Date: Mar 10, 2013 5:20 AM
Author: Hisanobu Shinya
Subject: Lindelof hypothesis for Hurwitz zeta-function with complex arguments

Dear Professors,

Hello.

I recentely came up with a result that the Lindelof
hypothesis (the well-known conjecture in analytic
number theory) is true for the Hurwitz zeta-function
zeta(s, a) = sum_{n \geq 1} (n + a)^(-s) with arg a > 0,
as Im(s) -> -oo.

Note that it does not make sense
to speak of the Lindelof hypothesis for zeta(s, a)
if, say arg a > 0 and Im(s) -> +oo; to see this,
for instance, consider one term (n + a)^(-a - it),
with t -> oo.

It is easy to see that

(n + a)^(-a - it)
= e^[(-a - it)log(n + a)]
= e^[(-a - it)( log|n + a| + i arg(n + a))]

But by the definition of the number a, we have
arg(n + a) > 0, and so the number

(-it)(i arg(n + a)) = t arg(n + a)

becomes positive. Thus (n + a)^(-a - it) has
the exponential magnitude as t goes to positive infinity.

Now, my question for you would be whether this result
is new or not. Considering the simplicity of the proof,
I have been a bit doubtful of its originality, but
well, I do not find it in Wikipedia or MathWorld.

If you would like to read the proof for this,
please visit the home page of the Journal of
Mathematics Research, and find an article titled
"On a Formula of Mellin and Its Application to the
Study of the Riemann Zeta-function", found in
Vol. 4, No. 6.

The URL for Table of Contents of that issue is

http://www.ccsenet.org/journal/index.php/jmr/issue/view/696

I am looking forward to your reply.

With regards,

H. Shinya

PS. Although the article claims the proof of
the Lindelof hypothesis for the Riemann zeta-function,
please ignore the part related to it; it contains
a fatal error.

An erratum with another method to avoid the error will
appear later, but whether you read or not is not of my
interest for now.

The journal is the so-called pay-to-publish one.
But, well, my first attitude toward this journal
was rather literally pay-to-publish, and since I
am an amateur who has no authority to criticize
the quality of journals, please choose not to
write about this issue.