-1
votes
0answers
64 views

Strategy verifying Riemann Hypothesis? [on hold]

The basic strategy for verifying the Riemann Hypothesis Count all of the zeros of $\zeta(t)$ for $0 < t < T$ Compute an upper bound on the number of zeros of the zeta function which lie in the ...
0
votes
0answers
23 views

$\dim \mathcal{S}_k(\Gamma_0(N))$

I'm looking for a formula which gives the dimension of $\mathcal{S}_k(\Gamma_0(N))$ the space of cusp forms of weight $k$ and level $N$. I found the following statement for $k\geq 4$ $$\dim ...
5
votes
1answer
179 views

Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ ...
1
vote
0answers
61 views

Are even Dirichlet series constants?

I consider a Dirichlet series with an absolute abscissa of convergence $\sigma$ that can be meromorphically extend to $\mathbb{C}$: $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ and the analytic ...
3
votes
2answers
45 views

How to find all the positive integer number $n$ such that $\sum_{i=1}^{n}a_{i}=0$ and $|a_{i}|=1$ has a solution in $\mathbb{C}$ with $a_i+a_j\neq 0$

Find all the $n$ for which there exist complex numbers $a_{1},a_{2},\cdots,a_{n}$ such that: (1):$$|a_{i}|=1,i=1,2,\cdots,n$$ (2):$$a_{1}+a_{2}+\cdots+a_{n}=0$$ (3): for any $i\neq ...
0
votes
2answers
60 views

Logic behind a Geometric Construction of regular heptadecagon

I'm reading a Chinese book "Methods of Mathematical Physics" by Wu Chongshi. During introduction of complex analysis, it explains a Geometric Construction of regular heptadecagon. Task: to achieve ...
5
votes
1answer
74 views

Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
0
votes
1answer
24 views

Bernoulli numbers identity with binomial coefficient

The generating function for the Bernoulli numbers $B_k$ is given by $f(z) = \frac{z}{e^z -1}= \sum_{k=0}^{\infty} \frac{B_k}{k!} z^k$. Applying the identity $$1 = \frac{e^z -1}{z} \cdot ...
2
votes
1answer
31 views

About the implicit funtion in a holomorphic situation.

Let $f(x,y)$ be a polonomial with integral coefficients which has a zero $(a,b)\in \mathbb{R}^2$ such that the partial derivative respect to $y$ at this point is nonzero. Then by the implicit function ...
16
votes
1answer
341 views

Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
1
vote
0answers
35 views

Zero to power Zero (Zero ^ Zero) indeterminable or not? [duplicate]

I want to know Zero power to Zero equal to 1 or Indeterminable. I think it cannot be exist. Please explain with proper mathematical definitions.
2
votes
0answers
88 views

On Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq ...
0
votes
0answers
11 views

When is the fourier transform of a quasi-character $\hat c(\alpha)=|\alpha|c^{-1}(\alpha)$?

This is from lemma $2.4.2$ of Tate's thesis. Let $c$ be a quasi-character on $k^{*}$, the multiplicative group of a number field completed at a non-archimedian place. Lemma 2.4.2 For $c$ in the ...
1
vote
1answer
34 views

If $\zeta$ is a function of characters what does it mean for it to be regular?

This is from lemma 2.4.1 of Tate's thesis. Lemma 2.4.1: A $\zeta$-function is regular in the "domain" of all quasi-characters of exponent greater than $0$. proof: We must show that for each ...
3
votes
1answer
47 views

Conditionally convergent products

Can someone explain why this occurs. I came across this in a book by Titchmarsh. $$\prod_{n=2}^{\infty}\left(1-\frac{e^{in\theta}}{\log(n)}\right)$$ this sum does not converge for any rational ...
0
votes
0answers
39 views

Number of lattice points in an annulus

Consider the lattice spanned by two nonzero complex numbers $\xi_{1}$ and $\xi_{2}$ such that their ratio is not real. Let $w = m\xi_{1} + n\xi_{2}$. Let $A(n)$ be the number of lattice points such ...
6
votes
1answer
170 views

The series $2+3x+5x^2+7x^3+11x^4+…$

It occurred to me to ask whether the power series whose coefficients are the primes has non-zero radius of convergence, and if so, what kind of function it produces. Wikipedia has some bounds on the ...
3
votes
0answers
116 views

Ratio of maximal to minimal jump in the set of angle multiples (corrected)

