All Questions
7 questions
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Function uniquely determined by its values at integer arguments
A smooth enough, slow growing real-valued function $f(x)$, is uniquely determined by its values at integer arguments. I don't remember the name of the theorem and the conditions for this to be true. ...
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Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?
In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...
- 5,470
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Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?
This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO.
For every ...
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Questions on Riemann's explicit formula
If we consider this version of the prime-counting function
$$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$
(with $\pi$ being the normal prime-counting function), then we can write $\...
- 749
41
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6
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"Long-standing conjectures in analysis ... often turn out to be false"
The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1
His example of a "long-standing conjecture" is the Riemann hypothesis,...
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Convolution integral of series involving the non-trivial zeros of $\zeta(s)$
Let us consider the convolution $$f\left(x\right):=\int_{2}^{x-2}\sum_{\rho_{1}}\frac{u^{\rho_{1}}}{\rho_{1}}\sum_{\rho_{2}}\frac{\left(x-u\right)^{\rho_{2}}}{\rho_{2}}du,\,x>4$$ where $\rho_{i},\,...
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3
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Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros
Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that $$I=\sum_{...
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