Do many universities allow students to get a PhD in mathematics without publishing a paper?
It is typical in the U.S. that mathematics Ph.D. students do not publish anything at all prior to earning a Ph.D., I think even at the elite places. Publication per se is not such a high priority, nor over-literal gauge of accomplishment, as it seems to be in some CompSci and Engineering disciplines (at least in the minds of some people).
That is, to be clear, especially for people who've thought, or been indoctrinated to think, that the measure of the value of something is the prestige of the conference or journal or ... in which it was published, the idea in mathematics still does seem to be that the thesis advisor and thesis committee decide whether the candidate has done sufficient work to earn the degree.
Seems a sane system to me, especially given the acceleration of freneticism elsewhere.
Yes, it is very common for students to earn a PhD in mathematics without publishing any papers before graduating. Here are a few pieces of context:
I have read a couple research articles that analyzed the Mathematical Review database, which is a very thorough listing of mathematics publications. One such article is "Patterns of Collaboration in Mathematical Research". It states that 42.7% of authors in the database have only one article (likely taken from their thesis) and another 14.6% have only two. I recommend that paper highly for more information on mathematics publishing.
Publication rates in mathematics vary significantly by subfield, because the threshold for the amount of progress needed for a new paper also varies by field. In some subfields, it is common for researchers to publish many (smaller) papers. In others, it can take years to write a single paper. Publication rates also vary by author: some prestigious faculty rarely publish, others are prodigious writers.
My general sense is that publication rates are increasing: mathematicians in general publish more than they used to.
I see much more emphasis on graduate student publication that I did in the past. While the majority of students publish nothing before their PhD, I think the number who do publish is going up. Similarly, there is more emphasis on undergraduate research, and this translates into more publications by undergraduates. In the 1970s, say, publications by undergraduates were much more rare.
The increase in publication is especially true at mid-level schools, e.g. non-elite state schools and some larger non-elite private colleges, which want to raise their research profile and so expect more research than in the past. Historically, say 30 years ago, one might have been able to get tenure with only one or two papers, and teaching was the most important criterion. Now, research is the primary criterion and many of these "aspiring" schools. On the other hand I had a job interview once with a school that explicitly emphasized they did not expect research - this trend toward more publication is not universal.
There is also a distinction between top students (e.g. those who are competitive for prestigious postdocs and for job offers from R-1 schools) versus typical students (who may be perfectly qualified for academic jobs, but are not competitive for R-1 positions). In my experience, top students often have research collaborations outside their thesis topic before graduation, and often have other publications before graduation. But these students make up a small percentage of PhD recipients overall. Even at prestigious schools, not all PhD students are on track to be competitive for R-1 positions. And publication rates again vary by research subfield.
Mathematics in particular is sometimes used for a "vocational" PhD, because it has always been possible for some to find work in industry (e.g. software or R&D) and government (e.g. the U.S. National Security Agency) with a PhD in mathematics. My colleagues and I often talk about bright researchers who we "lost" to non-academic jobs. For students who do not plan to continue in academia, I think there is a smaller incentive to publish extra papers beyond the PhD before graduating.
I think the answer will depend on the country, so let me answer for France.
From what I see, I would say that one is generally allowed to defend one's PhD without having published a paper, but only if the thesis contains the material for at least one international publication. This is judged by the two or three PhD referees and then confirmed by the PhD committee.