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As explained by Peter Humphries in the comments, the famous conjecture here is that the imaginary part of the non-trivial zeros of $\zeta(s)$ is transcendental (or equivalently, linearly independent o …

We have an elementary sharp lower bound for the regulator of a real quadratic field as a function of the discriminant $$R\geq \tfrac{1}{2}(\sqrt{d-4}+\sqrt{d})$$ It is sharp because the equality hol …

This is the status as far as I know. For dimension $\leq 2$ it is up to date. For higher dimensional representations I'm sure it is very incomplete, so feel free to edit or comment. Dimension 1. Known …

I'm aware of Shao-Ji Feng's result that Karatsuba's weaker conjecture ("conjecture 1") is true conditionally on the Lindelöf Hypothesis. Shao-Ji Feng, "On Karatsuba conjecture and the Lindelöf hypot …

I'll stay clear from the infamous $B_L'$ notation. Legendre's original (correct) statement is that $$\pi(x)=\frac{Bx}{\log x-A+o(1)}$$ where the so-called "Legendre's constant" is $A$. The incorre …

As Carlo Beenakker mentions in the comments, \begin{equation*} \begin{split} -\sum_{\beta'=1/2} \mathrm{Re} \left(\frac{1}{1/2+it-\rho'}\right)&=- \mathrm{Re}\left( \frac{1}{1/2+it-(1/2+it')}\right)\ …

Basically, the coefficients of an holomorphic cusp form are related to the number of points on a certain smooth projective variety over $\mathbb{F}_p$, and the Weil-Riemann hypothesis gives the necces …

The main reference here is the very useful User's Guide, C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, "User's Guide to PARI / GP" (2003) Particularly the sections about bnrstark (pp. 1 …

To be fair, the functional equation of $\zeta(s)$ is not really $\Lambda\left(1 - s\right)=\Lambda\left( s\right)$, but $$\zeta(1-s)=\frac{2}{(2\pi)^s}\Gamma(s)\cos \left(\frac{\pi s}{2}\right) \zeta …

The conditions to impose to the Euler product of an L-function in order for a generalized Riemann hypothesis to hold are well understood after Selberg's 1992 paper. In particular, given $L(s)=\prod_p …

In any event, this is essentially Hecke's original argument, the only other truly independent known proof being that of Tate. The original paper (in german) is, I believe, Erich Hecke "Über die Zeta …

They prove it in section 14 ("An Application") of their paper John Friedlander & Henryk Iwaniec, "The Illusory Sieve" (2005) As pointed out by Lucia in the comments, the result is conditional on t …

In this paper, van der Geer and Zagier come across a concrete Hilbert cusp form of weight $6$ on $\mathrm{SL}_2(\mathcal{O})$, namely $$f=-16(x+x^{-1})^3 (q+(27x^3-39x-39x^{-1}+27x^{-3})q^2+(285x^6-7 …

Iwaniec's original proof is avaible online Henryk Iwaniec, "Almost-Primes Represented by Quadratic Polynomials" (1978) In the same paper he also proves the following lower bound for the density of …

The analogy between zeta functions and trace formulas goes at least to Selberg, when he proved his famous trace formula for hyperbolic surfaces and the result turned out to resemble Weil's generalizat …