Sum function of a function

$ 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