1
vote
0answers
31 views

How many zero's does a general real entire function $f(z)$ have?

Let $f(z)$ be a real entire function. How do we find the number of solutions for $f(w)=0$ ? Can we express the number of zero's of $f$ in terms of its Taylor coëfficiënts ? Im not looking for the ...
12
votes
2answers
282 views

Why are people more interested in the Riemann hypothesis than Goldbach's conjecture? [closed]

One of my friends, a math professor, told me almost every one of his colleagues (in the math department) had attempted to prove the Riemann hypothesis at some point in their life (maybe secretly). ...
1
vote
0answers
31 views

Are there any linkage or transformation between some period transcendental numbers and algebraic irrational numbers?

Are there any linkage or transformation by combination of integral and algebraic function like in the definition of period number between some period transcendental numbers and algebraic irrational ...
1
vote
0answers
18 views

What is the definition of period number and any relation between Abelian integral and such a kind of period number

I recall there is a kind of real or complex number called period number which is defined by integral and algebraic function.But now,I search it again and again having gotten no result.Now any one can ...
2
votes
1answer
43 views

Problem about Eisenstein series on $\Gamma_1(N)$

I'm learning about Eisenstein series on $\Gamma_1(N)$ and it seems to me that I have misunderstood something. I imagine the following situation : Let $\nu$ be a function on $(\mathbb{Z}/ N ...
0
votes
0answers
16 views

Negative Weight meromorphic modular forms/ Sections of Line bundles

it is known, that we can see modular forms as section of line bundles on a Riemann surface. Especially, we know that a meromorphic modular form of weight 2 on SL(2,Z) corresponds to a meromorphic ...
0
votes
0answers
28 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
183 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
1answer
63 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
76 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
26 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
32 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 ...
20
votes
1answer
360 views

How to find $\int_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx$

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx. $$ (I have literature on this, if people want). Note, we can ...
1
vote
0answers
39 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
94 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
12 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
43 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 ...
7
votes
1answer
174 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
79 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
110 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
192 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
41 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
47 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
187 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
82 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
91 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
95 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
65 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
221 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
104 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), ...
10
votes
3answers
227 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
65 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
120 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 $z=0$.