  A private information retrieval scheme is a mechanism that allows a user to
retrieve any one out of $K$ messages from $N$ non-communicating replicated
databases, each of which stores all $K$ messages, without revealing anything
about the identity of the desired message index to any individual database. If
the size of each message is $L$ bits and the total download required by a PIR
scheme from all $N$ databases is $D$ bits, then $D$ is called the download cost
and the ratio $L/D$ is called an achievable rate. For fixed $K,N\in\mathbb{N}$,
the capacity of PIR, denoted by $C$, is the supremum of achievable rates over
all PIR schemes and over all message sizes, and was recently shown to be
$C=(1+1/N+1/N^2+\cdots+1/N^{K-1})^{-1}$. In this work, for arbitrary $K, N$, we
explore the minimum download cost $D_L$ across all PIR schemes (not restricted
to linear schemes) for arbitrary message lengths $L$ under arbitrary choices of
alphabet (not restricted to finite fields) for the message and download
symbols. If the same $M$-ary alphabet is used for the message and download
symbols, then we show that the optimal download cost in $M$-ary symbols is
$D_L=\lceil\frac{L}{C}\rceil$. If the message symbols are in $M$-ary alphabet
and the downloaded symbols are in $M'$-ary alphabet, then we show that the
optimal download cost in $M'$-ary symbols, $D_L\in\left\{\left\lceil
\frac{L'}{C}\right\rceil,\left\lceil \frac{L'}{C}\right\rceil-1,\left\lceil
\frac{L'}{C}\right\rceil-2\right\}$, where $L'= \lceil L \log_{M'} M\rceil$.
