also
6 notes

Definition

Rational trigonometric sums are sums of the form ${S_{\varphi }}(q)=\sum \limits _{x=1}^{q}{e^{2\pi i{\frac {\varphi (x)}{q}}}}$ ${\ displaystyle {S _ {\ varphi}} (q) = \ sum \ limits _ {x = 1} ^ {q} {e ^ {2 \ pi i {\ frac {\ varphi (x)} {q}} }}}$ where $\varphi (x)=\sum \limits _{k=0}^{n}{a_{k}x^{k}}$ ${\ displaystyle \ varphi (x) = \ sum \ limits _ {k = 0} ^ {n} {a_ {k} x ^ {k}}}$ Is a polynomial with integer coefficients, moreover $(a_{0},\dots ,a_{n},q)=1$ ${\ displaystyle (a_ {0}, \ dots, a_ {n}, q) = 1}$ (with a nontrivial greatest common divisor, the fraction can be reduced and reduced to a general form).

Some ratings

When evaluating rational trigonometric sums in mathematics, one usually considers the upper bound on the modulus of the sum, since it is much easier to estimate. In this regard, it is accepted that $a_{0}=0$ ${\ displaystyle a_ {0} = 0}$ , so multiplying such a sum by $e^{2\pi ia_{0}}$ ${\ displaystyle e ^ {2 \ pi ia_ {0}}}$ does not change its absolute value.

Special cases

Linear Amounts

If $\varphi (x)=ax$ ${\ displaystyle \ varphi (x) = ax}$ , then, using Iverson's notation , we can indicate that ${S_{\varphi }}(q)=q[{q\mid a}]$ ${\ displaystyle {S _ {\ varphi}} (q) = q [{q \ mid a}]}$ . The proof of this fact follows trivially from the fact that the sum of the roots of unity for any integer base is zero. Such amounts are called linear.

Gaussian Sums (Quadratic)

Rational trigonometric sums over polynomials of the form $\varphi (x)=ax^{2}$ ${\ displaystyle \ varphi (x) = ax ^ {2}}$ called Gaussian sums.

For such sums, the exact values of the absolute value are known, namely

|{S_{\varphi }}(q)|={\begin{cases}{\sqrt {q}},&q\equiv 1\mod 2\\{\sqrt {2q}},&q\equiv 0\mod 4\\0,&q\equiv 2\mod 4\end{cases}}

{\ displaystyle | {S _ {\ varphi}} (q) | = {\ begin {cases} {\ sqrt {q}}, & q \ equiv 1 \ mod 2 \\ {\ sqrt {2q}}, & q \ equiv 0 \ mod 4 \\ 0, & q \ equiv 2 \ mod 4 \ end {cases}}}

General ratings

Further, for the convenience of presentation, we take $n=\deg \varphi$ ${\ displaystyle n = \ deg \ varphi}$ .

Hua made an estimate $|{S_{\varphi }}(q)|<c(n)q^{1-{\frac {1}{n}}}$ ${\ displaystyle | {S _ {\ varphi}} (q) | <c (n) q ^ {1 - {\ frac {1} {n}}}}$ where $c(n)$ ${\ displaystyle c (n)}$ Is a constant depending only on $n$ ${\ displaystyle n}$ . I.e $|{S_{\varphi }}(q)|=O(q^{1-{\frac {1}{n}}})$ ${\ displaystyle | {S _ {\ varphi}} (q) | = O (q ^ {1 - {\ frac {1} {n}}})}$ at fixed $n$ ${\ displaystyle n}$ . ^[one]

If $\varphi (x)=ax^{n}$ ${\ displaystyle \ varphi (x) = ax ^ {n}}$ then with simple $q>2$ ${\ displaystyle q> 2}$ more accurate estimate is true $|{S_{\varphi }}(q)|\leq (n-1){\sqrt {q}}$ ${\ displaystyle | {S _ {\ varphi}} (q) | \ leq (n-1) {\ sqrt {q}}}$ . ^[2]

