All Questions
Tagged with analytic-number-theory complex-analysis
445
questions
3
votes
0
answers
42
views
How to prove a property of the product of Eisenstein Series
I want to prove that, if $$ E_i(\tau) = \frac{1}{2\zeta(2i)}\sum_{(m,n)\in\mathbb{Z}^2-\{(0,0)\}}(m\tau+n)^{-k}$$
is the normalized Eisenstein Series of weight $i$, then $E_i(\tau)E_j(\tau)\neq E_{i+j}...
- 39
7
votes
1
answer
711
views
What's the easiest way to prove the Riemann Zeta function has any zeros at all on the critical line?
I learned in my intro to complex analysis class that the Riemann Zeta has trivial zeros at the negative even integers; this follows from the Riemann Zeta Reflection identity and the behavior of the $\...
- 1,290
0
votes
0
answers
94
views
Expressing a function in terms of the nontrivial zeros of the Riemann zeta function
Consider the function $\phi(x)$:
$$ \phi(x)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{2}}\left(\left\{ \frac{x}{n}\right\}-\frac{1}{2}\right)$$
$\left\{\cdot\right\}$ being the fractional part function. ...
3
votes
1
answer
143
views
$1^\alpha+2^\alpha+3^\alpha+\cdots+n^\alpha$
Let $\alpha>0$ and $m$ be a positive integers, use Euler's summation formula we can prove that there exists a constant $C$ such that
$$ \sum_{k=1}^nn^\alpha=\frac{n^{\alpha+1}}{\alpha+1}+\frac{n^...
- 915
0
votes
0
answers
16
views
Estimates for the logarithmic derivative of $\Lambda(s,\chi)$
We have the following estimate about the logarithmic derivative of $\xi_0(s)=s(1-s)\dfrac{\zeta(s)}{\zeta_\infty(s)}$ for $s=\sigma+it$ and $\rho=\beta+i\gamma$:
$$
\frac{\xi_0'}{\xi_0}(s)=\sum_{\rho\...
- 117
0
votes
1
answer
82
views
Estimation of the absolute value of the $n$th non-real zero of the Riemann zeta function
Recently, I have been studying the oringinal proof of the prime number theory by Hadamard. I didn't get it on the estimation of the absolute value of the $n$th non-real zero of the $\zeta$ function by ...
1
vote
1
answer
62
views
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{...
- 135
2
votes
0
answers
80
views
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
$$ \sum_{n=0}^{\infty} e^{nx} = \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \...
- 17.5k
2
votes
1
answer
192
views
solution verification: Is $K(s)$ holomorphic on $\Bbb C$?
Consider the Mellin integral
$$K(s)=\int_{1/2}^1 \zeta\bigg(-\frac{1}{\log x}\bigg)~x^{s-1}~dx $$
Where $\zeta(\cdot)$ is the Riemann zeta function defined for real $1/e<x<1.$ $K(s)$ is ...
- 754
0
votes
1
answer
36
views
Proving $|\zeta(\sigma+it)|=O(\log t)$ uniformly for $1\le\sigma\le 2$ and $t\ge 10$
Prove that $|\zeta(\sigma+it)|=O(\log t)$ uniformly for $1\le\sigma\le 2$ and $t\ge 10$.
The solution given by my lecturer is as follows. Recall the approximate formula for zeta, given by $$\zeta(s)=\...
- 135
0
votes
1
answer
85
views
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 $0 < u < \infty$, $(u + a)^{s−1} ≪ |a|^{σ−1}$ ...
- 281
2
votes
1
answer
76
views
Meromorphic continuation of Euler product
Short version: What can be said about the meromorphic continuation of the Euler product $$\prod _{p}\left (1+\frac {p^{-s}}{p-2}\right )?$$
Longer version: I realise I have some misconceptions about ...
- 1,662
1
vote
1
answer
61
views
a small doubt in the proof of the quantitative form of the prime number theorem
I have been studying the proof of the prime number theorem in the quantitative form as in Theorem 6.9 of Montgomery & Vaughan's book "Multiplicative Number Theory, which focuses on proving ...
- 11
3
votes
0
answers
117
views
A natural intractable geometrically inspired double-sum
Consider the following geometric setup: at every $(m,n)$ with $m$, $n$ both even, place a unit circle. Then we invert this entire setup through the unit circle at the origin. How much of the area of ...
- 599
0
votes
0
answers
38
views
analytically continue functions on independent planes in $\Bbb R^3$
I know techniques to analytically continue some functions off the positive real line. I am less sure given I have 2 independent euclidean planes sitting in $\Bbb R^3$ with some function $f$ defined on ...
- 754