# Mean values are not the natural object to study

Moments of the Riemann -function were introduced as a tool to attack the Lindelöff Hypothesis, which asserts that for any . This is equivalent to

for any , where the implied constant depends on .

The above bound has only been established for and , and for those values an asymptotic formula is known for the more general integral

where and is a function which descreases rapidly at .

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 Dirichlet series

where is the number of divisors of . We have

But if then can only be expressed as an infinite product of -functions, and it has a natural boundary at .

An example which may be more closely related to the situation at hand is

where . Again has a natural boundary at , but is is possible to modify the coefficients of , only involving the terms divisible by a cube, so that the function has an analytic continuation. This is promising because a partial sum of is the diagonal contribution to the integrand of .

See Titchmarsh [ MR 88c:11049] for more background information.

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