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.