Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

learn more… | top users | synonyms

26
votes
1answer
843 views

Are Primes a Self-Fulfilling Prophecy?

Assume the following process: Let's start with the set of primes $\{p_k\}$ Then we use the Euler product being equivalent to Riemann's Zeta function $$ \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = ...
20
votes
1answer
519 views

Upper bound on differences of consecutive zeta zeros

The average gap $\delta_n=|\gamma_{n+1}-\gamma_n|$ between consecutive zeros $(\beta_n+\gamma_n i,\beta_{n+1}+\gamma_{n+1}i)$ of Riemann's zeta function is $\frac{2\pi}{\log\gamma_n}.$ There are many ...
2
votes
1answer
481 views

Primes and Riemann zeta function.

Primes numbers and Riemann zeta function. Question 1: Is there a proof of the infinitude of prime numbers using the Riemann Zeta function. Exboço could show me a proof of this where I could find it? ...
2
votes
1answer
180 views

Dirichlet character over Riemann zeta function

Let $\chi$ be a Dirichlet character mod q and let $$L(s,\chi)=\sum_{n\leq x} \frac{\chi(n)}{n^s}.$$ What is the value of $\displaystyle\lim_{s \rightarrow 1} \frac{L(s,\chi)}{\zeta(s)}$ for principal ...
8
votes
1answer
189 views

Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function.

Let $f(n)$ be a multiplicative function defined by $f(p^a)=p^{a-1}(p+1)$, where $p$ is a prime number. How could I obtain a formula for $$\sum_{n\leq x} f(n)$$ with error term $O(x\log{x})$ and ...
2
votes
1answer
255 views

Has it been ruled out that the Riemann hypothesis fails for only finite number of zeros?

Has it been ruled out that the Riemann hypothesis fails, but fails only for finite number of zeros?
1
vote
1answer
92 views

Another question about proof that $\zeta(s) \neq 0$ for $\Re(s) = 1$

This is a question distinct from but related to the question I wrote here: Question about proof that $\zeta(s) \neq 0$ for $\Re(s) = 1$, so assume the same things that I wrote there. The paper then ...
4
votes
1answer
203 views

Question about proof that $\zeta(s) \neq 0$ for $\Re(s) = 1$

I'm following this paper: http://mathdl.maa.org/images/upload_library/22/Chauvenet/Zagier.pdf Define $\Phi(s) = \displaystyle\sum_p \frac{\log p}{p^s}$. By taking a logarithm and differentiating ...
4
votes
3answers
915 views

Order of growth of $(s-1)\zeta(s)$

Again, order of growth problems. Show that the function $(s-1)\zeta(s)$ is an entire function of growth order $1$; or equivalently, $$|(s-1)\zeta(s)| \leq A_{\epsilon} \; \exp ...
3
votes
2answers
245 views

Approximate Riemann zeta function

Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$. In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function. My question is: Is there a Functional equation for ...
5
votes
0answers
217 views

How to simplify $\newcommand{\bigk}{\mathop{\vcenter{\hbox{K}}}}\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_k(s)}{g_k(s)}\right)^{-1}$

I'd like to simplify $$\newcommand{\bigk}{\mathop{\huge\vcenter{\hbox{K}}}}B(s)=\prod_{p\in\mathbb{P}}\left(1+\bigk_{k=1}^{\infty }\frac{f_{k}(s)}{f_{k}(s)}\right)^{-1}$$ to something of the form ...
13
votes
1answer
488 views

challenging integral involving $\zeta(5)$

I ran across a curious integral that seems to be rather tough that some on the site may enjoy. Show that $$\displaystyle \int_{0}^{1}\frac{\sqrt{1-x^{2}}}{1-x^{2}\sin^{2}(x)}dx = ...
5
votes
2answers
827 views

An identity involving the Möbius function

$$\sum_{n=1}^{\infty}\frac{1}{n^s}\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}=1$$ for $s>1$. How do I prove this identity?
3
votes
1answer
371 views

Why are Gram points for the Riemann zeta important?

Given the Riemann-Siegel function, why are the Gram points important? I say if we have $S(T)$, the oscillating part of the zeros, then given a Gram point and the imaginary part of the zeros (under the ...
10
votes
2answers
550 views

How to find $\zeta(0)=\frac{-1}{2}$ by definition?

I would like to know how we can find the following result: $$\zeta(0)=-\frac12$$ Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$ to find this?
3
votes
0answers
356 views

Can the openness of Gilbreath's conjecture be reduced to proof of the Riemann hypothesis?

As an information theoretician, it's a personal hobby of mine to find elegant analogies to open mathematical problems. After all, they have a profound impact on how I conduct my research and how I ...
19
votes
5answers
3k views

The main attacks on the Riemann Hypothesis?

Attempts to prove the Riemann Hypothesis So I'm compiling a list of all the attacks and current approaches to Riemann Hypothesis. Can anyone provide me sources (or give their thoughts on possible ...
8
votes
0answers
316 views

New generalization of Riemann Zeta?

I am interested in the following generalization of the Riemann Zeta function: $$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$ This is most closely related ...
2
votes
0answers
210 views

Why Pólya's method is wrong?

By truncating the Fourier transform, Pólya managed to prove that the Xi function on the critical line was approximately $$\xi(1/2+is) = (2\pi)^2 ( K_{9/4+is/2}( 2\pi) +K_{9/4-is/2}( 2\pi))$$ If this ...
1
vote
1answer
207 views

Can you provide a lower bound on $|\zeta (s) |$ for fixed $\mathrm{Re}(s) > 1$?

