Tagged Questions

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Proof that at least one of the nontrivial zeta zeroes has an irrational height (assuming RH) [on hold]

This seems quite simple so its likely someone has done this before (a few Google searches returned empty and I would be really grateful for a relevant link), but in case it's new, I wanted to check if ...
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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 ...
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Is it possible to show that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?

Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation $$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}.$$ Basic theorems about Dirichlet series ...
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Derivative of Riemann Zeta at nontrivial zeros

I would like to know whether the real part of the first derivative of the Zeta function at the non trivial zeros of Zeta is stricly positive and if so, is there a proof for it. Also, are there tables ...
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How “deep” is the unboundedness of the reciprocal of the Riemann zeta function on vertical lines in the critical strip?

I think it is probably well known that, for every $1/2<\sigma\leq 1$, the function $1/\zeta(\sigma+it)$ is unbounded. Yet, I cannot decide how deep this is. I imagine it could be proved using a ...
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expressing $\log(\left \lfloor x \right \rfloor!)$ in terms of zeta-zeros

referring to a question i posted on MS, I post it here, as I didn't get an answer: let $\psi(x)$ be the second Chebyshev Function. By the definition of this summatory function, and the fundamental ...
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$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function. This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$. Its convergence is unknown if $1/2< ... 2answers 570 views On extended Riemann Hypothesis and coefficients of Selberg Class L-functions There is the conjecture that Selberg Class L-functions satisfy RH. So that an L-function needs to have its coefficient multiplicatives (plus other conditions: functional equation,...) in order to ... 2answers 303 views What is known about the set$S$of couples of rationals$(q,q')$such that$\zeta(q+iq')$is rational? The question is the title. For example, if we could show that$S$is finite, then this would entail that every large enough integer$n$is such that$\zeta(2n+1)$is irrational and that, under RH, ... 0answers 81 views Closed forms for paired factors of a 4-factor infinite product. Is their shape fixed? As a follow up on this question, I have now composed the following model of closed forms for infinite products of pairs of factors. At the heart is a 4-factor infinite product shown in red at the ... 1answer 206 views Do we know an upper bound for the multiplicity of the non-trivial zeros of Zeta? In Are the non trivial zeros of Zeta simple?, I asked whether it was known that all non-trivial zeros of the Riemann Zeta function were simple or not. It appears that such a proof is missing. But are ... 0answers 198 views Are these valid expansions of the Riemann$\xi(s)$function in the Hadamard product? In this post I derived for$s=a + ti$, that assuming the RH, the following should be true: $$\displaystyle \frac{\xi(\frac12 - a + s)}{\xi(\frac12 - a)} = \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} ... 0answers 201 views Does the difference of two converging infinite series correctly induce the non-trivial zeros of \zeta(s)? The following analytic continuation for \zeta(s) towards \Re(s)>-1 derived here:$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(s+1+ \sum _{n=1}^{\infty } \left( {\frac ... 1answer 311 views 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 ... 0answers 233 views Definite integral probably equal to zeta with known (but unusable) closed form for the indefinite integral Related to this and this questions. Basically got definite integral that experimentally equals$\zeta(s)$both numerically and symbolically. Closed form for the indefinite integral is known, but I ... 