Hoping it is not too famous an open problem, I would suggest trying to (dis)prove that Euler's constant $\gamma$, defined as $\displaystyle{\lim_{n\to\infty}H_{n}-\log n}$ where $H_{n}$ is the $n$-th ...

I just read on Google+ that the paper will be published in 2018 in a Japanese journal whose editor-in-chief is Mochizuki himself. See https://plus.google.com/+johncbaez999/posts/DWtbKSG9BWD

In France again, IHES (Institut des Hautes Etudes Scientifiques), maybe a kind of Princeton's IAS "à la française".

Taniyama, Shimura and later Weil conjectured around 60 years ago that the L-function of an elliptic curve arises from a modular form. This conjecture was known to entail the last Fermat theorem after ...

Woodin's omega conjecture. See this pdf

Fermat's conjecture that all numbers of the form $F_{n} : =2^{2^{n}}+1$ are prime. Euler proved that $641\mid F_{5}$ .

What matters here is not how to help him/her to fulfill his/her potential, but how to help him/her be happy in life. Being different is not easy to deal with, and often leads to loneliness or social ...

Marek Wolf, a Polish physicist, is the author of several articles about the distribution of prime numbers. He studied jumping champions and provided a heuristic formula refining Cramer's conjecture, ...

Maybe not exactly what you're looking for, but in his book "Merveilleux nombres premiers" (in French), Jean-Paul Delahaye mentions the so-called Mills' constant $A=1.30637788386...$ which fulfills, ...

Please notice a few differences between French and English. "Un nombre positif" is a non negative number, "supérieur à" is "greater than or equal to". In English 0 is not a natural number, while in ...

Only a partial answer as it is too long for a mere comment. There are known connections between non vanishing of L-functions and some versions of the Goldbach conjecture. Gautami Bhowmik showed that ...

A good principle would be to apply this piece of advice from Hermann Weyl : to understand well a mathematical object, determine and study the structure of its group of automorphisms.

In the French edition, it is said that the considered constant does not exceed $4(1+9\lambda_{1}+\lambda_{1}\lambda_{2}/(2-\lambda_{2})^2)$ where $\lambda_{1}>0$, $0\leq \lambda_{2}<2$ are such ...

This recent preprint may be of interest for you, as the authors first consider L-functions and then find back the algebraic variety they come from.

In number theory, the notation $\log$ is commonly used, especially when asymptotics are considered. One also frequently uses the notation $\log_{k}$ for the $k$ -th iterate of this function. ...

This is related to the fact that the circle has constant curvature. Indeed the curvature of a curve obtained plotting $y=f(x)$ is $\mathcal{C}_{f}(x)=f''(x)(1+(f'(x))^{2})^{-3/2}$, which comes ...

I'd like to have on a usb key a user friendly software that could parse a math article to check the proofs in it without having to learn how to use stuff like Coq and highlight the possible gaps. But ...

Definitely not a pure interplay between two subfields of math, but the omnipresence of quaternions in physics is stunning. I even figured out a few years ago that writing down the equivalent of Cauchy-...

I finally managed to find back the article I was talking about. Just click on the green link in the first message of the following link: link text

Characterizing a class of integers sharing some property $P$ by defining an arithmetic function taking a single value $k_{P}$ at those integers and then give an equivalent of this arithmetic function. ...

Any $n$ which is of the form $4P$ where $P$ is a perfect number is good: just consider as a set $\{a_{i}\}$ the set of its divisors. As the sum of the reciprocals thereof equals $2$, this guarantees ...

I'm not really sure this is a suitable answer to your question, but I'd like to have my recently published novel "Sahelios", in which a Japanese highschool student named Satori (Japanese female given ...

Only a partial answer for now, as it is too long for a comment. From my comment and Carlo Beenakker's answer, it suffices to consider the case where $n$ and $m$ have different radicals but the same ...

Maybe not exactly what you're looking for, but you may be interested in this preprint by Kaczorowski and Perelli: arXiv:1506.07630 where the authors study the links between several kinds of ...

The integral does not converge. See https://academic.oup.com/blms/article-abstract/31/4/424/277640?redirectedFrom=fulltext. On the other hand, a proof that this integral is less than $K.t^{\alpha}$ ...

See https://www.researchgate.net/publication/321187136_Pair_Correlation_of_Zeros_of_the_Real_and_Imaginary_Parts_of_the_Riemann_Zeta-Function where the authors investigate the behavior of the real ...

Grothendieck wrote a text entitled "Structure de la psyché" that might help shed a light on the issues you consider. As a former member of Bourbaki, it seems highly plausible that he tried doing so to ...

The French publisher Assimil is very good (and famous) to get conversational skills in many foreign languages. As for the mathematical words, you can use Wikipedia, that's how I got to know that the ...

RH holds if and only if the group of isometries of the complex plane that preserve globally the multiset of non-trivial zeroes of the Riemann Zeta function is isomorphic to $Z/2Z$ (otherwise, it would ...