Partial Linear

Using the standard formula of the sum of a geometric progression , we can deduce that for $\varphi (x)={\frac {ax}{q}}$ ${\ displaystyle \ varphi (x) = {\ frac {ax} {q}}}$ done

$\left\vert {\sum \limits _{x=1}^{m}{e^{2\pi i\varphi (x)}}}\right\vert =\left\vert {\frac {e^{2\pi i{\frac {a}{q}}}-e^{2\pi i{\frac {a(m+1)}{q}}}}{1-e^{2\pi i{\frac {a}{q}}}}}\right\vert \leq {\frac {2}{\min \left({\left\lbrace {\frac {a}{q}}\right\rbrace ,1-\left\lbrace {\frac {a}{q}}\right\rbrace }\right)}}$ ${\ displaystyle \ left \ vert {\ sum \ limits _ {x = 1} ^ {m} {e ^ {2 \ pi i \ varphi (x)}}} \ right \ vert = \ left \ vert {\ frac {e ^ {2 \ pi i {\ frac {a} {q}}} - e ^ {2 \ pi i {\ frac {a (m + 1)} {q}}}} {1-e ^ { 2 \ pi i {\ frac {a} {q}}}}} \ right \ vert \ leq {\ frac {2} {\ min \ left ({\ left \ lbrace {\ frac {a} {q}} \ right \ rbrace, 1- \ left \ lbrace {\ frac {a} {q}} \ right \ rbrace} \ right)}}}$ ,

Where $\left\lbrace {x}\right\rbrace$ ${\ displaystyle \ left \ lbrace {x} \ right \ rbrace}$ means the fractional part of a number $x$ ${\ displaystyle x}$ .

The impossibility of some nontrivial estimates

A. A. Karatsuba proved ^[3] that for $n>\left({{\frac {1}{2\log {2}}}-\varepsilon }\right){\frac {p}{\log {p}}},\ \varphi (x)=ax^{n}$ ${\ displaystyle n> \ left ({{\ frac {1} {2 \ log {2}}} - \ varepsilon} \ right) {\ frac {p} {\ log {p}}}, \ \ varphi ( x) = ax ^ {n}}$ there are infinitely many simple $p$ ${\ displaystyle p}$ for which $|S_{\phi }(p)|>\left({1-\delta (\varepsilon )}\right)p$ ${\ displaystyle | S _ {\ phi} (p) |> \ left ({1- \ delta (\ varepsilon)} \ right) p}$ where $\delta (\varepsilon )\to 0$ ${\ displaystyle \ delta (\ varepsilon) \ to 0}$ at $\varepsilon \to 0$ ${\ displaystyle \ varepsilon \ to 0}$ , i.e. with $n$ ${\ displaystyle n}$ for the corresponding trigonometric sums, upper estimates necessary for most applications are impossible.

Application

The first proof of the quadratic reciprocity law (Gauss, 1795) used Gauss sums over a polynomial of the form $\varphi (x)={\frac {ax^{2}}{q}}$ ${\ displaystyle \ varphi (x) = {\ frac {ax ^ {2}} {q}}}$ .

Using rational trigonometric sums, Vinogradov deduced an approximate description of the distribution of quadratic residues and non-residues ^[2] .

The sums considered can also find application in proving Waring's problems by methods of analytic number theory.

History

Trigonometric sums were first applied by Gauss in 1795 to prove the quadratic reciprocity law .

Notes

↑ I. Vinogradov. The method of trigonometric sums in number theory. - Science, 1971.
↑ ¹ ² B.I. Segal. Trigonometric sums and some of their applications to number theory, volume 1. - UMN, 1946.
↑ A. A. Karatsuba, On estimates of total trigonometric sums, Mat. Notes, 1967, Volume 1, Issue 2, 199–208

[1] I. Vinogradov. The method of trigonometric sums in number theory. - Science, 1971.

[Segal-2] ¹ ² B.I. Segal. Trigonometric sums and some of their applications to number theory, volume 1. - UMN, 1946.

[3] A. A. Karatsuba, On estimates of total trigonometric sums, Mat. Notes, 1967, Volume 1, Issue 2, 199–208