Questions about the famous conjecture from Riemann saying that the non-trivial zeroes of the Riemann Zeta function all lie on the so-called critical line $\Re(s)=\dfrac{1}{2}$, its various generalizations and the different approaches towards its solution.

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6
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182 views

Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$

If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log ...
6
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221 views

Are all complex zeros of $\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$ on the critical line for all $z \lt 1$?

Numerical evidence suggests that all complex zeros residing in the critical strip $0 < \Re(s) < 1$ of: $$\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$$ are on the ...
6
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890 views

Questions on de Branges' work on the Riemann hypothesis

According to Wikipedia, Louis de Branges de Bourcia has obtained some notable results, such as a proof of the Bieberbach conjecture in 1985, which is now known as de Branges' theorem. Initially, his ...
3
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267 views

Definite integral of $\zeta(s)$ over the critical strip

Take the following definite integral: $$f(s):=\int_s^{1-s} \zeta(x) \mathrm{d} x$$ with $s \in \mathbb{C}$, $s=\sigma \pm ti$, $0<\sigma<1$ and $t,\sigma \in \mathbb{R}$. The graph of ...
2
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131 views

The influence of $\chi(s)$ on complex zeros of $\frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$

I was exploring the formula: $$g(s)_{\pm} := \displaystyle \frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$$ and found that for all $\Re(s) \ne \frac12$: ...
2
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0answers
330 views

Characterizing essential singularities

In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ...
1
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0answers
114 views

Except for a finite few outside the strip, do all complex zeros of $\zeta(a+s)\pm \zeta(a+1-s)$ reside on the critical line for all $a\lt 0$?

Assume $a \in \mathbb{R}$ and $s \in \mathbb{C}$. Numerical evidence suggests that all complex zeros, except for a finite few outside the strip, of: $$\zeta(a+s)\pm \zeta(a+1-s)$$ lie on the line ...
1
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0answers
80 views

Are all complex zeros of $Li_s(i)\, + \, Li_{1-\overline{s}}\,(-i)$ equal to the $\rho$'s?

Take the well known square relationship for polylogarithms: $$Li_s(z)\, + \, Li_{s}(-z)=2^{1-s}Li_s(z^2)$$ Assume $z=i$: $$Li_s(i)\, + \, Li_{s}(-i)=2^{1-s}Li_s(-1)=-2^{1-s}\,\eta(s)$$ with ...
1
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0answers
71 views

Are all complex zeros of $Li_s(z)\, \pm \, Li_{1-s}(z)$ on the critical line or outside the critical strip for $z \le -1$?

This question loosely builds on this one, however is a bit simpler and I found the results to be more robust. It seems that all zeros in the critical strip $0 \lt \Re(s) < 1$ of: $$Li_s(z)\, \pm ...
0
votes
0answers
142 views

Is RH equivalent to the following estimate?

This question is a follow up from About Goldbach's conjecture and comes from what I read about the Farey series related criterion for RH. Let $r_{k}(n)$ be the $k+1$-th potential typical primality ...
0
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0answers
221 views

Does there exist a Weierstrass/Hadamard factorization for $\chi(s)-1$ ?

Would like to build once more on this question. Take $s=\sigma + ti, s \in \mathbb{C}, 0<\Re(\sigma)<1$. Let's assume it is proven that: $$\zeta(1-s) - \zeta(s)$$ has all its zeros on the ...