Zeta Function In Real And Imaginary Parts

Question

The zeta function is defined for a complex input $s$:

$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$$

What I am wondering is if there are functions $f$ and $g$ such that:

$$\zeta(s) = f(s) + ig(s)$$

By using simple algebra on the Dirichlet series you can show that the real and imaginary components are separable into the form:

$$\zeta(\sigma + it) = \sum_{n=1}^{\infty} \frac{\cos(t\ln(n))}{n^\sigma} - i\sum_{n=1}^{\infty}\frac{\sin(t\ln(n))}{n^\sigma}$$

Are there any other functions like this? Can you separate the real and imaginary components on other definitions of the zeta function? Like the Hurwitz or other representations? Any input would be appreciated!

Answer 1

For any set $X$ and any function $f:X\to \mathbb{C}$, you can simply define $g,h:X\to\mathbb{R}$ by $$g(x)=\mathrm{Re}(f(x))\qquad h(x)=\mathrm{Im}(f(x))$$ and then you have $f(x)=g(x)+ih(x)$ for all $x$. Whether or not these functions have any interesting properties is another matter.

By the way, the equality $\zeta(s)=\sum\limits_{n=1}^\infty\frac{1}{n^s}$ is only true for complex numbers $s$ with $\mathrm{Re}(s)>1$. It is not a correct statement for other choices of $s$, the only correct definition is with analytic continuation.

Answer 2

You can use Hardy and Littlewood’s approximation (see Titchmarsh, Theorem 4.11), which is valid for $\sigma>0$:

$$\zeta(s)=\sum_{n\le x}\frac{1}{n^s}-\frac{x^{1-s}}{1-s}+O(x^{-\sigma}).$$

If you let $x=t$ (something the conditions of the cited theorem allows you to do), you can combine the second term with the error term to get an error term of $O(t^{-\sigma}+it^{-\sigma})$, unless I have made a mistake in my calculations. Now you can expand the summation in the same way that you expanded it for the Dirichlet series.