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Difference between revisions of "Additive divisor problem"

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The problem of finding asymptotic values for sums of the form:
 
The problem of finding asymptotic values for sums 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/a/a010/a010640/a0106401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1}
 +
\left .
 +
{ {\sum _ {m \leq  n} \tau _{ {k _ 1}} ( m ) \tau _{ {k _ 2}} ( m + a ) ,} \atop {\sum _ {m < n}\tau _{ {k _ 1}} ( m ) \tau _{ {k _ 2}} ( n - m ) ,}}
 +
\right \}
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a0106402.png" /> is the number of different factorizations of an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a0106403.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a0106404.png" /> factors, counted according to multiplicity. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a0106405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a0106406.png" /> are integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a0106407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a0106408.png" /> is a fixed integer different from zero and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a0106409.png" /> is a sufficiently large number. In particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a01064010.png" /> is the [[Number of divisors|number of divisors]] of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a01064011.png" />. Sums of the form (1) express the number of solutions of the equations
+
where $  \tau _{k} (m) $
 +
is the number of different factorizations of an integer $  m $
 +
in $  k $
 +
factors, counted according to multiplicity. Here $  k _{1} $
 +
and $  k _{2} $
 +
are integers $  \geq 2 $,  
 +
a $
 +
is a fixed integer different from zero and $  n $
 +
is a sufficiently large number. In particular $  \tau _{2} (m) = \tau (m) $
 +
is the [[Number of divisors|number of divisors]] of the number $  m $.  
 +
Sums of the form (1) express the number of solutions of the equations
  
<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/a/a010/a010640/a01064012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2}
 +
x _{1} \dots x _{ {k _ 2}} \  - \  y _{1} \dots y _{ {k _ 1}} \  = a ,
 +
$$
  
<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/a/a010/a010640/a01064013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3}
 +
x _{1} \dots x _{ {k _ 1}} \  + \  y _{1} \dots y _{ {k _ 2}} \  = \  n ,
 +
$$
  
respectively. Particular cases of the additive divisor problem (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a01064014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a01064015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a01064016.png" />) are considered in [[#References|[1]]]–[[#References|[3]]]. The additive divisor problem with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a01064017.png" /> and an arbitrary positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a01064018.png" /> was solved using the [[Dispersion method|dispersion method]] of Yu.V. Linnik [[#References|[4]]].
+
respectively. Particular cases of the additive divisor problem ( $  k _{1} = k _{2} =2 $,  
 +
$  k _{1} = 2 $
 +
and $  k _{2} = 3 $)  
 +
are considered in [[#References|[1]]]–[[#References|[3]]]. The additive divisor problem with $  k _{1} = 2 $
 +
and an arbitrary positive integer $  k _{2} $
 +
was solved using the [[Dispersion method|dispersion method]] of Yu.V. Linnik [[#References|[4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.E. Ingham,  "Some asymptotic formulae in the theory of numbers"  ''J. London Math. Soc. (1)'' , '''2'''  (1927)  pp. 202–208</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Esterman,  "On the representations of a number as the sum of two products"  ''Proc. London Math. Soc. (2)'' , '''31'''  (1930)  pp. 123–133</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Hooly,  "An asymptotic formula in the theory of numbers"  ''Proc. London Math. Soc. (3)'' , '''7'''  (1957)  pp. 396–413</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.V. Linnik,  "The dispersion method in binary additive problems" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.E. Ingham,  "Some asymptotic formulae in the theory of numbers"  ''J. London Math. Soc. (1)'' , '''2'''  (1927)  pp. 202–208</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Esterman,  "On the representations of a number as the sum of two products"  ''Proc. London Math. Soc. (2)'' , '''31'''  (1930)  pp. 123–133</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Hooly,  "An asymptotic formula in the theory of numbers"  ''Proc. London Math. Soc. (3)'' , '''7'''  (1957)  pp. 396–413</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.V. Linnik,  "The dispersion method in binary additive problems" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a01064019.png" /> is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a01064020.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010640/a01064021.png" />, cf. [[#References|[a1]]], Sect. 16.7.
+
The function $  \tau _{2} (m) = \tau (m) $
 +
is also denoted by $  d (m) $
 +
or $  \sigma _{0} (m) $,  
 +
cf. [[#References|[a1]]], Sect. 16.7.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Clarendon Press  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Clarendon Press  (1979)</TD></TR></table>

Latest revision as of 08:37, 6 February 2020


The problem of finding asymptotic values for sums of the form:

$$ \tag{1} \left . { {\sum _ {m \leq n} \tau _{ {k _ 1}} ( m ) \tau _{ {k _ 2}} ( m + a ) ,} \atop {\sum _ {m < n}\tau _{ {k _ 1}} ( m ) \tau _{ {k _ 2}} ( n - m ) ,}} \right \} $$

where $ \tau _{k} (m) $ is the number of different factorizations of an integer $ m $ in $ k $ factors, counted according to multiplicity. Here $ k _{1} $ and $ k _{2} $ are integers $ \geq 2 $, $ a $ is a fixed integer different from zero and $ n $ is a sufficiently large number. In particular $ \tau _{2} (m) = \tau (m) $ is the number of divisors of the number $ m $. Sums of the form (1) express the number of solutions of the equations

$$ \tag{2} x _{1} \dots x _{ {k _ 2}} \ - \ y _{1} \dots y _{ {k _ 1}} \ = \ a , $$

$$ \tag{3} x _{1} \dots x _{ {k _ 1}} \ + \ y _{1} \dots y _{ {k _ 2}} \ = \ n , $$

respectively. Particular cases of the additive divisor problem ( $ k _{1} = k _{2} =2 $, $ k _{1} = 2 $ and $ k _{2} = 3 $) are considered in [1][3]. The additive divisor problem with $ k _{1} = 2 $ and an arbitrary positive integer $ k _{2} $ was solved using the dispersion method of Yu.V. Linnik [4].

References

[1] A.E. Ingham, "Some asymptotic formulae in the theory of numbers" J. London Math. Soc. (1) , 2 (1927) pp. 202–208
[2] T. Esterman, "On the representations of a number as the sum of two products" Proc. London Math. Soc. (2) , 31 (1930) pp. 123–133
[3] C. Hooly, "An asymptotic formula in the theory of numbers" Proc. London Math. Soc. (3) , 7 (1957) pp. 396–413
[4] Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc. (1963) (Translated from Russian)

Comments

The function $ \tau _{2} (m) = \tau (m) $ is also denoted by $ d (m) $ or $ \sigma _{0} (m) $, cf. [a1], Sect. 16.7.

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press (1979)
How to Cite This Entry:
Additive divisor problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_divisor_problem&oldid=44382
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article