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Trigonometric sums depending on two integer parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r0772201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r0772202.png" />:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r0772203.png" /></td> </tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r0772204.png" /> runs over all non-negative integers less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r0772205.png" /> and relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r0772206.png" />. The basic properties of Ramanujan sums are multiplicity with respect to the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r0772207.png" />,
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Trigonometric sums depending on two integer parameters  $  k $
 +
and $  n $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r0772208.png" /></td> </tr></table>
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$$
 +
c _ {k} ( n)  = \sum _ { h }
 +
\mathop{\rm exp}
 +
\left (
  
and also the representation in terms of the [[Möbius function|Möbius function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r0772209.png" />:
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\frac{2 \pi n h i }{k}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722010.png" /></td> </tr></table>
+
\right )  = \
 +
\sum _ { h } \cos 
 +
\frac{2 \pi n h }{k}
 +
,
 +
$$
  
Ramanujan sums are finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722012.png" /> is finite. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722013.png" />.
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when  $  h $
 +
runs over all non-negative integers less than  $  k $
 +
and relatively prime to  $  k $.
 +
The basic properties of Ramanujan sums are multiplicativity with respect to the index  $  k $,
 +
 
 +
$$
 +
c _ {k k  ^  \prime  } ( n)  =  c _ {k} ( n) c _ {k  ^  \prime  } ( n) \ \
 +
\textrm{ if }  ( k , k  ^  \prime  ) = 1 ,
 +
$$
 +
 
 +
and also the representation in terms of the [[Möbius function|Möbius function]]  $  \mu $:
 +
 
 +
$$
 +
c _ {k} ( n)  = \
 +
\sum _ {d \mid  ( k , n ) } \mu \left (
 +
\frac{k}{d}
 +
\right ) d .
 +
$$
 +
 
 +
Ramanujan sums are finite if  $  k $
 +
or $  n $
 +
is finite. In particular, $  c _ {k} ( 1) = 1 $.
  
 
Many multiplicative functions on the natural numbers (cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]]) can be expanded as series of Ramanujan sums, and, conversely, the basic properties of Ramanujan sums enable one to sum series of the form
 
Many multiplicative functions on the natural numbers (cf. [[Multiplicative arithmetic function|Multiplicative arithmetic function]]) can be expanded as series of Ramanujan sums, and, conversely, the basic properties of Ramanujan sums enable one to sum series of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722014.png" /></td> </tr></table>
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$$
 +
\sum _ { n= 1} ^  \infty 
 +
 
 +
\frac{c _ {k} ( q n ) }{n  ^ {s} }
 +
f ( n) ,\ \
 +
\sum _ { k= 1} ^  \infty 
 +
 
 +
\frac{c _ {k} ( q n ) }{k  ^ {s} }
 +
f ( k) ,
 +
$$
 +
 
 +
where  $  f $
 +
is a multiplicative function and  $  q $
 +
is an integer. In particular,
 +
 
 +
$$
 +
\sum _ { k= 1 }^  \infty 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722015.png" /> is a multiplicative function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722016.png" /> is an integer. In particular,
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\frac{c _ {k} ( n) }{n  ^ {s} }
 +
  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722017.png" /></td> </tr></table>
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\frac{\sigma _ {1- s} ( n) }{\zeta ( s) }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722018.png" /> is the Riemann [[Zeta-function|zeta-function]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722019.png" /> is the sum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722020.png" />-th powers of the divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722021.png" />. Such sums are closely connected with special series for certain additive problems in number theory (cf. [[Additive number theory|Additive number theory]]); for example, the representation of a natural number as an even number of squares. S. Ramanujan [[#References|[1]]] obtained many formulas involving Ramanujan sums.
+
where $  \zeta $
 +
is the Riemann [[Zeta-function|zeta-function]] and $  \sigma _ {a} $
 +
is the sum of the $ a$-th powers of the divisors of $  n $.  
 +
Such sums are closely connected with special series for certain additive problems in number theory (cf. [[Additive number theory|Additive number theory]]); for example, the representation of a natural number as an even number of squares. S. Ramanujan [[#References|[1]]] obtained many formulas involving Ramanujan sums.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Ramanujan,  "On certain trigonometrical sums and their applications in the theory of numbers"  ''Trans. Cambridge Philos. Soc.'' , '''22'''  (1918)  pp. 259–276  ((Also: Collected papers, Chelsea, reprint, 1962.))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,  "Note on Ramanujan's trigonometrical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077220/r07722022.png" /> and certain series of arithmetical functions"  ''Proc. Cambridge Philos. Soc.'' , '''20'''  (1920–1921)  pp. 263–271</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy (ed.)  et al. (ed.) , ''Collected papers of S. Ramanujan'' , Chelsea, reprint  (1962)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  B. Volkmann,  "Verallgemeinerung eines Satzes von Maxfield"  ''J. Reine Angew. Math.'' , '''271'''  (1974)  pp. 203–213</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of the Riemann zeta-function" , Clarendon Press  (1951)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Levin,  "The life and work of the Indian mathematician Ramanujan"  ''Istoriko-Mat. Issled.'' , '''13'''  (1960)  pp. 335–378  (In Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. Ramanujan,  "On certain trigonometrical sums and their applications in the theory of numbers"  ''Trans. Cambridge Philos. Soc.'' , '''22'''  (1918)  pp. 259–276  ((Also: Collected papers, Chelsea, reprint, 1962.))</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,  "Note on Ramanujan's trigonometrical function $c_q(n)$ and certain series of arithmetical functions"  ''Proc. Cambridge Philos. Soc.'' , '''20'''  (1920–1921)  pp. 263–271 {{ZBL|48.0151.04}}</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy (ed.)  et al. (ed.) , ''Collected papers of S. Ramanujan'' , Chelsea, reprint  (1962)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  B. Volkmann,  "Verallgemeinerung eines Satzes von Maxfield"  ''J. Reine Angew. Math.'' , '''271'''  (1974)  pp. 203–213</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top">  E.C. Titchmarsh,  "The theory of the Riemann zeta-function" , Clarendon Press  (1951) {{ZBL|0042.07901}}</TD></TR>
 +
<TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Levin,  "The life and work of the Indian mathematician Ramanujan"  ''Istoriko-Mat. Issled.'' , '''13'''  (1960)  pp. 335–378  (In Russian)</TD></TR>
 +
</table>

