Questions tagged [analytic-number-theory]
Questions on the use of the methods of real/complex analysis in the study of number theory.
3,997
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Lower bound for the prime zeta function
The prime zeta function is defined for $\mathfrak{R}(s)>1$ as
$P(s)=\sum_{p} \ p^{-s}$, where $p \in \mathbb{P}$.
It is well-know this series converges whenever $\mathfrak{R}(s)>1$.
Now, ...
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Inverse Mellin transform of $\Delta(t):=\sum_{s=1}^\infty d(s)\sqrt{\frac{t}{s}}M_1(4\pi \sqrt{ts})$
Define
$$ \Delta(t):=\sum_{s=1}^\infty d(s)\sqrt{\frac{t}{s}}M_1(4\pi \sqrt{ts}) $$
where $M_1(z)=-Y_1{(z)}-\frac{2}{\pi}K_1(z)$ and $d(s)$ is the divisor function.
What is the inverse Mellin ...
- 754
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Applications of the de Bruijn-Newman constant outside of the Riemann Hypothesis
According to Wikipedia,
The de Bruijn–Newman constant, denoted by Λ and named after
Nicolaas Govert de Bruijn and Charles Michael Newman, is a
mathematical constant defined via the zeros of a certain ...
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Dirichlet series and Laplace transform
Let $\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}$ be a Dirichlet series. It can be represented as a Riemann-Stieltjes integral as follows:
$$\displaystyle\sum_{n=1}^\infty \dfrac{a_n}{n^s}=\int_1^\...
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Proving that $\left|\sum_{n<x}\mu(n)\right|\ll x\exp(-c\sqrt{\log x})$ for some $c>0$
Assume that for $\sigma\ge 1-\frac{1}{(\log(2+|t|)^2}$ we have $$|\zeta(\sigma+it)|\gg\frac{1}{(\log(2+|t|))^2}.$$ Using Perron's formula and moving the line of integration to $\textrm{Re}(s)=1-\frac{...
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Does the limit of the exponential mobius exponential series asymptotically equal its regularized power series?
Context:
Consider the function $\sum_{n=0}^{\infty} e^{nx}$. An extremely unrigorous manipulation of this series would yield
$$ \sum_{n=0}^{\infty} e^{nx} = \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \...
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solution verification: Is $K(s)$ holomorphic on $\Bbb C$?
Consider the Mellin integral
$$K(s)=\int_{1/2}^1 \zeta\bigg(-\frac{1}{\log x}\bigg)~x^{s-1}~dx $$
Where $\zeta(\cdot)$ is the Riemann zeta function defined for real $1/e<x<1.$ $K(s)$ is ...
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Proving $|\zeta(\sigma+it)|=O(\log t)$ uniformly for $1\le\sigma\le 2$ and $t\ge 10$
Prove that $|\zeta(\sigma+it)|=O(\log t)$ uniformly for $1\le\sigma\le 2$ and $t\ge 10$.
The solution given by my lecturer is as follows. Recall the approximate formula for zeta, given by $$\zeta(s)=\...
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How to bound $\mu (\{ x \le X : |ψ(x) − x| \ge εx^{1/2} (\log x)^2 \})$ from above?
From the following estimate $$ \int_0^X |ψ(x) − x|^{2k} dx ≪ (ck^2)^k X^{k+1} \tag{1}$$ where $c$ is an absolute constant, I want to prove the following estimate $$ µ ( \{ x \le X : |ψ(x) − x| \ge ...
- 281
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Gauss sum over imaginary quadratic field
Define the Gauss sum
$$ G\left( \frac{a}{m}\right) = \sum_{n(\text{mod}\ m)} e\left( \frac{a n^2}{m}\right) ,$$
then I know the following result:
For any $a,m$ with $(2a,m) = 1$,
$$ G\left( \frac{a}{...
- 63
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Perron's formula in the region of conditional convergence
I am a bit confused about the proof of Perron's formula. It states that for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ and real numbers $c > 0$, $c > \sigma_c$, $x > 0$ we have
$$\...
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Understanding the proof of Theorem 10.2 in Montgomery & Vaughan's Multiplicative Number Theory
In Theorem 10.2 of the book of Montgomery & Vaughan's Multiplicative Number Theory there are two claims comes without any explanation:
1- For $0 < u < \infty$, $(u + a)^{s−1} ≪ |a|^{σ−1}$ ...
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Estimating number of product of $k-$primes upto given $x$
We know that up to $x$ the number of primes can be estimated by
$$\pi(x) = x \prod_{p \leq \sqrt{x}} \left(1 - \frac{1}{p}\right)$$
Can we extend this type of argument to estimate the number of terms ...
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Is there an analytic continuation of the Legendre Chi function $\chi_2(z)$ for $z > 1$?
The Legendre Chi function $\chi_2(z)$ is define as
$$
\chi_2(z) = \sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)^2}
$$
for $-1 \le z \le 1$. But $z > 1$ the series diverges. For real value of $z$ is ...
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Additive characters over a Number Field
Given any non-zero
integral ideal $\mathfrak{b}$ of $K$, an additive character modulo $\mathfrak{b}$
is defined to be a non-zero function $\sigma$ on $\mathfrak{o}/\mathfrak{b}$ which satisfies
$$
\...
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