about ksivah, there are many issues. To name but a few offhand:
There is no “maaseh ksivah”.
The writing does not last (derabanan, and combined with other things could be muttar because of shvus dshvus and/or safek derabanan)
Writing on a computer or cell phone screen is actually made of of thousands of tiny dots. According to some, this is not considered writing.
According to some kosev is only with ksav ashuris
When you are mitztaref these various reasons you can come up with a heter.
Another thing, it might be a gramma since pushing the button causes a chain reaction in the phone, first the signal is sent to the computer within the phone, then it is sent to another part of the computer, the output, then from the output to the screen input, and finally to the screen itself. So many steps happen, which could be considered gramma.