The russian invented the theory to rise the russian nationalism,they said that the russian jews are khazars,which means that they are ignorant and all the problem of russia is because of them,then they said that lenin and stalin were their puppets and blah blah,then the theory was spread to ukraine and east europe and then to west europe,the communist at that time help to spread that theory too and the anti-semite europe of the 20 century "hugged" the theory,and also expanded it to "all european jews are khazars and this is why the jews are ignorant race".
Now let's go back to the past, in a world dominated by two major religions-christianity and islam,the khazarian king want to adopt a neutral religion In order not to quarrel with the two major religions,he finally chose judaism,and then he and his elite were converted to judaism,some say that only him or his family converted and some say that all the khazarian elitists
so numbers are between 1-500 khazars who converted.
The khazars and the jews were not consider the converted as part of the jewish people/nation,they considered them as khazarian who believes in the Jewish faith.
After the fall of their kingdom, they were gone, no more khazars.
About the ashkenazim -
Historical and genetic studies recently have shown what the Jews said all the time
Ashkenazi origin is from the levant.
Khazar theory today mainly used by anti-semites or certain interest groups,not taking into consideration that the jews didn't consider themselves as pure race.