$f$ is a function on the set of complex numbers such that $f(z)=1/(z*)$, where $z*$ is the complex conjugate of $z$. $S$ is the set of complex numbers $z$ such that the real part of $f(z)$ lies between $1/2020$ and $1/2018$. If $S$ is treated as a subset of the complex plane, the area of $S$ can be expressed as $m× \pi$ where $m$ is an integer. What is the value of $m$? 
1. Given the function \( f(z) = \frac{1}{\overline{z}} \), where \( \overline{z} \) is the complex conjugate of \( z \). Let \( z = x + iy \) where \( x \) and \( y \) are real numbers, then \( \overline{z} = x - iy \).

2. We need to find the real part of \( f(z) \). First, compute \( f(z) \):
   \[
   f(z) = \frac{1}{\overline{z}} = \frac{1}{x - iy}
   \]
   To simplify, multiply the numerator and the denominator by the conjugate of the denominator:
   \[
   f(z) = \frac{1}{x - iy} \cdot \frac{x + iy}{x + iy} = \frac{x + iy}{x^2 + y^2}
   \]
   Therefore,
   \[
   f(z) = \frac{x}{x^2 + y^2} + i \frac{y}{x^2 + y^2}
   \]
   The real part of \( f(z) \) is:
   \[
   \text{Re}\left( f(z) \right) = \frac{x}{x^2 + y^2}
   \]

3. We are given that the real part of \( f(z) \) lies between \( \frac{1}{2020} \) and \( \frac{1}{2018} \):
   \[
   \frac{1}{2020} \leq \frac{x}{x^2 + y^2} \leq \frac{1}{2018}
   \]

4. To find the set \( S \), we need to solve the inequality:
   \[
   \frac{1}{2020} \leq \frac{x}{x^2 + y^2} \leq \frac{1}{2018}
   \]
   This can be rewritten as:
   \[
   \frac{x}{x^2 + y^2} = a \quad \text{where} \quad \frac{1}{2020} \leq a \leq \frac{1}{2018}
   \]
   Rearrange to get:
   \[
   x = a(x^2 + y^2)
   \]
   \[
   x - ax^2 - ay^2 = 0
   \]
   \[
   x^2 + y^2 = \frac{x}{a}
   \]

5. This represents a circle with center \(\left(\frac{1}{2a}, 0\right)\) and radius \(\frac{1}{2a}\). For \( a = \frac{1}{2020} \) and \( a = \frac{1}{2018} \), we get two circles:
   \[
   \text{Circle 1:} \quad \left(1010, 0\right) \quad \text{with radius} \quad 1010
   \]
   \[
   \text{Circle 2:} \quad \left(1009, 0\right) \quad \text{with radius} \quad 1009
   \]

6. The area \( S \) is the area between these two circles. The area of a circle is given by \( \pi r^2 \). Therefore, the area of the region between the two circles is:
   \[
   \text{Area} = \pi (1010^2 - 1009^2)
   \]
   Simplify the expression:
   \[
   1010^2 - 1009^2 = (1010 + 1009)(1010 - 1009) = 2019 \cdot 1 = 2019
   \]
   Thus, the area is:
   \[
   \text{Area} = 2019 \pi
   \]

The final answer is \( \boxed{2019} \).