Selberg-Delange method

The Selberg-Delange method is a technique from analytical number theory . It is used to determine the mean order of a number theoretic function . It is named after Atle Selberg and Hubert Delange .

The classes and ${\displaystyle T}$ ${\displaystyle P}$

Be If there is a Dirichlet series with a convergence half- plane , then this belongs to the class if the Dirichlet series ${\displaystyle z\in \mathbb {C} ,c_{0}>0,0<\delta \leq 1,M>0.}$ ${\displaystyle \textstyle F(s)=\sum a(n)/n^{s}}$ ${\displaystyle \operatorname {Re} (s)>1}$ ${\displaystyle P(z;c_{0},\delta ,M)}$

${\displaystyle G(s;z):=F(s)\zeta (s)^{-z}}$

represents a holomorphic function in the whole area and also the inequality there ${\displaystyle \left\{\operatorname {Re} (s)>1-{\tfrac {c_{0}}{1+\max\{0,\log(|\operatorname {Im} (s)|)\}}}\right\}}$

${\displaystyle |G(s;z)|\leq M(1+|\operatorname {Im} (s)|)^{1-\delta }}$

enough. Here denotes the Riemann zeta function . If there is a sequence with and the series belongs to the class , then by definition it is even in the class Then the following modified function can be written locally as a Taylor series at the origin : ${\displaystyle \zeta (s)}$ ${\displaystyle b(n)}$ ${\displaystyle |a(n)|\leq b(n)}$ ${\displaystyle \textstyle \sum b(n)/n^{s}}$ ${\displaystyle P(w;c_{0},\delta ,M)}$ ${\displaystyle F(s)}$ ${\displaystyle T(z,w;c_{0},\delta ,M).}$

${\displaystyle {\frac {s^{z}F(s+1)}{s+1}}=\sum _{k=0}^{\infty }\mu _{k}(z)s^{k},\quad |s|<\min\{c_{0},1\}.}$

statement

If it is in the class , the following already applies to : ${\displaystyle F(s)}$ ${\displaystyle T(z,w;c_{0},\delta ,M)}$ ${\displaystyle x\geq 3,N\geq 0,A>0,|z|,|w|\leq A}$

${\displaystyle \sum _{n\leq x}a(n)=x(\log x)^{z-1}\left(\sum _{0\leq k\leq N}{\frac {\mu _{k}(z)}{\Gamma (z-k)(\log(x))^{k}}}+O\left(M(\mathrm {e} ^{-c_{1}{\sqrt {\log(x)}}}+({\frac {c_{2}N+1}{\log(x)}})^{N+1})\right)\right).}$

The positive constants and the implicit constant in the Landau symbol depend at most on the choice of and . An important special case is . Then follow whenever applies. This makes it possible to choose such that the error term is minimized. You can reach the statement by choosing ${\displaystyle c_{1},c_{2}}$ ${\displaystyle c_{0},\delta }$ ${\displaystyle A}$ ${\displaystyle z\in \mathbb {Z} }$ ${\displaystyle 1/\Gamma (z-k)=0}$ ${\displaystyle k\geq z}$ ${\displaystyle N}$ ${\displaystyle N:=\lfloor \log(x)/\mathrm {e} c_{2}\rfloor }$

${\displaystyle \sum _{n\leq x}a(n)=x(\log(x))^{z-1}\left(\sum _{0\leq k\leq z-1}{\frac {\mu _{k}(z)}{\Gamma (z-k)(\log(x))^{k}}}+O(M\mathrm {e} ^{-c_{1}{\sqrt {\log(x)}}})\right).}$

Advantages and disadvantages

Advantages of the Selberg-Delange method are the rather explicit specification of an error term and the lack of necessity that they always have to be non-negative. However, the required vertical estimation (which cannot be omitted!) Can represent a hurdle. So if less detailed information about the mean order is needed, one can also fall back on Taubers' theorems , which are already valid under significantly weaker assumptions, but do not allow any estimation of the error terms. ${\displaystyle a(n)}$

Individual evidence

1. ^ Gérald Tenenbaum : Introduction to analytic and probabilistic number theory . AMS, Rhode Island 1990, p. 281 .