Continuing the previous post, it has struck me as interesting that the frequency of haplogroups R1a and Q in Altaian Turkic speakers is about 46/17 or a ratio of ~2.7, while in Ashkenazi Jews it is 12/5, or a ratio of ~2.4.
I haven't been able to dig up any reliable information about the ultimate origins of the Khazars in Asia, and many Turkic-speaking groups lack haplogroup Q, which is on the other hand found among Uralic-speakers, both in Siberia and in Hungary. Haplogroup Q is found at traces in Anatolia, making it possible that eastward moving Turkic speakers may have possessed this haplogroup. Even in Hungary, the Uralic-speaking Magyars were at the head of a confederacy which included Turkic speakers.
So, it seems likely at first hand that Turkic elements may have possessed some blend of R1a and Q in addition to other lineages. What the proportion of these in Khazars was is anyone's guess, but the similarity of the ratio of the two haplogroups in Ashkenazi Jews and Altaians is interesting.
Altaians also have 22.4% of haplogroup C, which is lacking in Ashkenazi, but this is probably due to the later westward moving of Altaic Mongolians in Genghis Khan times, which followed migrations of Turkic speakers.
Altaians also have traces of haplogroup N, which has also been detected in Ashkenazi, and is yet another piece of evidence of some "Asian" input.
If Proto-Khazars were similar to present-day Altaians minus haplogroup C, then they would have a frequency of about 59% R1a and 22% Q.
Therefore, it seems reasonable that an overall 5/22 = ~22% of such Proto-Khazar elements into the Ashkenazi Jewish population may be likely. But, the Khazars of Khazaria may themselves have been somewhat mixed with Western Eurasian elements, which would decrease their frequency of haplogroup Q.
All this is very speculative, and one would have to take a look at Altaian and Ashkenazi Jewish haplotypes within R1a and Q to check whether this theory is valid, but if it is, I would guess that about 1/4 male-mediated influence from a Khazarian source (with wide error margins!) may be just about right.
PS: The silence of researchers about the presence of haplogroup Q in the Ashkenazi is puzzling. I hope that someone is writing a paper on it, since Q is much more of a "smoking gun" of a Khazar connection than R1a which is much more problematic due to its fairly wide distribution in West Eurasia.