On the distribution of pseudoprimes

Source abstract

Let P ( x ) \mathcal {P}(x) denote the pseudoprime counting function. With $$L ( x ) = exp ⁡ { log ⁡ x log ⁡ log ⁡ log ⁡ x / log ⁡ log ⁡ x } , L(x) = \exp \{ \log x\log \log \log x/\log \log x\} ,$$ we prove P ( x ) ⩽ x ∙ L ( x ) − 1 / 2 \mathcal {P}(x) \leqslant x \bullet L{(x)^{ - 1/2}} for large x , an improvement on the 1956 work of Erdös. We conjecture that P ( x ) = x ∙ L ( x ) − 1 + o ( 1 ) \mathcal {P}(x) = x \bullet L{(x)^{ - 1 + o(1)}} .