# How to generalize Newman's simplification of O-Tauberian theorem?

Can you prove the next theorem:

Let $$f$$ be Dirichlet series with real, positive coefficients $$(a_n>0)$$. If $$f$$ is holomorphic on $$\Re(z)\ge1$$, but has one singularity at $$z=1$$, then

$$\lim_\limits{z \to 0^+} \left[f(z)-c(z-1)^k\right]$$ exists $$\iff$$ $$\lim_\limits{x \to \infty} \frac{A(x)\ln(x)^{k+1}}{x}=c$$

where $$c$$ and $$k$$ are some real constants and

$$A(x)=\sum_{n=1}^x a_n$$

Behind the theorem:

I'm exploring complex analysis methods in number theory, specifically for determining asymptotic growth of partial sums of multiplicative functions. I have read Newman's proof of PNT, and I tried to use these ides to find the asymptotic growth of the next sum:

$$\sum_{n=1}^{x} \frac{1}{d(n)}$$

where $$d(n)$$ is the number of divisors of $$n$$. It seems to be, and computer approves it, that the growth is $$\frac{cx}{\sqrt{\ln(x)}}$$ for some real constant $$c$$. Now, in order to prove it, I defined $$f$$ to be Dirichlet series with coefficients $$a_n=\frac{1}{d(n)}$$. I proved that $$f(z)-c\sqrt{\zeta(z)}$$ is holomorphic for $$\mathfrak{R}(z)>0$$. Also, I have that $$f(z)-\frac{c}{\sqrt{z-1}}$$ is holomorphic for $$\Re(z)\ge1$$ except for $$1$$. It is not holomorphic at the point $$1$$, so I can't use Newman's analytic theorem. But, if we could ignore it, we would get that,

$$\int_1^\infty \frac{A(t)-ct\sqrt{\ln(t)}}{t^2}dt$$

converges. (That comes out when we differentiate $$f(z)-\frac{c}{\sqrt{z-1}}$$). Here I used

$$A(x)=\sum_{n=1}^{x} \frac{\ln(n)}{d(n)}$$.

Then, using Newman's idea, we can get that $$A(x)$$~$$cx\sqrt{\ln(x)}$$. From there, using partial summation, we get that the first sum is asymptotically equal to what it is expected to be, $$\frac{cx}{\sqrt{\ln(x)}}$$.

Please, help me, this is going to be in my undergraduate work, I need to know these things. Help me to prove this asymptotic growth.

• I would suggest to take a look at korevaar's tauberian theory bible and see what's known – Conrad Mar 27 at 19:17
• @Conrad Can you please give me a link for that? – donaastor Mar 27 at 19:31
• This is the link to the official version of the book - any decent university library should have it and of course, I am sure that unofficial versions exist too. springer.com/us/book/9783540210580 – Conrad Mar 27 at 22:28

It is Ryo-Kato's generalization of Kable's generalization of Wiener-Ikehara Tauberian theorem. Kable states for $$k$$ of form $$\frac{1}{n}$$, and Ryo-Kato states it for all rational $$k$$. The real $$k$$ is still a mystery, though.