3answers 1k views Does this infinite sum provide a new analytic continuation for$\zeta(s)$? It is well known that the infinite sum: $$\displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$ only converges for$\Re(s)>1$. The Dirichlet 'alternating' sum: $$\displaystyle \zeta(s) = ... 1answer 471 views Are the zeros of the sum/difference of these integrals all on the critical line? The answers given to the question whether all zeros in the critical strip of \zeta(s)\pm\zeta(1-s) lie on the critical line, suggest that this can indeed be proven, however only for those zeros ... 0answers 169 views Naive conjecture about zeros and local extrema of \Re \zeta(\sigma+i t) (resp. \Im \zeta(\sigma+ it)) for 0 \le \sigma \le \frac12 Based on limited numerical evidence, I suspect this conjecture. Conjecture: Fix 0 \le \sigma \le \frac12 and let t > 0. Between consecutive local extrema of \Re \zeta(\sigma+i t) (resp. ... 2answers 313 views Consequences of a bound on possible counterexamples to Riemann hypothesis The Riemann hypothesis has many strong consequences in number theory. The question is: would a bound on the number of zeros of Riemann zeta-function in the critical strip with real part not equal 1/2 ... 1answer 395 views Is there always a zero between consecutive local extrema of \Re \zeta(1/2+it) (or \Im \zeta(1/2+i t) Based on limited numerical evidence, I am inclined to suspect that there is always zero of \Re \zeta(1/2+it) between consecutive local extrema of \Re \zeta(1/2+it) (and the same for \Im ... 3answers 251 views Inequality for the modulus of Riemann zeta on horizontal lines and alleged partial result of Maple According to a conjecture p.4 |\zeta(\frac12 -\Delta + it))| > |\zeta(\frac12 + \Delta + i t| for 0 < \Delta < \frac12 and |t| > 2 \pi +1. Since \zeta(\overline{s}) = ... 1answer 243 views Finite sum seemingly related to nontrivial zeta zeros For t \in \mathbb{R} define$$ F(t) = \sum_{n=1}^{[t]} \frac{(-1)^{(n-1)}}{n^{\frac12 + it}}$$Let \operatorname{Arg}(t) be \operatorname{atan2}(\Im t , \Re t) - basically this is \arctan, ... 0answers 224 views Differential equation for zeta on the critical line Edit Major rewrite since Johan Andersson observed the original question is trivial because of vanishing of coefficients. From this question$$ \zeta(1-x) = ... 0answers 311 views $\zeta(x)$in terms of$\zeta'(x),\zeta'(1-x),\Gamma,\psi$By differentiating$\xi$and solving for$\zeta(1-x)$: $$\zeta(1-x) = \frac{2(\zeta'(x)\Gamma(x/2)+\Gamma((1-x)/2) \zeta'(1-x)\pi^{x-1/2}) )}{\Gamma((1-x)/2) \pi^{-1/2+x}(2\log\pi ... 0answers 199 views computing a certain contour integral [closed] I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ... 0answers 182 views Can \zeta(s) for \Re(s)>1 be split into two factors that each can be analytically continued? Assuming the RH and s \in \mathbb{C}, \rho_n =\frac12 \pm i\gamma_n, the following (altered) Hadamard product:$$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n i ... 0answers 196 views Complex zeros of$\zeta'(s)/\zeta(s) + \zeta'(1-s)/\zeta(1-s) $= simpler expression (except at zeta zeros) Let$ G(s) := \frac{\zeta'(s)}{\zeta(s)} + \frac{\zeta'(1-s)}{\zeta(1-s)}$where$s$is not a zero of zeta.$G$has real zeros and a pair of complex zeros near$\frac12 \pm 6i$. Partial results: By ... 1answer 375 views How is “large” defined in an equality for the modulus of Riemann zeta? This paper p.4 claims: Corollary C. Assume RH. For all large$t$we have $$|\zeta(\frac12 +it)| \le \exp\left(\frac38 \frac{\log{t}}{\log{\log{t}}}\right) \qquad (1)$$$t$a Gram points often ... 2answers 305 views On the critical line$ \Re \zeta'(s)/\zeta(s) =? 1/2 \log(\pi) - 1/2 \Re \psi(s/2)$? For$\Re s = 1/2$numerical evidence suggest: $$\Re \zeta'(s)/\zeta(s) = 1/2 \log(\pi) - 1/2 \Re \psi(s/2) \qquad (1)$$ How this was found. Consider the symmetrized zeta function$\zeta^*(x)= ...
Let $0 < k_1 < k_2 < k_3 < \cdots$ be all the zeros of the Riemann zeta function on the critical line: $$\zeta(\frac{1}{2} + i k_j) = 0$$ Let $f$ be the Fourier transform of the sum ...