An upper bound for the[inline-graphic not available: see fulltext] norm of a GCD-related matrix

We find an upper bound for the[inline-graphic not available: see fulltext] norm of the[inline-graphic not available: see fulltext] matrix whose[inline-graphic not available: see fulltext] entry is[inline-graphic not available: see fulltext], where[inline-graphic not available: see fulltext] and[inline-graphic not available: see fulltext] are the greatest common divisor and the least common multiple of[inline-graphic not available: see fulltext] and[inline-graphic not available: see fulltext] and where[inline-graphic not available: see fulltext] and[inline-graphic not available: see fulltext] are real numbers. In fact, we show that if[inline-graphic not available: see fulltext] and[inline-graphic not available: see fulltext], then[inline-graphic not available: see fulltext] for all positive integers[inline-graphic not available: see fulltext], where[inline-graphic not available: see fulltext] is the Riemann zeta function.