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For each non‐principal Dirichlet character  χ , let  n   χ   be the least  n  with  χ ( n )∉{0, 1}. For each prime  p , letχ   p    (  ⋅  )  =  (   .  p   )be the quadratic character modulo  p . In 1961, Erdős showed thatn    χ   ppossesses a finite mean value as  p  runs over the odd primes in increasing order. We show that as  q →∞, the average of  n   χ   over  all  non‐principal characters  χ  modulo  q  is ℓ( q )+ o (1), where ℓ( q ) denotes the least prime not dividing  q . Moreover, if one averages over all non‐principal characters of moduli at most  x , the average approaches (as  x →∞) the limiting value∑   ℓℓ   2∏   p  ⩽  ℓ    (  p  +  1  )    ≈  2.5350541804  ,where the sum is over primes ℓ and the product is over primes  p ⩽ℓ.  One can also view Erdős's theorem as giving the average size of the least non‐split prime in the quadratic field of conductor  p , where again  p  runs over odd primes. Similar results with the average taken over all quadratic fields were recently proved by the second author. In this paper, we prove a result of this type for cubic number fields: if one averages over all cubic fields  K , ordered by discriminant, then the mean value of the least rational prime that does not split completely in  K  is∑   rr  (  5   /   6  +  1   /   r  +  1   /    r   2    )    1  +  1   /   r  +  1   /    r   2∏   p  <  r1   /   6    1  +  1   /   p  +  1   /    p   2≈  2.1211027269  ,where the sum is over all primes  r . Similar average values are obtained for the least rational prime with other specified factorizations in  K , although some are conditional on the generalized Riemann hypothesis.

The average least character non‐residue and further variations on a theme of Erdős