# Tagged Questions

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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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### 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 ...
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Is there an L-function ($L_s=\sum_{n =1}^{\infty} \frac{a_n}{n^s}$) having a functional equation coming from a relation of the type [1]: $\sum_{n =1}^{\infty} a_n \; e^{-2\pi nx}= \frac{A}{x^k} ... 1answer 720 views ### Riemann hypothesis and Kakeya needle problem The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ... 1answer 2k views ### What if the Riemann Hypothesis were false? There are lots of known and interesting consequences of the Riemann Hypothesis being true. Are there any known and interesting consequences of the Riemann Hypothesis being false? 1answer 923 views ### A reformulation of the Riemann Hypothesis I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates$\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where$N$is product of distinct primes. Let's define ... 5answers 4k views ### Quasicrystals and the Riemann Hypothesis 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 ... 1answer 364 views ### Heuristic for Montgomery's conjecture This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is believed to be true. ... 2answers 680 views ### Effective Chebotarev without Artin's conjecture Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Artin's$L$-function and ... 2answers 480 views ### Best bounds toward Serre's uniformity conjecture If$E$is a non-CM elliptic curve over$Q$, then it is a famous theorem of Serre that there is some integer$M(E)$such that for any prime$\ell > M(E)$, the image of the Galois representations ... 4answers 2k views ### Are there refuted analogues of the Riemann hypothesis? The classical Riemann Hypothesis has famous analogues for function fields and finite fields which have been proved. It has by now very many analogues, many of them still open. Are there important ... 2answers 331 views ### Mertens function limits using$\phi+2$n As we can see in the plot below, Mertens function:$M(x)\equiv \sum_{n=1}^{x}\mu(n)$has wild swings from positive to negative and back again. When we use: $$x=\frac{1}{2+\frac{1}{\phi+2}}\text{, ... 4answers 1k views ### Good uses of Siegel zeros? The short version of my question goes: What is known to follow from the existence of Siegel zeros? A longer version to give an idea of what I have in mind: The "expectional zeros" of course first ... 1answer 447 views ### Is there a connection between the closed forms of these two infinite products? Take the following two infinite products that have closed forms. Assume: \gamma_n > 0 \in \mathbb{R};s,a,x \in \mathbb{C}; x \ne 0,a \pm ix\gamma_n \ne 0 The first product:$$\displaystyle ... 3answers 562 views ### A closed form of infinite products of complex zeros involving$\Im(\rho_n)$. Does a proof of this closed form imply RH? Building on this question scaling the imaginary part of$\rho$s in infinite products, I like to conjecture that: $$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu_n} \right) \left(1- ... 1answer 894 views ### Is there a Montgomery's conjecture for Dirichlet characters and Artin representations ? Edit: as GH noticed, the way I tried to state Montgomery's conjecture is wrong. There were some mistakes in the references I used, which compounded with some mistakes of mine, gave a very poor post. ... 4answers 443 views ### What happens to \zeta(s) when all its \Im(\rho_n) are “scaled” linearly? I found that the following infinite product with \mu = a +n b i and a,b real, s \in \mathbb{C}:$$\displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\mu} \right) \left(1- \frac{s}{1-\mu} ... 1answer 300 views ###$\zeta(2k+1)$expressed in a product of two infinite products of non-trivial zeros. Take the Hadamard product for$\zeta(s)$: $$\displaystyle \zeta(s) = \pi^{\frac{s}{2}} \dfrac{\prod_\rho \left(1- \frac{s}{\rho} \right) \left(1- \frac{s}{1-\rho} ... 0answers 119 views ### The influence of \chi(s) on complex zeros of \frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})} I was exploring the formula:$$g(s)_{\pm} := \displaystyle \frac{\zeta(\overline{s})}{\zeta(1-s)} \pm \frac{\zeta(s)}{\zeta(1-\overline{s})}$$and found that for all \Re(s) \ne \frac12: ... 0answers 217 views ### Does there exist a Weierstrass/Hadamard factorization for \chi(s)-1 ? Would like to build once more on this question. Take s=\sigma + ti, s \in \mathbb{C}, 0<\Re(\sigma)<1. Let's assume it is proven that:$$\zeta(1-s) - \zeta(s)$$has all its zeros on the ... 1answer 649 views ### Interplay between Riemann and Swinnerton-Dyer Hello everyone, After reading RH ( Riemann's Hypothesis ) and Swinnerton-Dyer conjecture, I asked myself why can't RH hold for L-Functions ( Hasse-Weil L-function ). In particular the GRH imposes ... 1answer 221 views ### Deriving the Riemann non-trivial zeros from \zeta_{H}(s,a) + \zeta_{H}(s,1-a) The Hurwitz zeta function:$$\zeta_{H}(s,a)$$reduces to \zeta(s) when a=1 and to (2^s-1)\zeta(s) when a=\frac12. However, I stumbled upon a peculiar third connection:$$\zeta_{H}(s,a) + ... 1answer 838 views ### What happens when infinite values of$\zeta_{H}(s,z)$approach$\zeta(s)$? Take the following Hurwitz zeta: $$\zeta_{H}(s,z)$$ with$s=\sigma \pm ti$and$\displaystyle z=1 \pm \frac{i}{a}$and$t,a \in \mathbb{R}$. In the critical strip$0 \lt \sigma \lt 1$, this Hurwitz ... 0answers 266 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 ... 2answers 412 views ### Are all zeros of ζ^{k}(s)±ζ^{k}(1−s) on the critical line (k=k-th derivative)? The non-trivial zeros of$\zeta^{k}(s)$, with$k=k^{th}$derivative, do not lie on a line and seem to be distributed randomly in the region$\sigma > \frac12$. However the non-real zeros in the ... 1answer 463 views ### Zeros of the function$\zeta(s) \pm \zeta(\overline s)$Building on this question: Zeros of$\zeta(s) \pm \zeta(1-s)$, I experimented further with: $$\zeta(s) \pm \zeta(\overline s)$$ Assuming$s=\sigma + ti$, I observed that this function also has many ... 5answers 1k views ### Are the 'semi' trivial zeros of$\zeta(s) \pm \zeta(1-s)$all on the critical line? The proof that$\Gamma(z)\pm \Gamma(1-z)$only has zeros for$z \in \mathbb{R}$or$z= \frac12 +i \mathbb{R}$has been given here: Are all zeros of$\Gamma(s) \pm \Gamma(1-s)$on a line with real ... 1answer 549 views ### Is there information about the$\rho$'s hidden in the zeros of$\Re(\chi(s))$? Take the symmetrical form of the completed Zeta-function:$\displaystyle \chi(s) \zeta(s) = \chi(1-s) \zeta(1-s)$with$\chi(s)=\pi^{-(\frac{s}{2})} \Gamma(\frac{s}{2})$. For$s=\sigma + ti$, I ... 3answers 554 views ### Possible locations for non trivial zeroes lying off the critical line It has been proven that: 1) if$s$is a non trivial zero$\rho$of$\zeta(s)$then so is$1−s$. 2)$\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$3)$ 0 < \Re(\rho) ...
$$\sigma(n) < e^\gamma n \log \log n$$ In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin ...