Search Results

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 …

The answer seems to be no. It is hard to find direct confirmation, but every time their names are mentioned together it is made clear that there's no relationship. The Russian mathematician Askold …

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (an …

The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact: There a …

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 …

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 …

This is explained for example in Iwaniec & Kowalski's "Analytic Number Theory", as an standard application of Selberg's $\Lambda^2$ sieve. See chapter 6, Elementary sieve methods. In particular you ge …

It is an open problem, only known for $k=1,2$. Both the conjecture and the known cases are due to Schinzel. You can find a nice survey here: "On the third iterates of the φ- and σ-functions" H. Maie …

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 don't see the relation between the first sentence and the question, so perhaps I'm missing something, but the answer to the question is yes, regardless. Fourier expansion is a very important tool i …

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 …

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 …

No explicit values are known. For a recent reference, see "A still sharper region where $π(x) − \text{li}(x)$ is positive", by Y. Saouter, T.S. Trudgian and P. Demichel (2014). An extract from the la …

The Weil conjectures have to do with local zeta-function over finite fields. The Riemann zeta function and the multiple zeta functions are defined over $\mathbb{Q}$ So Weil conjectures are simply th …

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 …