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