Questions tagged [analytic-number-theory]
Questions on the use of the methods of real/complex analysis in the study of number theory.
4,235 questions
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Prove that $\int_0^1\operatorname{Li}_2\left(\frac{1-x^2}{4}\right)\frac{2}{3+x^2}\,\mathrm dx= \frac{\pi^3 \sqrt{3}}{486}$
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})= \...
- 49
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Convergence of Integral Expressions for ξ(s) Involving ψ'(x) and ψ''(x) in the Complex Plane
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}...
- 11
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Complex Analysis Reading [closed]
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 ...
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Does analytic continuation preserve equality after rearranging a Dirichlet-type double sum when one side is made absolutely convergent by a weight?
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 ...
- 61
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Clean version of inequality for $\Gamma(z)$ - known?
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, ...
- 1,997
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Technical step in the proof of Linnik's theorem in Iwaniec-Kowalski (18.82)
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 ...
- 41
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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?
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 ...
- 119
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Analytic resummation of a series involving modified Bessel functions $K_\nu $
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) \;+\; ...
- 167
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Approximating $\mathrm{Li}(x)$ (logarithmic integral) with $H_M(x)$
Let
$$
u(x)=e^{1/\ln x}-1 \qquad (x>1),
$$
and consider
$$
H_5(x)=x\Big(u+\tfrac12 u^2+\tfrac{4}{3}u^3+\tfrac{11}{3}u^4+\tfrac{223}{15}u^5\Big).
$$
This comes from the more general family
$$
H_M(x)=...
- 1,189
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Bessel satisfies linear pde but can't find any references for it
The form
$$\Phi_s(p)= \int_0^\infty e^{-px} e^{-s/x} \, dx = 2\sqrt{\frac sp} K_1(2\sqrt{sp})$$
is a standard representation for the $K_\nu(\cdot)$ Bessel function ($\nu=1$). It appears in analytic ...
- 1,189
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Do the nontrivial zeros of $\zeta(1/2 + iy)$ correspond to the zeros of a wave-like Dirichlet series $\Psi(1/2, y)$?
I am exploring an alternative interpretation of the nontrivial zeros of the Riemann zeta function $\zeta(s)$, based on wave interference.
I define the complex function:
$$
\Psi(x, y) = \sum_{n=1}^{\...
- 39
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Change-of-sign of the $GL_2$ Maass form Fourier coefficients $\rho_f(m)$ and $\rho_f(-m)$
Suppose that $f$ is a half-integral-weight $GL_2$ Maass form of level $q$ and nebentypus $\chi$. If $\rho_f(m)$ denotes the $m^{th}$ Fourier coefficient of $f$, then it is known that $$\rho_f(-m)=\pm\...
- 255
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Meaning of the integral $\frac{2}{\pi}\int_{0}^{\infty} \frac{\Xi}{t^{2}+\frac{1}{4}} t^{n}\cosh{\alpha t}\ dt$ in the proof by Hardy on $\infty$ 0s.
I have been studying Hardy's proof on the infinite zeros of the Riemann Zeta Function from The Theory of Riemann Zeta Function by E.C. Titchmarsh, and I have understood the proof, but I am unable to ...
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Is $S_k(x)$ non-empty for all $k$ for sufficiently large $x$?
Let $x$ be large and
$$
A = \{1,3,5,\dots,\le x\}
$$
be the odd integers $\le x$. For each odd prime $p \le x$ and for each integer $k$, remove from $A$ all integers
$$
n \equiv \frac{p-9}{2} \pmod p \...
- 63
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Book Recommendation On Analytic Number Theory
I am planning to study number theory, and as preparation I have studied high school–level differential and integral calculus (primarily single-variable), high school algebra, and a little abstract ...
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Growth rate of $\displaystyle\sum_{n=1}^\infty E_1(2n\log x)-\sum_\rho\mathrm{li}(x^\rho)+\mathrm{li}(x^{\bar\rho})$ as $x\to\infty$
I was experimenting with the following approximation of the prime counting function $\pi(x)$ under the assumption of the RH $$\pi^*(x)=\sum_{m\leq z}\frac{\mu(m)}{m}\mathrm{li}(x^{1/m}),\hspace{0.5cm}\...
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Showing that $\operatorname{Li}(x) \sim\frac{x}{\log x}$.
I want to show that $\newcommand{\Li}{\operatorname{Li}} \Li(x) \sim \frac{x}{\log x}$.
We define $\sim$ as
$f(x) \sim g(x)$ iff $\frac{f(x)}{g(x)} \to 1$ as $x \to \infty$.
$\Li(x)$ is called the ...
- 279
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How to prove $f'(x) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} \cos(x/k)\overset{?}{=} \Omega(\log \log x)$?
While investigating $f(x) = \sum_{k=1}^{\infty} (-1)^{k+1} \sin(x/k)$ such as on these questions on its unboundedness and its infinite number of real zeros, I referenced one of the first post on MSE ...
- 1,695
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Simultaneous prime-free short intervals modulo small $q$
Fix a large parameter $X$. For $0<\delta<\theta<\tfrac12$, set $H := X^{\theta}$.
Question. Is it true that there exist infinitely many values of $X$ and, for each such $X$, a modulus $q\le X^...
user1675092
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Single-AP two-form dispersion beyond $1/2$ on short-interval averages?
Fix an even integer $h \geq 12$ and set $W = \prod_{p < h} p$. Choose a residue class $b \pmod{W}$ with the covering property that for every $s$ in the range $1 \leq s \leq h-1$, there is a prime $...
user1675092
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$\mathsf{GCD}$ of special integer forms [closed]
Given $a,b\in2\mathbb N_{\geq0}+1$ and $a<b$, is there a bound of the form $k=O(\log\log(ab))$ or $k=O(\log(ab))$ on the minimum $k\in\mathbb N$ satisfying
$$\mathsf{GCD}\big({2q^ka+1},{2q^kb+1}\...
- 6,341
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Proof that $\mathcal{M}_x\left[\displaystyle\sum_{n \le x} a_n\right](s) = \displaystyle\sum_{n=1}^\infty \frac{a_n}{n^s}$?
When looking into the derivation of Perron's formula, I found that it seems to come from using the inverse Mellin transform of the equation
$$\mathcal{M}_x\left[\sum_{n \le x} a_n\right](s) = \frac{1}{...
- 2,386
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Finding the Dirichlet coefficients of a function
Let $A(s)$ be an approximation for $\zeta^{(n)}(s)\zeta^{(n)}(1-s)$. What are the Dirichlet coefficients? More precisely, let $$A(s)=\sum_{n\le X}\frac{a_n}{n^s}\approx \zeta^{(n)}(s)\zeta^{(n)}(1-s).$...
- 87
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Scaled complex conjugates of non-trivial zeros of Riemann Zeta function
Let $s=\frac12+iz$ be a non-trivial zero of the Riemann Zeta function $\zeta(s)$.
Let $f(s,a)=\frac12+iaz$ where $s=\frac12+iz$ be the $a$ complex conjugate of $s$.
Eg: $f(s,1)=s$ and $f(s,-1)=\...
- 6,341
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A problem of the Vaaler's approximation formula about $\psi (x)$ function
This is a simplified expression of the theorem:
Theorem 5 (Vaaler). Let $M \in \mathbb{N}$. Then there exists a trigonometrical polynomial
$$
\psi^*(x):=\sum_{1 \leqslant|m| \leqslant M} a_M(m) e(m x)
...
- 327