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Let $F(x) = \sum_{1 < n \leq x} (-1)^{\pi (n)}$ where $\pi (n) $ is the prime counting function. I am trying to understand how $ F(x) $ behaves as $ x \to \infty$. In particular, what are the asympt …

An open problem in analytic number theory is to determine whether or not the following infinite series converges: $\sum_{n=2}^{\infty} \frac {(-1)^{\pi(n)}}{n log n}$ Here, $\pi(n)$ is the prime cou …

Euclid proved that there are infinitely many primes via a clever algebraic argument that most of us are familiar with. Euler proved that $\lim_{s \to 1^+} \sum_p \frac{1}{p^s} = +\infty$ which gives …

One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to k …

During his investigation of zeta Riemann defined the $\xi$ function as $\xi(s):= \Gamma(\frac{s}{2})(s-1)\pi^{-s/2}\zeta(s)$ which is an entire function that is invariant under the substitution $s \to …

I am investigating whether or not there exist $\epsilon > 0$ such that $\zeta(s) \neq 0$ on the strip $1-\epsilon < \Re(s) \leq 1$. Suppose not. Then given $\delta > 0$ there exists a zero of zeta $\ …

It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ …

So what happens if there is a non-trivial zero of the Riemann zeta function off the critical line? Has there been any work in the following direction: We know from Landaus theorem that there is a po …

Erdős asked1 whether the series $$\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}$$ converges. Here, $p_n$ denotes the n-th prime. I can show that this series converges simultaneously with the series $\sum_{ …

Is there a heuristic argument behind the exponent in the circle problem? The problem that I am referring to is the following: Consider a circle of radius $R$ centered at the origin in the plane and …

Most (if not all) of the proofs of the Prime Number Theorem that I have seen in the literature rely on the fact that the Riemann zeta function, $\zeta(s)$, does not vanish on the line ${\rm Re}(s) = 1 …

I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers. Let: $\ \ E_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 …

Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi (z …

Can the prime number theorem be obtained from the explicit formula, $\psi(x)=x-\sum_{\zeta(\rho)=0}\frac{x^\rho}{\rho}+O(1)$? Here, $\psi(x)=\sum_{k=1}^\infty\sum_{p^k<x}\log p$

Is there a heuristic argument that supports the validity of the Riemann hypothesis or are we just relying on numerical evidence? Moreover, what is the strongest theorem that supports the validity of …