(This is the corrected version of the question I asked here: Ratio of maximal to minimal jump in the set of angle multiples.) Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times ...
1
vote
1answer
23 views

Ratio of maximal to minimal jump in the set of angle multiples

Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times S^1\to\mathbb{R}$ be the distance function given by the arc length. Let $\theta\in S^1$ be an element of infinite order, that is ...
1
vote
0answers
43 views

A problem with cosine function

I try to understand something from number theory and the author gave this as an excersise: Prove that $z\longmapsto 2\sqrt{p}\cos z$ is a bijection of a set ...
0
votes
2answers
60 views

On finding the zeros of a polynomial

What is the zero (real) of the polynomial $$x^{k+1}-2x^{k}+1=0$$ If there is such, how can I find it or what method can I use?
0
votes
0answers
78 views

What is the explicit formula for the nth prime? [duplicate]

The explicit formulas for the second chebyshev function or the prime counting function (in terms of Riemann zeta zero's) are well known. But what is the explicit formula for the nth prime ? For ...
1
vote
2answers
108 views

Higher dimensional analogues of the argument principle?

I know there are higher dimensional analogues of the argument principle. (See http://en.wikipedia.org/wiki/Variation_of_argument) But I do not have books about it and I cannot find anything of value ...
5
votes
2answers
181 views

Solving an integral coming from Perron's formula

In analytic number theory, Perron's formula says that $$ \sum_{1 \leq k < n} a_k + \frac{1}{2}a_n = \int_{c - i\infty}^{c+i\infty} f(s)\frac{n^s}{s}ds, $$ where $f(s) = \sum_{k \geq 1} a_k/k^s$ ...
1
vote
0answers
40 views

Proof that Hecke operators on modular forms commute

I am working on Hecke operators on modular forms and would like to prove that these commute. Specifically, I am trying to prove that $$ T_nT_m=\sum_{d\vert (n, m)} d^{k-1} T_{\frac{nm}{d^2}}=T_mT_n, ...
1
vote
0answers
45 views

Computing a lower bound for partition function

Let $$F(x)= \sum_{n=0}^{\infty}p(n)x^n=\prod_{n=1}^{\infty}\frac1{1-x^n}$$ be the generating function for the partitions. I showed that $$\log F(x) \sim \frac{\pi^2}{6(1-x)} \quad as\quad x\to 1,\quad ...
3
votes
1answer
180 views

Elementary bound on the Riemann zeta function

I am currently preparing for a course in analytic number theory and I wanted to get a heads start. In my preparation, I came across the following problem: Show that for $|y|\geq 2$, $|\zeta(1+iy)| ...
1
vote
1answer
30 views

path of the integral in the initial definition of gamma function

Can the path of the integral in the initial definition of gamma function be altered to a straight line starting from $0$ to $\infty;e^{ia},a<\pi/2$)?
3
votes
0answers
81 views

special values of zeta function and L-functions

I was reading in some lectures notes about the Riemann zeta-function which takes on special values: $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ In fact, we can compute even values of the ...
3
votes
1answer
89 views

Convergence of Rademacher's formula: Extending the partition numbers to complex index

Consider the famous formula of Rademacher (actually Hardy, Ramanujan, and Rademacher): $$p(n) = \frac{1}{\pi \sqrt{2}} \sum_{k=1}^{\infty} \sqrt{k}\ A_k(n)\ F_k'(n)$$ $$A_k(n) = \sum_{0 \le m < k, ...
0
votes
1answer
48 views

Multiplicity of a zero of an L-function and covering spaces

This question may not be suitable for MathOverflow due to its relative vagueness, hence I ask it here. I just read in Wikipedia that there was a bijective correspondence between the path connected ...
2
votes
1answer
93 views

Absolute convergence of Euler products and infinite series

We know that given a multiplicative function $f$ for which the series $\sum_{n=1}^\infty f(n)$ converges absolutely then so does the Euler product $\prod_{p}\sum_{k=0}^\infty f(p^k)$, but does the ...
0
votes
0answers
106 views

Zeros of a power series

Suppose we have a power series with (real or complex) coefficients $\sum_{n \geq 0} a_n x^n$ (that has nonzero radius of convergence). Can one say something about its zeros in terms of the ...
2
votes
1answer
64 views

Confused about the explicit formula for $\psi_0(x)$

In the explicit formula for $\psi_0(x)$ used in the PNT proof : $$\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}) $$ In particular the ...
6
votes
1answer
213 views

