Search Results

Consider the function $$\vartheta(x;q,a)=\sum_{p \leq x ,q|(p-a)}\log p=\frac{x}{\phi(q)}+O(\frac{x}{(\log x)^C}).$$ If the Riemann Hypothesis is ture, in the case $q=a=1$ we have $$|\psi(x)-x|<\frac{ …

Suppose $x>0$ and let $f(x)=\sum_{k\le x}\frac{1}{\varphi(k)}$, where $\varphi(k)$ is the Euler totient function. It is well known that $\sum_{k\le x}\frac{1}{k}\sim\log x$. What is the asymptotic beh …

Consider the function $$f(x)=-\frac{x}{2}+\sum_{p\le x}\log(p)-\frac{1}{x}\sum_{p\le x}p\cdot \log(p).$$ In my recent work, I need to get an explicit [rather than asymptotic] upper bound of this funct …

Let $a,b$ be positive integers, $F=\mathbb{Q}(a^{1/3},b^{1/2})$. Let $E$ be the elliptic curve defined over $F$ by the cubic equation $$y^2=x^3+3a^{1/3}x+2b^{1/2}.$$ Then the $j$-invariant $j(E) = \fr …

Let $a,b,c$ be coprime nonzero positive integers such that $a+b=c$. The ABC conjecture states that for any $\varepsilon>0$, we have $$c < C_{\varepsilon}\operatorname{rad}(abc)^{1+\varepsilon}.$$ Ther …

It is well-known that the Mertens function $M(n)=\sum_{k=1}^n\mu(k)$ changes sign infinitely many times when $n\rightarrow +\infty$. Let $f(n)=\sum_{k=1}^n\frac{\mu(k)}{k}$, then $\lim\limits_{n\right …

Consider the tensor product of finite extensions of a field $F$ of characteristic zero. (I am interested in the case $F=\mathbb{Q}_p$.) $(1)$ If $M$ is a finite Galois extension of $F$ with Galois gro …

Let $p\in\{11,17,19,37,43,67,163\}$ be a prime number. In [1], B. Mazur proves that there are only finite number of elliptic curves $E$ [over $\mathbb{Q}$] having an isogeny of degree $p$. Here is my …