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For my Bachelor's thesis, I am investigating divergent series. This is (yet another) question on this topic.

Apparently, a divergent series $$ S = \sum_{n=1}^{\infty} a_{n} $$ can be summed by means of analytic continuation of the corresponding dirichlet series $$ f(s) = \sum_{n=1}^{\infty} \frac{ a_{n} }{n^{s}} $$ at $s=0$.

Howevever, I can not find any examples of divergent series being summed by means of this method (for example, I can't find anything in Hardy's Divergent Series). Can you please provide me with some examples and/or theory which show how this method can be used to sum divergent series? References are also very much appreciated.

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    $\begingroup$ I'm just checking the really nice book of Konrad Knopp on infinite series, where the chap 13 deals with divergent series. Writing about general Dirichlet's series in the style of your example he mentions, that Riesz together with G.H.Hardy has written a small book about summability of the general Dirichlet-series. Knopp calls it the "Riesz-summation method". Perhaps there is something in that book for you. $\endgroup$ – Gottfried Helms Aug 30 '14 at 21:17
  • $\begingroup$ It seems to me, that the second part in this answer mathoverflow.net/questions/168789/… to my question there might be interesting because it explicates a summation of a dirichlet series which in the (my) question was also considered in the light of Cesaro-summation $\endgroup$ – Gottfried Helms Sep 1 '14 at 10:45
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For example, for any Dirichlet character $\chi$, the sums $\sum_{n=1}^\infty \chi(n)$ can be summed by analytic continuation of $\sum_n \chi(n)/n^s$, which has a meromorphic continuation.

Similarly, for many other arithmetical functions (such as coefficients of modular forms) the analogous sum has a meromorphic continuation, so is summable in this sense.

However, not every reasonable sequence of coefficients, even if admitting reasonable-looking arithmetic descriptions, gives a Dirichlet series with a meromorphic continuation. Some have natural boundaries, as was discovered by Estermann c. 1928. An account of some relatively elementary examples is in my course notes, linked-to from the HTML with title "Estermann phenomenon", from my course-notes page at http://www.math.umn.edu/~garrett/m/mfms/

In fairly immediate situations, and in general, the question of whether a Dirichlet series has a meromorphic continuation (beyond a fairly obvious half-plane) is a difficult open question.

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Example.

The power series $f(x)=\sum_{n=1}^\infty (2^{m_2(n)+1}-1)x^n$, where $m_2(n)$ is the multiplicity of the factor $2$ of $n$ (the $2$-adic order of $n$), diverges for all $x\neq 0$. As $f(x)-2f(x^2)=x+x^2+x^3+\cdots=x/(1-x)$ for $|x|<1$, its iterated elementary Ramanujan sum is $1/2$ (definition, also here).

The Dirichlet series $F(s)=\sum_{n=1}^\infty (2^{m_2(n)+1}-1)n^{-s}$ converges in the half-plane $\sigma>2$. It takes the form $$ F(s)=\frac{1}{1-2^{1-s}}\zeta(s) $$ (here). Therefore, $F(0)=-\zeta(0)=1/2$.

Edit. Change notation: $f_2(x)=f(x)$, $f_1(x)=x+x^2+x^3+\cdots $ and $f_0(x)=x-x^2+x^3-\cdots $.

We know that $f_0(x)=x/(1+x)$ for $|x|<1$ is Abel summable to $\sum_{n=1}^\infty (-1)^{n+1}=1/2$ at $x=1$ (Grandi series) (here), $f_1(x)-2f_1(x^2)=f_0(x)$ and $f_2(x)-2f_2(x^2)=f_1(x)$.

In general, for $f(x)=\sum_{n=0}^\infty a_n x^n$, $f^{(N)}(x)$ is defined by $$ f^{(N)}(x)=\left(x\frac{d}{dx}\right)^N f(x)~. $$

We have $f_1^{(N)}(x)-2^{N+1}f_1^{(N)}(x^2)=f_0^{(N)}(x)$ and $f_2^{(N)}(x)-2^{N+1}f_2^{(N)}(x^2)=f_1^{(N)}(x)$.

We know that the elementary Ramanujan sum of $f_1^{(N)}(x)$ at $x=1$ is $$ f_1^{(N)}(1)=\frac{f_0^{(N)}(1)}{1-2^{N+1}}=\zeta(-N)~. $$

The iterated elementary Ramanujan sum of $f_2^{(N)}(x)$ at $x=1$ is $$ f_2^{(N)}(1)=\frac{f_1^{(N)}(1)}{1-2^{N+1}}=\frac{1}{1-2^{1+N}}\zeta(-N) =F(-N) $$ (also here).

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