All Questions
9 questions
1
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0
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181
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Behavior of Dirichlet L-functions at the edge of the critical strip
Given a Dirichlet L-function $L(\chi, s)$ of a primitive character $\chi$, what is the asymptotic behavior of $L(\chi, 1+it)$ for real $t$? I am looking for as many answers for the same question. This ...
- 565
2
votes
1
answer
757
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Does asymptotic Goldbach imply GRH?
It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/...
- 6,984
5
votes
1
answer
2k
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Does $ M(x)=O(\sqrt{x}) $ if and only if the De Bruijn-Newman constant is negative?
The Riemann hypothesis is equivalent to the assertion that the De Bruijn-Newman constant $ \Lambda $ , as defined in https://www.sciencedirect.com/science/article/pii/S0001870809001133/pdf?md5=...
- 6,984
10
votes
1
answer
1k
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How does Riemann hypothesis implies estimates?
In Iwaniec, Luo and Sarnak article (precisely (4.23)), it is said that GRH for $L(s, \mathrm{sym}^2(f))$, for a holomorphic cusp newform $f$ of level $N$ and weight $k$, implies
$$\sum_{p \nmid N} \...
- 551
0
votes
0
answers
112
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Explicit formula for k-central numbers
Given a positive integer $ n $ and assuming Goldbach's conjecture, let $r_{0}(n)$ denote the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are primes. Let $k_{0}(n)$ denote 'the ...
- 6,984
2
votes
0
answers
270
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On a sequence of L-functions having same zeros in critical strip and GRH
I had an idea on GRH involving a sequence of L-functions having same zeros, then at one step I need a bound on these function and I wonder if this bound is in fact not as hard as GRH itself ?
Let's ...
- 1,199
3
votes
1
answer
426
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On link between Riemann hypothesis and partial GRH
Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...
- 1,199
12
votes
1
answer
969
views
Montgomery's pair correlation function without RH?
In the theory of the Riemann zeta function, Montgomery's Pair correlation function is defined as
$$
F(\alpha) = \frac{1}{N(T)}
\sum_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} \...
- 123
38
votes
4
answers
6k
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Modular forms and the Riemann Hypothesis
Is there any statement directly about modular forms that is equivalent to the Riemann Hypothesis for L-functions?
What I'm thinking of is this: under the Mellin transform, the Riemann zeta function $...
- 889