Ramanujan sum

As Ramanujan's sum is in number theory , a branch of mathematics , a certain finite sum , the value of the natural number and the integer depends called. She is going through ${\displaystyle c_{q}(n)}$ ${\displaystyle q}$ ${\displaystyle n}$

${\displaystyle c_{q}(n)=\sum _{a=1 \atop (a,q)=1}^{q}e^{2\pi i{\frac {a}{q}}n}}$

Are defined. The notation means the greatest common divisor of and the summation thus extends over the numbers with which to be prime. The summands in the sum are powers of a fixed complex root of unity . ${\displaystyle (a,q)}$ ${\displaystyle a}$ ${\displaystyle q}$ ${\displaystyle a}$ ${\displaystyle 1\leq a\leq q}$ ${\displaystyle q}$

S. Ramanujan introduced these sums in 1916. They play an important role in the circle method of Hardy , Littlewood and Vinogradov . → See also trigonometric polynomial .

With Ramanujan sums one can get interesting representations for number theoretic functions that allow an analytical continuation of these functions.

Spellings

For a clear presentation, abbreviations are used in number theory and the function is called the number-theoretical exponential function . ${\displaystyle e(x)=e^{2\pi ix}}$ ${\displaystyle e}$

With the number theoretic exponential function, the Ramanujan sum can be expressed as ${\displaystyle c_{q}(n)}$

${\displaystyle c_{q}(n)=\sum _{a=1 \atop (a,q)=1}^{q}e\left({\frac {a}{q}}\cdot n\right)}$ write.

For integers and one writes , read “a divides b”, if there is an integer with which there is no such number, one writes , read “a does not divide b”. The summation symbol means that the summation index runs through all positive factors of . For a prime power and an integer one writes (read “ divides b exactly”), but if , in other words, if . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a\mid b}$ ${\displaystyle c}$ ${\displaystyle b=a\cdot c}$ ${\displaystyle a\nmid b}$ ${\displaystyle \textstyle \sum _{d\,\mid \,m}f(d)}$ ${\displaystyle d}$ ${\displaystyle m}$ ${\displaystyle p^{k},k\in \mathbb {N} \setminus \lbrace 0\rbrace }$ ${\displaystyle b}$ ${\displaystyle p^{k}\parallel b}$ ${\displaystyle p^{k}}$ ${\displaystyle p^{k}\mid b}$ ${\displaystyle p^{k+1}\nmid b}$ ${\displaystyle (p^{k+1},b)=p^{k}}$

Elementary properties

If you hold one of the variables or in Ramanujan's sum fixed, the result is a number theoretic function as a function of other variables, must as for that term variable to be limited. With fixed the function is - periodic , that is, it is true ${\displaystyle q}$ ${\displaystyle n}$ ${\displaystyle c_{q}(n)}$ ${\displaystyle n}$ ${\displaystyle n\in \mathbb {N} \setminus \lbrace 0\rbrace }$ ${\displaystyle q}$ ${\displaystyle n\mapsto c_{q}(n)}$ ${\displaystyle q}$

${\displaystyle c_{q}(m)=c_{q}(n)}$if . ${\displaystyle m\equiv n{\pmod {q}},n,m\in \mathbb {Z} }$

If one leaves out the condition of coprime numbers in the summation, one obtains

${\displaystyle \sum _{a=1}^{q}e\left({\frac {a}{q}}\cdot n\right)={\begin{cases}q\quad {\text{falls}}\;n\equiv 0{\pmod {q}}\\0\quad {\text{sonst,}}\end{cases}}}$

because then the left side is a geometric sum. If you sort the sum according to the greatest common divisor of and , you get a Dirichlet convolution of the number theoretic function with the constant function : ${\displaystyle q}$ ${\displaystyle a}$ ${\displaystyle q\mapsto c_{q}(n)}$ ${\displaystyle I^{0}(q)=1}$

${\displaystyle \sum _{a=1}^{q}e\left({\frac {a}{q}}\cdot n\right)=\sum _{d\mid q}\sum _{a=1 \atop (a,q)=d}^{q}e\left({\frac {a}{q}}\cdot n\right)=\sum _{d\mid q}c_{q/d}(n)}$.

