On the r-th root partition function, II

Text For any positive real number r, let p r ( n ) be the number of solutions of the equation n = ⌊ a 1 r ⌋ + ⋯ + ⌊ a k r ⌋ with integers 1 ≤ a 1 ≤ ⋯ ≤ a k . Recently, Luca and Ralaivaosaona gave an asymptotic formula for p 2 ( n ) . In Part I, it is proved that, for any real number r > 1 , we have exp ⁡ ( τ 1 n r / ( r + 1 ) ) ≤ p r ( n ) ≤ exp ⁡ ( τ 2 n r / ( r + 1 ) ) for two positive constants τ 1 and τ 2 (depending only on r). In this paper, we prove that, for any real number r > 1 , p r ( n ) = exp ⁡ ( c 1 n r / ( r + 1 ) + c 2 n ( r − 1 ) / ( r + 1 ) + ⋯ + c l n ( r − l + 1 ) / ( r + 1 ) + O ( n 1 / ( r + 1 ) ) ) , where l is the integer with l r ≤ l + 1 and c 1 , c 2 , … , c l are computable constants depending only on r. In particular, c 1 = ( 1 + r − 1 ) ( r ζ ( r + 1 ) Γ ( r + 1 ) ) 1 / ( r + 1 ) . For 0 r 1 , we pose a conjecture for the asymptotic formula for p r ( n ) . Video For a video summary of this paper, please visit https://youtu.be/Z9HiSbr9eJY .