Questions tagged [analytic-number-theory]
Questions on the use of the methods of real/complex analysis in the study of number theory.
4,233 questions
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Estimate for number of squarefree integers up to a given number using Perron's formula
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 ...
- 107
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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})= \...
- 79
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Convergence of Poincaré Series
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 $k$. ...
- 25
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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}...
- 21
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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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Getting confused about log of the zeta function...
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 ...
- 3,147
7
votes
1
answer
205
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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, ...
- 2,017
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Why does this surplus-weighted Dirichlet sum reproduce the Riemann zeta function so closely?
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 $\...
2
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1
answer
107
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Technical step in the proof of Linnik's theorem in Iwaniec-Kowalski (18.82)
Going through the proof of Linnik's theorem in Iwaniec and Kowalski's Analytic number theory, I came across an affirmation I don't really understand. On Page 440, starting from the explicit formula ...
- 41
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116
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Abel summation for$ \frac{1}{n\log n}$
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 ...
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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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votes
1
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70
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Explicit bound on prime gaps assuming RH (Cramer's Theorem)
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 ...
- 13
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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) \;+\; ...
- 305
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49
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Can we get modest bounds on $\pi(x)$ by inverting $F_M(s)$?
Start with
$$
H_M(x)=x\sum_{k=1}^{M} a_k\,u^k,\qquad
u=e^{1/\ln x}-1
$$
where $a_k$ are Stirling coefficients, and $H_M(x)$ matches the first $M$ terms in the asymptotic expansion of $\mathrm{Li}(x)$.
...
- 1,209
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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)=...
- 1,209