From this it follows with Möbius' inverse formula :

${\displaystyle c_{q}(n)=\sum _{{d\mid q} \atop {n\equiv 0{\pmod {d}}}}\mu (q/d)\cdot d=\sum _{d\mid (q,n)}\mu (q/d)\cdot d.}$

It then follows:

• The Ramanujan sum always takes real and even integer values, ${\displaystyle c_{q}(n)}$
• it is , , ${\displaystyle c_{q}(n)=c_{q}(-n)}$ ${\displaystyle c_{q}(0)=\varphi (q)}$
• at fixed it is a multiplicative number theoretic function of , that is ${\displaystyle n}$ ${\displaystyle q}$

from follows ${\displaystyle (q,r)=1}$ ${\displaystyle c_{qr}(n)=c_{q}(n)\cdot c_{r}(n)}$

• and it always applies . ${\displaystyle c_{q}(n)=c_{q}((q,n))}$
• The Ramanujan sum can be represented by Euler's φ function and the Möbius function : ${\displaystyle \varphi }$ ${\displaystyle \mu }$

${\displaystyle c_{q}(n)=\varphi (q)\cdot {\frac {\mu (q/(q,n))}{\varphi (q/(q,n))}}}$(for one sets , more generally as positive GCD), ${\displaystyle n=0}$ ${\displaystyle (q,0)=q}$ ${\displaystyle (q,n)}$

• their values ​​are limited by fixed amounts , ${\displaystyle q}$ ${\displaystyle \varphi (q)}$
• is not square-free , so is . ${\displaystyle \!\,q/(q,n)}$ ${\displaystyle c_{q}(n)=0}$

Ramanujan sums for representing number theoretic functions

Ramanujan already showed for some important special cases that one can obtain interesting representations for number theoretic functions with his sums. For this purpose, a special kind of discrete Fourier transformation is introduced for number theoretic functions of the greatest common divisor:

Be and a number theoretic function. Then is called ${\displaystyle n\in \mathbb {Z} ,\;q\in \mathbb {N} }$ ${\displaystyle f:\mathbb {N} \rightarrow \mathbb {C} }$

${\displaystyle F_{f}(n,q)=\sum _{a=1}^{q}f((a,q))\cdot e\left(-{\frac {a}{q}}\cdot n\right)}$

discrete Fourier transform of . ${\displaystyle f((n,q))}$

The following applies to this Fourier transform:

1. ${\displaystyle F_{f}(n,q)=(f*c_{\underline {\quad }}(n))(q)=\sum _{d\mid q}f\left({\frac {q}{d}}\right)\cdot c_{d}(n)}$ and
2. ${\displaystyle f((n,q))={\frac {1}{q}}\sum _{a=1}^{q}F_{f}(a,q)e\left({\frac {a}{q}}\cdot n\right)}$for the inverse transform .

In these transformations, the determining equations only have to take into account a finite number of coefficients with a positive index by forming the greatest common divisor.

Examples

• Greatest common divisor:

${\displaystyle (n,q)=\sum _{a=1}^{q}e\left({\frac {a}{q}}\cdot n\right)\cdot \sum _{d\mid q}{\frac {c_{d}(a)}{d}}}$. This representation allows an analytic continuation of the greatest common divisor in the first place on the entire function ! ${\displaystyle n}$ ${\displaystyle n\in \mathbb {C} }$

• Euler's φ function:

${\displaystyle \varphi (q)=\sum _{a=1}^{q}(a,q)\cdot e\left(-{\frac {a}{q}}\right)}$. The trigonometric relations follow from this by dividing them into real and imaginary parts

${\displaystyle \sum _{a=1}^{q}(a,q)\cdot \cos \left(2\pi \cdot {\frac {a}{q}}\right)=\varphi (q)\quad }$ and ${\displaystyle \sum _{a=1}^{q}(a,q)\cdot \sin \left(2\pi \cdot {\frac {a}{q}}\right)=0.}$

