If it is a logo, do not use gradients. Simplify, abstract, reduce.

But let us put that aside.

An easy way to produce gradients is by **blending shapes**.

 1. Divide your circle into the required fractions. And draw two trapezoidal shapes. You can refine the shapes later.

2. Blend them https://helpx.adobe.com/illustrator/using/blending-objects.html

3. And duplicate the shapes. You can do the same for the external shadow.

[![enter image description here][1]][1]

There are other methods for gradients, but this one is basic, easy and you have control over it.
 
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Another method is using **gradient meshes**. Just do not be too greedy making a grid too big.

You can use the gradient mesh for the triangle section I did before.

https://helpx.adobe.com/illustrator/using/meshes.html

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[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/43H86.png
  [2]: https://i.sstatic.net/FSv5A.png