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\vskip 1cm{\LARGE\bf A Function Related to the \\
\vskip .1in
Rumor Sequence Conjecture}
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\large
Bruce Dearden, Joel Iiams, and Jerry Metzger\\
University of North Dakota\\
Department of Mathematics\\
Witmer Hall Room 313\\
101 Cornell Street Stop 8376\\
Grand Forks, ND 58202-8376\\
USA\\
\href{mailto:bruce.dearden@und.edu}{\tt bruce.dearden@und.edu}\\
\href{mailto:joel.iiams@und.edu}{\tt joel.iiams@und.edu}\\
\href{mailto:jerry.metzger@und.edu}{\tt jerry.metzger@und.edu}\\
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\begin{abstract}
 For an integer $b\geq 2$ and for $x\in [0,1)$, define 
 $\rho_b(x) = \sum_{n=0}^{\infty} \frac{\{\mskip -5mu\{b^nx\}\mskip -5mu\}}{b^n}$,
 where $\{\mskip -5mu\{t\}\mskip -5mu\}$ denotes the fractional part of the real number $t$. 
A number of 
properties of $\rho_b$ are derived, and then a connection between $\rho_b$ and the rumor
conjecture is established.  To form a rumor sequence $\{z_n\}$, first select integers
$b\geq 2$ and $k\geq 1$. Then  
select an integer $z_0$, and for $n\geq 1$ let $z_n = bz_{n-1} \bmod{(n+k)}$, where the right 
side is the least non-negative residue of $bz_{n-1}$ modulo  $n+k$. The rumor sequence
conjecture asserts that all such rumor sequences are eventually $0$. A condition on
$\rho_b$ is shown to be equivalent to the rumor conjecture. 
\end{abstract}

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