Questions tagged [analytic-number-theory]

Question 1

I would like to prove that $$ \int_0^1 \text{Li}_2(\frac{1-x^2}{4}) \frac{2}{3+x^2}dx= \frac{\pi^3 \sqrt{3}}{486}$$ It's known that $\Im \text{Li}_3(e^{2πi/3})= \frac{2\pi^3}{81}$, but I struggle to ...

Question 2

I am reading Iwaniec’s Topics in Classical Automorphic Forms Chapter 3 on Poincaré Series. The set up is for some Fuchsian group of first kind $\Gamma$, some multiplier system $\theta$ of weight $k$. ...

Question 3

Riemann's Zeta Function Page 17: note $\psi(x)=\sum_{n=1}^{+\infty}e^{-n^{2}\pi x}$ \begin{equation} \xi(1/2+ti)=4\int_{1}^{+\infty}\frac{d[x^{3/2}\psi^{\prime}(x)]}{dx}x^{-1/4} \cos\left(\frac{t\ln x}...

Question 4

Let $m$ and $N$ be two integers such that $m<N$ . Then find the number of integers less than or equal to $m$ and relatively prime to $N$?

Question 5

I am currently reading Stein’s Complex Analysis and have just finished Chapter 1 and 2. I have completed all the basic exercises, mostly on my own, and have attempted one or two of the more advanced ...

Question 6

Given some general arithmetic function $A(n)$ with a known asymptotic formula for $\sum_{n \le x}A(n)$, is there any way to figure out the asymptotic formula for $$\sum_{\substack{n \le x \\ \gcd(n,q) ...

Question 7

Let $F(n,d)$ be an arithmetic function of two variables (for example $F(n,d)$ could involve $\mu(d)$, $\lambda(d)$, divisor functions). Then let $$ A(n) := \sum_{d \le n} F(n,d) $$ is always a finite ...

Question 8

I am getting very confused with taking complex logarithm... I know that the $\zeta(s)$ has the Euler product $$\zeta(s) = \prod_p (1 - p^{-s})^{-1}$$ for $Re(s) > 1$. Furthermore, we can deduce ...

Question 9

Let $z=x+i y$, $x\geq 1/2$. Is the following inequality true? $$|\Gamma(z)|\leq (2\pi)^{1/2} |z|^{x-1/2} e^{-\pi |y|/2}$$ If you allow a fudge factor such as $e^{\frac{1}{6|z|}}$ on the right side, ...

Question 10

I’ve been experimenting with a surplus-weighted Dirichlet series that seems to replicate the local behavior of the Riemann zeta function. Below is the definition and a set of plots comparing it to $\...

Question 11

Going through the proof of Linnik's theorem in Iwaniec and Kowalski's Analytic number theory, I came across an affirmation I don't really understand. On Page 440, starting from the explicit formula ...

Question 12

I'm studying analytic number theory and am currently solving some problems that consist of applying Abel's summation formula to prove certain series are equal to something well-known. I'm a bit stuck ...

Question 13

I am reading about exponential sums in Analytic Number Theory by Iwaniec and Kowalski, page 199. At one point, they use the inequality $ \sum_{1 \le m \le q/2} \frac{1}{m} \le \log q. $ I understand ...

Question 14

Cramer, along with his conjecture $$g_n=O(\log^2 p_n)$$ also proved, assuming Riemann Hypothesis, $$g_n=O(\sqrt{p_n}\log p_n)$$ However no explicit estimates were provided. Have anyone made it ...

Question 15

I am interested in whether the following infinite series can be resummed or expressed in a more compact analytic form: \begin{align} S(t) = \sum_{n = 0}^{\infty} \left[ K_0\!\big((1 + 2n)t\big) \;+\; ...

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Questions tagged [analytic-number-theory]

Prove that $ \int_0^1 \text{Li}_2(\frac{1-x^2}{4}) \frac{2}{3+x^2}dx= \frac{\pi^3 \sqrt{3}}{486}$

Convergence of Poincaré Series

Convergence of Integral Expressions for ξ(s) Involving ψ'(x) and ψ''(x) in the Complex Plane

Determine the number of integers less than or equal to $m$ and relatively prime to $N$ for $m<N$. [closed]

Complex Analysis Reading [closed]

Sum of an arithmetic function upto x with gcd(n, q) = 1 for fixed q [closed]

Does analytic continuation preserve equality after rearranging a Dirichlet-type double sum when one side is made absolutely convergent by a weight?

Getting confused about log of the zeta function...

Clean version of inequality for $\Gamma(z)$ - known?

Why does this surplus-weighted Dirichlet sum reproduce the Riemann zeta function so closely?

Technical step in the proof of Linnik's theorem in Iwaniec-Kowalski (18.82)

Abel summation for$ \frac{1}{n\log n}$

Why does $\sum_{1 \le m \le q/2} \frac{1}{m} \le \log q$ hold on page 199 of Analytic Number Theory by Iwaniec and Kowalski?

Explicit bound on prime gaps assuming RH (Cramer's Theorem)

Analytic resummation of a series involving modified Bessel functions $K_\nu $

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