• The divisor function can be explicitly represented as a series using Ramanujan sums: ${\displaystyle \sigma _{k}}$ ${\displaystyle k>0}$

${\displaystyle \sigma _{k}(n)=\zeta (k+1)n^{k}\sum _{m=1}^{\infty }{\frac {c_{m}(n)}{m^{k+1}}}.}$

The calculation of the first values ​​of shows the fluctuation around the "mean value" (the average order of magnitude ) : ${\displaystyle c_{m}(n)}$ ${\displaystyle \zeta (k+1)n^{k}}$

${\displaystyle \sigma _{k}(n)=\zeta (k+1)n^{k}\left[1+{\frac {(-1)^{n}}{2^{k+1}}}+{\frac {2\cos {\frac {2\pi n}{3}}}{3^{k+1}}}+{\frac {2\cos {\frac {\pi n}{2}}}{4^{k+1}}}+\cdots \right]}$

• A kind of orthogonality for Ramanujan sums: Let the number-theoretic one function, i.e. the neutral element of the convolution operation with ${\displaystyle \eta (n)}$

${\displaystyle \eta (n)={\begin{cases}1\quad (n=1)\\0\quad \left(n\in \mathbb {N} \setminus \lbrace 1\rbrace \right).\end{cases}}}$

Then it follows by inverse Fourier transform for ${\displaystyle n\in \mathbb {Z} \setminus \lbrace 0\rbrace ,q\in \mathbb {N} :}$

${\displaystyle \eta ((n,q))={\frac {1}{q}}\sum _{a=1}^{q}c_{q}(a)\cdot e\left({\frac {a}{q}}\cdot n\right).}$

That means: Exactly when the sum on the right does not vanish, the numbers and are prime. Then the right side of the equation has the value 1. ${\displaystyle n}$ ${\displaystyle q}$

literature

• Jörg Brüdern : Introduction to analytical number theory . Springer, Berlin, Heidelberg, New York 1995, ISBN 3-540-58821-3 .
• Godfrey Harold Hardy : Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work . American Mathematical Society / Chelsea, Providence 1999, ISBN 978-0-8218-2023-0 .
• Godfrey Harold Hardy, Edward Maitland Wright : An Introduction to the Theory of Numbers . 5th edition. Oxford University Press , Oxford 1980, ISBN 978-0-19-853171-5 .
• John Knopfmacher: Abstract Analytic Number Theory . New edition. Dover Publications, 2000, ISBN 0-486-66344-2 .
• Srinivasa Ramanujan: On Certain Trigonometric Sums and their Applications in the Theory of Numbers . In: Transactions of the Cambridge Philosophical Society . tape 22 , no. 15 , 1918, pp. 259-276 .
• Srinivasa Ramanujan: On Certain Arithmetical Functions . In: Transactions of the Cambridge Philosophical Society . tape 22 , no. 9 , 1916, pp. 159-184 .
• Srinivasa Ramanujan: Collected Papers . American Mathematical Society / Chelsea, Providence 2000, ISBN 978-0-8218-2076-6 .
• Robert Charles Vaughan : The Hardy-Littlewood Method . 2nd Edition. Cambridge University Press, Cambridge 1997, ISBN 0-521-57347-5 .
• Wolfgang Schramm: The Fourier Transform of Functions of the Greatest Common Divisor . In: Integers: Electronic Journal of Combinatorical Number Theory . tape 8 , no. 50 , 2008 ( emis.de [PDF]).
• Ivan Matveevitch Vinogradov: The Method of Trigonometrical Sums in the Theory of Numbers . Translated from the Russian and annotated by Klaus Friedrich Roth and Anne Ashley Davenport. New York, Dover 2004.

Individual evidence

1. ↑ Ramanujan (1916)
2. ^ Vaughan (1997)
3. ↑ Brüdern (1995) p. 20.
4. ↑ Brüdern (1995) Lemma 1.3.1
5. ↑ ^{a b c} Schramm (2008)
6. ↑ E. Krätzel: Number theory . VEB Deutscher Verlag der Wissenschaften, Berlin 1981, p. 130 .