Abstract

The classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper, we consider the analogous multiplicative setting of the cyclic group $$(\mathbb {Z}/ q\mathbb {Z})^{\times }$$ and prove a similar result. For all suitably large primes $$q$$ we define $$P_\eta $$ to be the set of primes less than $$\eta q$$, viewed naturally as a subset of $$(\mathbb {Z}/ q\mathbb {Z})^{\times }$$. Considering the $$k$$-fold product set $$P_\eta ^{(k)}=\{p_1p_2\cdots p_k:p_i\in P_\eta \}$$, we show that, for $$\eta \gg q^{-{1}/{4}+\epsilon },$$ there exists a constant $$k$$ depending only on $$\epsilon $$ such that $$P_\eta ^{(k)}=(\mathbb {Z}/ q\mathbb {Z})^{\times }$$. Erdös conjectured that, for $$\eta = 1,$$ the value $$k=2$$ should suffice: although we have not been able to prove this conjecture, we do establish that $$P_1 ^{(2)}$$ has density at least $$\frac {1}{64}(1+o(1))$$. We also formulate a similar theorem in almost-primes, improving on existing results.

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