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Difference between revisions of "Sum function of a function"

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m (fixing subscripts)
 
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of the function  $  f $
 
of the function  $  f $
 
on the set of natural numbers  $  n \leq  x $,  
 
on the set of natural numbers  $  n \leq  x $,  
$  \sum _ {n\leq } x f( n) $.  
+
$  \sum _ {n\leq x }f( n) $.  
 
Sum functions are one of the basic means of expressing various properties of sequences of numbers.
 
Sum functions are one of the basic means of expressing various properties of sequences of numbers.
  
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$$  
 
$$  
\sum _ { n\leq } x f( n)  =   
+
\sum _ { n\leq   x} f( n)  =   
 
\frac{1}{2 \pi i }
 
\frac{1}{2 \pi i }
 
  \int\limits _ {c- i \infty } ^ { c+  i \infty } F( s)  
 
  \int\limits _ {c- i \infty } ^ { c+  i \infty } F( s)  
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$$  
 
$$  
\sum _ { n\leq } x f( n)  =  R( x) + O ( x ^ {\{ ( \alpha + 1) ( 2 \eta - 1) / ( 2
+
\sum _ { n\leq   x} f( n)  =  R( x) + O ( x ^ {\{ ( \alpha + 1) ( 2 \eta - 1) / ( 2
 
\eta + 1) \} + \epsilon } ) ,
 
\eta + 1) \} + \epsilon } ) ,
 
$$
 
$$

Latest revision as of 01:34, 26 April 2022


$ f $

The function of $ x \geq 1 $ that denotes the sum of the values $ f( n) $ of the function $ f $ on the set of natural numbers $ n \leq x $, $ \sum _ {n\leq x }f( n) $. Sum functions are one of the basic means of expressing various properties of sequences of numbers.

Examples of sum functions: the number of prime numbers $ \leq x $; $ \psi ( x) = \sum _ {n\leq x }\Lambda ( n) $— the Chebyshev function; the number of divisors of all $ n \leq x $, etc. (see [1], [2]).

The basic problem is to find as accurate as possible an expression of the sum function, and for a sum function which does not have asymptotics, to find the best estimate of its modulus for large values of $ x $.

The Cauchy integral theorem and Dirichlet series of the form

$$ F( s) = \sum _ { n= 1} ^ \infty f( n) n ^ {- s} $$

form the basis of the analytic methods of studying sum functions. If such a series converges absolutely for $ \mathop{\rm Re} s > \sigma _ {0} \geq 1 $, then for a non-integer $ x $, and $ c > \sigma _ {0} $, the identity

$$ \sum _ { n\leq x} f( n) = \frac{1}{2 \pi i } \int\limits _ {c- i \infty } ^ { c+ i \infty } F( s) \frac{x ^ {s} }{s} ds $$

holds; a corresponding estimate of the sum function of $ f $ is obtained from this by analytic continuation of $ F( s) $ by shifting the integration path to the left to a certain $ \mathop{\rm Re} s = \sigma _ {1} < 0 $ and estimating the integral along the new path. If $ f( n) = \Lambda ( n) $, for example, the integration can be shifted to $ \mathop{\rm Re} s = - \infty $, which gives the Riemann–von Mangoldt formula for $ \psi ( x) $. Of the common applications of the method, the following theorem is known.

Assumptions:

$ f( n) $, $ l _ {n} $ are complex numbers, $ \alpha \geq 0 $, $ \alpha _ {r} $, $ \gamma _ {r} $ are real numbers, $ \sigma _ {r} $, $ \beta _ {r} $ are positive numbers, $ \mu $ and $ \nu $ are integers $ \geq 1 $, $ \Gamma $ is the gamma-function, and $ \lambda _ {1} < \lambda _ {2} < \dots $.

