Questions tagged [analytic-number-theory]

Question 1

It's a result of Davenport that \begin{equation} \sum_{n\leq x}\mu(n)e^{i\alpha n}\ll\frac{x}{\log^Ax} \end{equation} for any $A>0$, and uniformly for $\alpha\in\mathbb{R}$. My question is: does ...

Question 2

I have been studying imaginary quadratic fields with class number $1$. The Heegner-Stark theorem states there are $9$ fundamental discriminants with this property: $-3$, $-4$, $-7$, $-8$, $-11$, $-19$,...

Question 3

We have $\left\lfloor x\right\rfloor$ as the floor function and $\left\{x \right\}$ as the fractional part. Looking for the asymptotic expansion as $x \rightarrow \infty$ of $$DW (x)=\sum_{n \le x} \...

Question 4

I'd like to prove that $$2\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x.$$ Ok, someone said that this holds, but I tried really ...

Question 5

Let $Q(x)$ denote the number of squarefree integers up to $x$. I want to show using Perron's formula that $$ Q(x) = \frac{x}{\zeta(2)} + O(x^{1/2} \exp(-c \sqrt{\log x})), $$ where $c$ is a positive ...

Question 6

I would like to prove that $$\int_0^1 \operatorname{Li}_2 \left(\frac{1-x^2}{4}\right) \frac{2}{3+x^2} \,\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}.$$ It’s known that $\Im \operatorname{Li}_3(e^{2πi/3})= \...

Question 7

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 8

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 9

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 10

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 11

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 12

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 13

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 14

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 15

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 ...

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Questions tagged [analytic-number-theory]

Cancellation in exponential sum involving Möbius? [closed]

Are the discriminants $-12$, $-16$, $-27$, $-28$ known to have class number $1$?

Asymptotic expansion as $x \rightarrow \infty$ of $\sum_{n \le x} \lfloor\frac{x}{n}\rfloor \left\{{2\sqrt{\lfloor\frac{x}{n}\rfloor}}\right\}$

Prove that $2\int_0^1 \frac{\operatorname{Li}_2(x(1-x))}{1-x+x^2}\mathrm{d}x=\int_0^1 \frac{\ln x \ln(1-x)}{1-x+x^2}\mathrm{d}x$

Estimate for number of squarefree integers up to a given number using Perron's formula

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

Convergence of Poincaré Series

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

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?

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