# Questions tagged [riemann-hypothesis]

Questions on the Riemann hypothesis, a conjecture on the behavior of the complex zeros of the Riemann $\zeta$ function. You might want to add the tag [riemann-zeta] to your question as well.

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### Bounds for the $n^{th}$-prime number and the inverse logarithmic integral?

Are there any bounds for the $n^{th}$-prime number $p(n)$ and the inverse logarithmic integral $\text{ali}(n)=\text{li}^{-1}(n)$ under the Riemann Hypothesis? $$|p(n)-\text{ali}(n)|<?$$ here is ...
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### If $\sum_{n=1}^{\infty} a_n n^h=\sum_{n=1}^{\infty}a_nn^{-h}=0$ for nonzero $a_i$, is there a way to show $h=0$ is the unique solution? [closed]

If I have $$\sum_{n=1}^{\infty} a_n n^h = 0\quad\mathrm{and}\quad\sum_{n=1}^{\infty} a_n n^{-h} = 0\\ \mathrm{with}\quad a_1,a_2,...\neq0\\ a_n = \frac{(-1)^{n}}{\sqrt{n}}\sin(t \ln(n))$$ Is there a ...
1answer
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### Is the value of the following limit with an infinite number of variables equal to zero or is it not decidable?

According to the alternating series test a general term $b_n$ in the sum: $\sum\limits_{n=1}^{\infty}(-1)^{n+1}b_n$ should converge to zero: $\lim\limits_{n \rightarrow \infty} b_n = 0$ From reuns ...
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### I am looking for class of math problems which are provable in ZF if and only if they are provable in ZFC

I know that P vs NP and Riemannian hypothesis are of this class but could not find any article on that. I would also appriciate links or books on related theme. My question is: what are some other ...
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### Does this disprove the Riemann Hypothesis?

I found interesting results when I put in certain values for z into zeta(z). I believe my findings do not disprove the Riemann hypothesis, but I am not so sure. When I input z = 1/3 + i into the ...
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### What happens if you compose the zeta function with two circle inversion mappings in this specific way?

Let $I^a_b : \mathbb{C} \to \mathbb{C}$ be defined as circle inversion mapping of a circle at the point $a \in \mathbb{C}$ of radius $b = \frac{1}{r}$ What happens to the zero's of the zeta function ...
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### Has anyone had studied the Riemann hypothesis with non-standard models?

In symbolic logic, non-standard numbers is common nomenclature for a model of numbers which satisfies the axioms of some arithmetical system, but doesn't consist of what people would normally think of ...
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### How many counterexamples the Riemann Hypothesis, if false, can have?

Are there papers, which show, that there can not be infinite many counter-examples for the Riemann Hypothesis? Can you give me references for these papers? I enjoy thinking and learning. I mean no ...
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### Bounding summation

Let $T$ be a sufficiently large real number and $\gamma$ be imaginary part of non-trivial zeros of Riemann zeta function. I would like to bound the sum like $$\sum_{0<\gamma\leq T}\gamma^{-1/4}.$$ ...
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### Inequality for divisor sigma $\sigma_{\nu}(n)$ function.

The divisor sigma function is defined as $$\sigma_{\nu}(n):=\sum_{d|n}d^{\nu}\textrm{, }n\in\textbf{N}$$ It is known that under Riemann's hypothesis it holds the following inequality due to ...
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### Are there any known rigid properties of the counterexample zeroes to RH assuming the failure of RH?

One often suggested "route" for the Riemann Hypothesis is through the study of quasicrystals. In particular, assuming RH, the nontrivial zeroes form a 1-dimensional quasicrystal. If we can ...
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### Proving Lindelof hypothesis using Phragmen-Lindelof theorem and Riemann Hypothesis

Apparently, assuming the Riemann Hypothesis to be true, we could use Phragmen-Lindelof theorem to deduce the Lindelof Hypothesis. I searched for a while but could not find a proof. Could anyone share ...