1) For any $ \epsilon > 0 $, $ f( n) \ll n ^ {\alpha + \epsilon } $;

2) the function

$$ F( s) = \sum _ { n= 1 }^ \infty f( n) n ^ {- s} , $$

defined for $ s = \sigma + it $, $ \sigma > 1 + \alpha $, is meromorphic in the whole plane, and has a finite number of poles in the strip $ \sigma _ {1} \leq \sigma \leq \sigma _ {2} $;

3) the series $ \sum _ {n= 1} ^ \infty l _ {n} \mathop{\rm exp} ( \lambda _ {n} s) $ converges absolutely when $ \sigma < 0 $;

4) for $ \sigma < 0 $,

$$ \prod _ { r= 1} ^ \mu \Gamma ( \alpha _ {r} + \beta _ {r} s) F( s) = $$

$$ = \ \prod _ { r= 1} ^ \nu \Gamma ( \gamma _ {r} - \delta _ {r} s) \sum _ { n= 1} ^ \infty l _ {n} \mathop{\rm exp} ( \lambda _ {n} s); $$

5) $ \beta _ {1} + \dots + \beta _ \mu = \delta _ {1} + \dots + \delta _ \nu $;

6) if one assumes that

$$ \sum _ { r= 1} ^ \nu \gamma _ {r} - \sum _ { r= 1} ^ \mu \alpha _ {r} + \frac{1}{2} ( \mu - \nu ) = \eta , $$

then $ \eta \geq \alpha + 1/2 $.

For a fixed strip $ \sigma _ {1} \leq \sigma \leq \sigma _ {2} $ there is a constant $ \gamma = \gamma ( \sigma _ {1} , \sigma _ {2} ) $ such that for $ \sigma _ {1} \leq \sigma \leq \sigma _ {2} $ and large $ | t | $ the estimate $ F( s) \ll \mathop{\rm exp} ( \gamma | t | ) $ holds.

Conclusion.

For any $ \epsilon > 0 $,

$$ \sum _ { n\leq x} f( n) = R( x) + O ( x ^ {\{ ( \alpha + 1) ( 2 \eta - 1) / ( 2 \eta + 1) \} + \epsilon } ) , $$

where $ R( x) $ is the sum of the residues of the function $ F( s) x ^ {s} /s $ over all its poles in the strip

$$ ( \alpha + 1) \frac{2 \eta - 1 }{2 \eta + 1 } < \sigma \leq \alpha + 1. $$

References

[1] E.C. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1951)
[2] L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 1 : 2 (1959) (Heft 13, Teil 1)

Comments

The Riemann–von Mangoldt formula, or von Mangoldt formula, for $ \psi ( x) $, $ x> 1 $, is

$$ \psi ( x) = x - \sum _ { p } \frac{x ^ {p} }{p} + \sum _ { n } \frac{x ^ {- 2n }}{2n} + \textrm{ const } . $$

This is von Mangoldt's reformulation of Riemann's main formula

$$ J( x) = \mathop{\rm Li} ( x) - \sum _ { p } \mathop{\rm Li} ( x ^ {p} ) - \mathop{\rm log} 2+ \int\limits _ { x } ^ \infty \frac{dt}{t( t ^ {2} - 1) \mathop{\rm log} t } , $$

where $ x> 1 $, the $ J $-function is

$$ J( x) = \frac{1}{2} \left ( \sum _ {p ^ {n} < x } \frac{1}{n} + \sum _ {p ^ {n} \leq x } \frac{1}{n} \right ) $$

and $ \mathop{\rm Li} ( x) $ is the logarithmic integral

$$ \mathop{\rm Li} ( x) = \lim\limits _ {\epsilon \downarrow 0 } \left [ \int\limits _ { 0 } ^ { {1-\epsilon } } \frac{dt}{ \mathop{\rm log} t } + \int\limits _ {1+ \epsilon } ^ { x } \frac{dt}{ \mathop{\rm log} t } \right ] . $$

References

[a1] H.M. Edwards, "Riemann's zeta function" , Acad. Press (1974) pp. Chapt. 3
How to Cite This Entry:
Sum function of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sum_function_of_a_function&oldid=52292
This article was adapted from an original article by A.F. Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article