Motivic Euler products in motivic statistics

We formulate and prove an analog of Poonen's finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil-Wood, who treated the case of smooth hypersurface sections. In fact, our techniques give analogs in motivic statistics of all known results in arithmetic statistics that have been proven using Poonen's sieve, including work of Bucur-Kedlaya on complete intersections and Erman-Wood on semi-ample Bertini theorems. A key ingredient is the use of motivic Euler products, as introduced by the first author, to write down candidate motivic probabilities. We also formulate a conjecture on the uniform convergence of zeta functions that unifies motivic and arithmetic statistics for varieties over finite fields.