# 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 ...
Suppose $0^+_\zeta$ is the set of non-trivial zeros of the Riemann zeta function $\zeta(s)$ which lie on or to the right of the critical line and above the $x$-axis, i.e, $$0^+_\zeta = \{s \in \mathbb{... 0answers 81 views ### Smooth number pairs satisfying a congruence Let \mathcal P=\{p_1,\dots,p_{2t}\} be 2t primes between 2^\ell and 2^{\ell+1} and fix an exponent bound a\in\mathbb Z_{\geq2}. Fix N\in\mathbb N whose prime factors p satisfy p>2^{\... 0answers 120 views ### \frac{1}{\pi} \int_{0}^\infty \frac{\log|\left(\frac{1}{2}+it\right)\zeta\left(\frac{1}{2}+it\right)|}{\frac{1}{4}+t^2}dt  Consider,$$I=\frac{1}{\pi} \int_{0}^\infty \frac{\log|\left(\frac{1}{2}+it\right)\zeta\left(\frac{1}{2}+it\right)|}{\frac{1}{4}+t^2}dt $$where \zeta is the Riemann Zeta function. Since , Hardy (... 1answer 217 views ### Why is the Barban-Halberstam-Davenport theorem important? I have a slightly open-ended question about the Barban-Halberstam-Davenport theorem and hope that it is not off-topic. The theorem itself states that for any A>0 and Q lying the range x\log^{-... 0answers 68 views ### Is there only one holomorphically induced parametrization \mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0}? A function f:\mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0} is called a parametrization if for any n\in \mathbb{Z}_{\geq 0} there exists k\in \mathbb{Z}_{\geq 0} such that n=f^{\circ k}(0). A ... 0answers 42 views ### Finding generator versus finding order for multiplicative groups modulo composites Let N be a composite and let \lambda(N) be Carmichael Lambda of N. It is known finding \lambda(N) reveals information about factors. Is there a polynomial time (under suitable assumptions) or ... 1answer 346 views ### Splitting of small primes in number fields generated by the torsion of elliptic curves Suppose E/\mathbb Q is a non CM elliptic curve and we look at the number field K_d generated by the d-torsion of E. What is known about the (complete) splitting of small primes in K_d? More ... 1answer 235 views ### Proof of Mertens' formula$$\sum_{p\leq x}\frac{1}{p} = \log(\log x) + a+ O\left(\frac{1}{\log x}\right) a=\int_{2}^{\infty}R(u)\frac{du}{u(\log u)^2} +1- \log(\log 2) e^{-1/p}(1-1/p)^{-1}= 1+O(1/p^2)$$Below is ... 1answer 315 views ### On infinite sum containing logarithmic derivative of Zeta function and Möbius function: Consider the following function:$$F(s)= \sum_m \mu(m) \sum_n \frac{e^{-n/2}\zeta^\prime (mns)}{n \zeta(mns)}$$Now, we can see, that function has simple poles {\left[\frac{1}{n}\right]}_{n=1}^\... 0answers 29 views ### Asymptotics of a sum involving multiplicative partitions of an integer n into k possibly non-distinct parts ≥2 Let x\in\left(0,1\right). For each integer n\geq2, let \Omega\left(n\right) denote the number of prime factors of n, counted according to multiplicities; thus \Omega\left(2\right)=1, \Omega\... 0answers 106 views ### Applications of Jensen's Formula to entire functions of finite order I am trying to understand a frequently omitted technical detail in applications of Jensen's Formula to bound the number of zeros of entire functions of finite order. We say that an entire function f(... 2answers 849 views ### Correlations of \phi(n)/n We know that the average of \phi(n)/n is approximated by a constant. Here \phi  is the Euler quotient function. One can furthermore show asymptotics with a secondary main term, at least for the ... 1answer 554 views ### Where should I learn about the p-adic L-functions of elliptic curves? Where is the best place to learn about the p-adic L-functions of Elliptic Curves? Doing a bit of research I have found books like "An Introduction to Cyclotomic Fields" by Washington, but ... 0answers 93 views ### Degree of compositum of all number fields under given discriminant For an integer n\geq 1 define f(n) to be the degree of the compositum of all number fields with discriminant at most n. What bounds are known on f(n)? 2answers 861 views ### On modified Euler product I know this forum is for professional Mathematicians. I didn't know if the following question is fit or not fit for the site. If not, feel free to delete it. Consider the modified Euler product as ... 2answers 304 views ### Deriving an asymptotic for \pi(x) directly from \log \zeta(s)? Denote by \pi(x) the number of primes p\leq x. We generally give approximations for \pi(x) by first approximating \psi(x) = \sum_{n\leq x} \Lambda(n). Part of the reason is presumably that, if ... 2answers 158 views ### Richaud-Degert type quadratic extensions A Richaud-Degert type real quadratic field is a number field of the form K = \mathbb{Q}(\sqrt{d}) where d = {(an)}^2 + ka > 0 for positive integers a, n and k \in \{ \pm 1, \pm 2, \pm 4 \}, ... 1answer 97 views ### Sum of inverse squares of numbers divisible only by primes in the kernel of a quadratic character Let \chi be a primitive quadratic Dirichlet character of d modulus m, and consider the product$$\prod_{\substack{p \text{ prime} \\ \chi(p) = 1}} (1-p^{-2})^{-1}. What can we say about the ...
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 \...