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 [TEX]\delta[/TEX] such that there exists x and [TEX]\tau[/TEX] such that
[TEX]\sum_{n=1}^x \frac{1}{n^{1+\delta+i \tau}}<0[/TEX]


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


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

[QUOTE=mart_r;511235]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 [TEX]\delta[/TEX] such that there exists x and [TEX]\tau[/TEX] such that
[TEX]\sum_{n=1}^x \frac{1}{n^{1+\delta+i \tau}}<0[/TEX]


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


[SPOILER](My state of knowledge is that [TEX]\delta[/TEX] > 1/8 if I did the math right.)[/SPOILER][/QUOTE]The sums you indicate are likely to be non-real. Perhaps you mean the real part?

The following might be pertiment: [url=https://maths-people.anu.edu.au/~brent/pd/rpb246.pdf]A NOTE ON THE REAL PART OF THE RIEMANN ZETA-FUNCTION[/url]

[QUOTE=Dr Sardonicus;511348]The sums you indicate are likely to be non-real. Perhaps you mean the real part?

The following might be pertiment: [url=https://maths-people.anu.edu.au/~brent/pd/rpb246.pdf]A NOTE ON THE REAL PART OF THE RIEMANN ZETA-FUNCTION[/url][/QUOTE]

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

Thanks a lot!