Newman's “Natural proof”(Analytic) of Prime Number Theorem (1980)

I am trying to understand this short proof by newmann. I faced some problems while grasping this very proof. Please help me out. 1 . I am not clear, why in step (1)'s proof he says that from unique ...
3
votes
2answers
103 views

Equality involving $\sum_n \sin(\gamma_n \log x)/\gamma_n$

This is I think an algebra confusion about an equality of Littlewood, $$\frac{\psi(x) - x}{\sqrt{x}} = -2\sum_{1}^{\infty}\frac{\sin( \gamma_n\log x)}{\gamma_n} + O(1).\hspace{20mm}(1)$$ He refers ...
3
votes
1answer
80 views

Problem with the proof that $\zeta(s)$ has no zeros for $\mathrm{Re}(s) = 1$

Almost every proof I read says that If $\zeta(s)$ has a zero of order $\mu$ in $1 + ai$ ($\mu \geq 0$) then $$\lim_{\epsilon \to 0}\; \epsilon \frac{\zeta'(1+\epsilon +ai)}{\zeta(1+\epsilon ...
2
votes
1answer
60 views

Limit approaching a pole of $\phi(s)=-\frac{\zeta'(s)}{\zeta(s)} - \sum_p \frac{\log p}{p^s(p^s -1)}$

If: $$\phi(s) = -\frac{\zeta'(s)}{\zeta(s)} - \sum_p \frac{\log p}{p^s(p^s -1)},$$ where $\zeta(s)$ is the riemann zeta function, why is: $$\lim_{\epsilon \to 0} \epsilon\phi(1+\epsilon) = 1\quad ?$$ ...
1
vote
0answers
50 views

$\theta(x) = O(x)$ in the prime number theorem

In the Newman short proof of the prime number theorem (http://www.maths.dur.ac.uk/~dma0hg/prime_number_theorem_zagier.pdf) Zagier states that the fact that $2^{2n} >= e^{\theta(2n) - \theta(n)}$ ...
2
votes
1answer
48 views

Difficulty with a meromorphic extension.

I'm trying to understand the prime number theorem, but never having followed a course in complex analysis, I have some difficulties. (the article is this: ...
4
votes
0answers
149 views

Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
9
votes
3answers
223 views

Where is the fallacy in the argument using Prime Number Theorem

I am reading about Prime Number Theorem from book by Ingham. As as application of PNT I found the following theorem: Now my doubt is at the step $\frac{\log(y)}{\log(x)}\rightarrow 1$, we can say ...
0
votes
2answers
63 views

Why is $\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n) $?

I'm trying to understand the equation: $$\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n).$$ Here $x\in \mathbb{R}, x\geq 0$, and $C = \{s:\operatorname{Re}(s) = ...
4
votes
1answer
119 views

How does it follow $s\int_1^{\infty}\frac{\psi(x)}{x^{s+1}}dx$?

I have two relations: 1)$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{1}^{\infty}\frac{\Lambda(n)}{n^s}$. 2)$\psi(x)=\sum_{n\leq x}\Lambda(n)$. From these two how does it follow that ...
5
votes
1answer
108 views

A case where $z^z = 0$ where $z$ is complex number

Is there any case where $z^z = 0$ where $z$ is complex number? The case excludes the case where $z=0$.
5
votes
1answer
146 views

Analytically continue a function with Euler product

I would like to estimate the main term of the integral $$\frac{1}{2\pi i} \int_{(c)} L(s) \frac{x^s}{s} ds$$ where $c > 0$, $\displaystyle L(s) = \prod_p \left(1 + \frac{2}{p(p^s-1)}\right)$. ...
0
votes
1answer
85 views

complex analysis poles and residues

I am trying to understand a lemma on the (end of the first page - second page) on this link: http://www.math.uga.edu/~pollack/infprimes-final.pdf Basically, they end up with $$\sum_{d \geq ...
5
votes
1answer
163 views

Dirichlet L-series and Gamma function question

Could someone help me, please, with this exercise? Consider a sequence of complex numbers $\{a_n\}$ such that $a_n=a_m $ iff $ n\cong m $ mod $q$ for some positive integer $q$. Define the ...
6
votes
0answers
157 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
4
votes
2answers
174 views

How to show $e^{2 \pi i \theta}$ is not algebraic.

I was wondering if someone could possibly help me figure out how to show $e^{2 \pi i \theta}$ is not algebraic when $\theta$ is irrational. Thanks!