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Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...

[I'm going to ask this question very carefully as a question similar to this received a critical response on this platform. I myself am very skeptical about this but I want to know, from the experts' ...

Consider the following function: $$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$ Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant. The following three conditions ...

Given a positive integer $n$, consider the diophantine equation $4x^2+y^2+4x+y=2n$ with solutions in non-negative integers $x$ and $y$. Define the proportion $$\delta_n=\frac{\#\{(x,y)\in\mathbb{Z}^2_{...

Consider the following function : $$F(z) = \omega(z){\sin^2\left(\frac{c\Gamma(z)}{z}\right)}$$ Here, $\omega(z)$ is a weight we are going to consider The following two conditions should meet for $\...

Consider the function $F(x)$ defined in following manner: $F(n)$ is finite (likely $F(x)\in[0,1]$) if $n$ is prime and zero otherwise: It has to satisfy following conditions: (1) $F(x)$ is ...