sum (1/n^s), negative values of

mart_r · 2019-03-20, 16:34

On the wing of an interest in the zeta function, I've been working on this conundrum in the past two days:

Find the largest $\delta$ such that there exists x and $\tau$ such that
$\sum_{n=1}^x \frac{1}{n^{1+\delta+i \tau}}<0$

The idea behind it is while $\zeta(\sigma+i\tau)$ is >0 for $\sigma$ >1, the intermediate summation terms can still be <0. What is the largest $\sigma$ for this to happen?

(My state of knowledge is that [TEX]\delta[/TEX] > 1/8 if I did the math right.)

Dr Sardonicus · 2019-03-21, 19:04

Quote:

Originally Posted by mart_r

On the wing of an interest in the zeta function, I've been working on this conundrum in the past two days:

Find the largest $\delta$ such that there exists x and $\tau$ such that
$\sum_{n=1}^x \frac{1}{n^{1+\delta+i \tau}}<0$

The idea behind it is while $\zeta(\sigma+i\tau)$ is >0 for $\sigma$ >1, the intermediate summation terms can still be <0. What is the largest $\sigma$ for this to happen?

(My state of knowledge is that [TEX]\delta[/TEX] > 1/8 if I did the math right.)

The sums you indicate are likely to be non-real. Perhaps you mean the real part?

The following might be pertiment: A NOTE ON THE REAL PART OF THE RIEMANN ZETA-FUNCTION

mart_r · 2019-03-21, 22:03

Quote:

Originally Posted by Dr Sardonicus

The sums you indicate are likely to be non-real. Perhaps you mean the real part?

The following might be pertiment: A NOTE ON THE REAL PART OF THE RIEMANN ZETA-FUNCTION

That looks just like the thing I was looking for.
So $\zeta$ ( $\sigma$ +i $\tau$ ) can be <0 when $\sigma$ >1 - must have gotten out of my sight.

Thanks a lot!

2019-03-20, 16:34	#1
mart_r Dec 2008 you know...around... 297₁₆ Posts	sum (1/n^s), negative values of On the wing of an interest in the zeta function, I've been working on this conundrum in the past two days: Find the largest $\delta$ such that there exists x and $\tau$ such that $\sum_{n=1}^x \frac{1}{n^{1+\delta+i \tau}}<0$ The idea behind it is while $\zeta(\sigma+i\tau)$ is >0 for $\sigma$ >1, the intermediate summation terms can still be <0. What is the largest $\sigma$ for this to happen? (My state of knowledge is that [TEX]\delta[/TEX] > 1/8 if I did the math right.) Last fiddled with by mart_r on 2019-03-20 at 16:42

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