# Questions tagged [analytic-number-theory]

A beautiful blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

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### Advice: What topics to study now in analytic number Theory( And if there are video lectures( Open Online course) / Course notes available on website)

I am a person living in a 3rd world country and completed my masters in mathematics in July 2020. Then I began to study some additional topics in Pure Mathematics as I was applying for Ph.D. abroad( ...
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### Complexity of a Diophantine equation having $\leq1$ solutions

We are provided a single Diophantine equation $$f(x_1,\dots,x_n)=0$$ having degree $\geq2$ and having the promise it has $\leq1$ solutions in the set $\{0,\dots,m-1\}^n$ and $t$ is the number of terms ...
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### A question on assigning finite values to divergent sums involving expression of primes

We know the following: $$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$ This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$. ...
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### Integers having the same digits as their prime factors

Say an integer is "digitally conservative" in base $b$ if the set of its digits coincides with the set of the digits of its prime factors, like $37127=137\times 271$ in base $10$, and denote ...
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### Lower bounds for class number of function fields with fixed $q$, growing $g$

Let $X$ be a smooth project curve of genus $g$ over the finite field with $q$ elements. Let $h$ be $\# \mathrm{Pic}^0(X)(\mathbb{F}_q)$. Weil showed that $h \geq (\sqrt{q}-1)^{2g}$. Lachaud and Martin-...
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### What can one say about $\sum\limits_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$?

Denoting by $p_i$ the $i$-th prime, is it known that $\displaystyle \sum_{i=1}^\infty \frac{1}{p_{i+1}^2-p_i^2}$ converges? Can one compute a few digits based on euristic considerations or plausible ...
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### Reference for behavior of Artin $L$-functions at $\Re(s) = 1$

Would anyone know a reference that proves the basic facts about Artin $L$-functions at $\Re(s) = 1$? Namely, the non-vanishing and holomorphicity for non-trivial characters. I assume this was done in ...
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### Obtain a series expansion of $a^2(q)a^2(q^4)$

Let $a(q)$ denote the Borwein function $$a(q)=\sum_{m,n=-\infty}^\infty q^{n^2+nm+m^2}.$$ In this research paper the author has obtained the series expansion for $a(q)a(q^4)$. I want the series ...
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### Stabilizers of points in the upper half-plane

Suppose that $\Gamma$ is a group acting discontinuously on $\mathcal{H} = \{ z \in \mathbb{C} : \operatorname{Im}(z) > 1\}$. In order to keep things simple, suppose that \$\Gamma \subseteq \...