Latest revision as of 14:27, 17 March 2023


Trigonometric sums depending on two integer parameters $ k $ and $ n $:

$$ c _ {k} ( n) = \sum _ { h } \mathop{\rm exp} \left ( \frac{2 \pi n h i }{k} \right ) = \ \sum _ { h } \cos \frac{2 \pi n h }{k} , $$

when $ h $ runs over all non-negative integers less than $ k $ and relatively prime to $ k $. The basic properties of Ramanujan sums are multiplicativity with respect to the index $ k $,

$$ c _ {k k ^ \prime } ( n) = c _ {k} ( n) c _ {k ^ \prime } ( n) \ \ \textrm{ if } ( k , k ^ \prime ) = 1 , $$

and also the representation in terms of the Möbius function $ \mu $:

$$ c _ {k} ( n) = \ \sum _ {d \mid ( k , n ) } \mu \left ( \frac{k}{d} \right ) d . $$

Ramanujan sums are finite if $ k $ or $ n $ is finite. In particular, $ c _ {k} ( 1) = 1 $.

Many multiplicative functions on the natural numbers (cf. Multiplicative arithmetic function) can be expanded as series of Ramanujan sums, and, conversely, the basic properties of Ramanujan sums enable one to sum series of the form

$$ \sum _ { n= 1} ^ \infty \frac{c _ {k} ( q n ) }{n ^ {s} } f ( n) ,\ \ \sum _ { k= 1} ^ \infty \frac{c _ {k} ( q n ) }{k ^ {s} } f ( k) , $$

where $ f $ is a multiplicative function and $ q $ is an integer. In particular,

$$ \sum _ { k= 1 }^ \infty \frac{c _ {k} ( n) }{n ^ {s} } = \ \frac{\sigma _ {1- s} ( n) }{\zeta ( s) } , $$

where $ \zeta $ is the Riemann zeta-function and $ \sigma _ {a} $ is the sum of the $ a$-th powers of the divisors of $ n $. Such sums are closely connected with special series for certain additive problems in number theory (cf. Additive number theory); for example, the representation of a natural number as an even number of squares. S. Ramanujan [1] obtained many formulas involving Ramanujan sums.

References

[1] S. Ramanujan, "On certain trigonometrical sums and their applications in the theory of numbers" Trans. Cambridge Philos. Soc. , 22 (1918) pp. 259–276 ((Also: Collected papers, Chelsea, reprint, 1962.))
[2] G.H. Hardy, "Note on Ramanujan's trigonometrical function $c_q(n)$ and certain series of arithmetical functions" Proc. Cambridge Philos. Soc. , 20 (1920–1921) pp. 263–271 Zbl 48.0151.04
[3] G.H. Hardy (ed.) et al. (ed.) , Collected papers of S. Ramanujan , Chelsea, reprint (1962)
[4] B. Volkmann, "Verallgemeinerung eines Satzes von Maxfield" J. Reine Angew. Math. , 271 (1974) pp. 203–213
[5] E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951) Zbl 0042.07901
[6] V.I. Levin, "The life and work of the Indian mathematician Ramanujan" Istoriko-Mat. Issled. , 13 (1960) pp. 335–378 (In Russian)
How to Cite This Entry:
Ramanujan sums. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ramanujan_sums&oldid=18643
This article was adapted from an original article by K.Yu. Bulota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article