Moments of the Riemann -function were introduced as a
tool to attack the
Lindelöff Hypothesis, which asserts that
for any .
This is equivalent to
The above bound has only been established for and ,
and for those values an asymptotic formula is known for
the more general integral
When or , the function can be continued to a meromorphic function. But if is larger, then is (conjecturally) not continuable to a meromorphic function, and in fact it has a natural boundary. Diaconu, Goldfeld, and Hoffstein have shown that the function continues to a sufficiently large region that standard conjectures for the moments of -functions should be able to be recovered from the polar divisors of . However, the fact that the function under consideration is not entire suggests that it may not be the correct object to study.
Problem: Find a natural way to modify so that it becomes an entire (meromorphic) function.
The fact that
is not a nice function for
is an aspect of the ``Estermann Phenomonon.'' Consider the
An example which may be more closely related to the situation at hand is
See Titchmarsh [ MR 88c:11049] for more background information.
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