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Suppose I have an integral Dirichlet series $f(s) = \sum c_n n^{-s}$, $c_n \in \mathbb{Z}$, with at least one non-zero term $c_N$. Suppose furthermore that this series converges absolutely and uniformly for $\mathrm{Re}(s) > 1 + \delta$ for any $\delta > 0$ (so that $c_n$ grows slower than $n^\epsilon$ for any $\epsilon$).

I want $f(s)$ to be "small" (in terms of $N$) for fixed $\mathrm{Re}(s)$ and a range of $\mathrm{Im}(s)$. How small can I get it over any given range?

I think one can take sums of $(N + \Delta n)^{-s}$ for $\Delta n \approx O(\ln N)$ and cancel out terms in the Taylor series up to order $N^{-(s + O(\ln N))}$, all while keeping the coefficients polylog in N for $Im(s) \approx O(\ln N)$. Can we do any better? What about larger values of $s$?

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Are you trying to keep $f(s)$ small over the entire range of Im($s$), or do you just want it small at one point in the range? - In general, this would be a better question if you included more details (about the construction you allude to in the last paragraph, for example, or about the specific parameters of the problem you want to consider). – Greg Martin Oct 6 '11 at 19:49
I do not believe it is possible to keep it small for all Im($s$). I am looking for statements of the form, "for $|Im(s) - \sigma | \approx O(g(N))$ the following construction gives $|f(s)| \approx O(N^{-(Re(s) + h(N))})$". As for the construction I alluded to, consider that $N^{-s} - (N+1)^{-s} = s N^{-(s+1)} + O(s^2 N^{-(s+2)})$. Furthermore if we pick any two $k,k'$, we can find a linear combination of $N^{-s}$, $(N+k)^{-s}$ and $(N+k')^{-s}$ that will be of order $s^2 N^{-(s+2)}$. We can keep doing this approximately $ln N$ times at least before the error terms start blowing up. – Craig Oct 6 '11 at 20:47
up vote 1 down vote accepted

I alluded to one construction that gives small values for $f(s)$ in a given range of Im(s). I will elaborate on that construction here.

Consider the Taylor expansion of $(N+x)^{-s}$. This is

$(N+x)^{-s} = N^{-s} * (1 - \frac{sx}{N} + \frac{s(s+1)x^2}{2N^2} - \ldots )$.

We are going to truncate this at some finite order $k$, and allow $x$ to take on different integer values. We then obtain a number of terms polynomial in $s$ (times $N^{-s}$) and we want a linear combination of these terms to vanish for all $s$ -- that implies $x$ takes on at least $k$ different values.

Define the $c_n$ to be the coefficients in this sum. We would like the remaining error term in the Taylor series (the terms of order $s^{(k+1)}/N^{k+1}$ and higher) to be $o(N^{-k})$. This implies that $\sum c_n (n-N)^{k+1} s^{(k+1)} / k!$ is $o(N)$. Remember that we have at least $k-1$ nonzero terms in this sum, and so there is at least one $n$ with $|n-N| \geq (k-1)/2$.

To ensure that this sum is $o(N)$, we should take $k$ to be $o(\ln N)$ and similarly for $s$. We can readily find a set of $n$ and $c_n$ which will keep the sum at $o(N)$ -- I believe the $k$th-order difference should work, although I am not certain.

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