It's easy to prove, for example, that $|\zeta(2 + it)| > 2 - \frac{\pi^2}{6}$. However, there is some $\sigma > 1$ for which $\zeta ( \sigma ) = 2$, and it is more difficult to obtain a lower ...
10
votes
2answers
489 views

Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$

Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as $$ D(n) = \sum_{k=1}^{n}d(k) , $$ where $$ d(n) = \sum_{k|n}^{n}1. $$ One can observe the following pattern in the values of ...
15
votes
2answers
1k views

Logarithmic derivative of Riemann Zeta function

Given the logarithmic derivative of the zeta function $\dfrac{\zeta^\prime (s)}{\zeta(s)}$ how does it behave near $s=1$? I mean if for $s=1$ the Laurent series for the logarithmic derivative becomes ...
6
votes
2answers
694 views

Why does the Riemann zeta function have zeros in the complex plane? How is it possible to find them?

I ask this because, according to Euler's product formula, Riemann's zeta function =(1/something), so how could that be zero? Also, how could one find zeros that are on the negative side and find a ...
2
votes
1answer
1k views

How is $\zeta(0)=-1/2$? [duplicate]

Possible Duplicate: Why does $1+2+3+\dots = {-1\over 12}$? Fermat's Dream by Kato et al. gives the following: $\zeta(s)=\sum\limits_{n=1}^{\infty}\frac{1}{n^s}$ (the standard Zeta ...
7
votes
1answer
254 views

Riemann's $\zeta$ function and the uniform distribution on $[-1,0]$

It seems that the $n$th cumulant of the uniform distribution on the interval $[-1,0]$ is $B_n/n$, where $B_n$ is the $n$th Bernoulli number. And also $-\zeta(1-n) = B_n/n$, where $\zeta$ is Riemann's ...
8
votes
1answer
270 views

An infinite series involving the Zeta Function

I am wondering if anyone knows how to evaluate either of the following sums in terms of known constants: $$\sum_{k=2}^{\infty}-\frac{\zeta^{'}(k)}{\zeta(k)},$$ and ...
9
votes
2answers
550 views

Order of Zeros of the Riemann Zeta Function

Is it true that all zeros of the Riemann Zeta Function are of order 1? Let $h(z) = \frac{\zeta'(z)}{\zeta(z)}\frac{x^z}{z}$, where $x$ is a positive real number ($x > 1$, probably?) , and $\zeta$ ...
13
votes
4answers
622 views

Do these series converge to logarithms?

It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$ If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$ Starting: ...
4
votes
0answers
109 views

Order of growth of real $x_{n}$ such that $\zeta(x_{n}) = 1 + 1/2^{n}$

On a lark, I decided to calculate (via Newton's method and using mpmath) the real $x_{n}$ such that $\zeta(x_{n}) = 1 + 1/2^{n}$ for as many $n\in\mathbb{N}$ as I could. What sort of surprised me is ...
12
votes
1answer
428 views

elliptic generalizations of Euler's trick

So Euler employed the following identity $$\sin(z) = z \prod_{n=1}^{\infty} \left[1-\left(\frac{z}{n\pi}\right)^{2}\right]$$ to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$ I'm curious if there's been ...
3
votes
1answer
130 views

Multiplication of coefficients in Dirichlet series

This appears to be a relationship: $\sum\limits_{p\;\text{prime}} \frac{1}{p^s} = \log\zeta (s) - \sum\limits_{n=1}^{\infty}\frac{\sqrt{a_{n}b_{n}}}{n^{s}}$ where $a_{n}$ is a sequence of fractions ...
13
votes
3answers
5k views

Analytic continuation- Easy explanation?

Today, as I was flipping through my copy of Higher Algebra by Barnard and Child, I came across a theorem which said, The series $$ 1+\frac{1}{2^p} +\frac{1}{3^p}+...$$ diverges for $p\leq 1$ and ...
5
votes
0answers
376 views

Is this a relation between the Riemann zeta function and the Prime zeta function?

I am partly repeating myself here again. Is this a correct relation between the Riemann zeta function and the Prime zeta function? $$ \zeta (s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^{s}}$$ ...
1
vote
0answers
152 views

Determining well-definedness for functions

How does one determine well-definedness in analytical continuation for $\Gamma(s)\zeta(s)$ function? Firstly: $$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1$$ ...
135
votes
11answers
19k views

Why does $1+2+3+\cdots = -\frac{1}{12}$?

$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$. Why should analytically continuing to $\zeta(-1)$ give the right answer?
9
votes
3answers
944 views

Values of the Riemann Zeta function and the Ramanujan Summation - How strong is the connection?

The Ramanujan Summation of some infinite sums is consistent with a couple of sets of values of the Riemann zeta function. We have, for instance, $$\zeta(-2n)=\sum_{n=1}^{\infty} n^{2k} = 0 ...
2
votes
0answers
77 views

L-function of product

Given two varieties of finite type over a finite field, what is the $L$-function of their product in terms of the $L$-function of the factors?
10
votes
2answers
542 views

Integrating $\frac{x^k }{1+\cosh(x)}$

In the course of solving a certain problem, I've had to evaluate integrals of the form: $$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x $$ for several values of k. I've noticed that that, for k a ...
8
votes
1answer
263 views

What is known about the pattern for $\zeta(2n+1)$?

Related to the question Does $\zeta(3)$ have a connection with $\pi$?: It is well known that $$\zeta(2n) = f(2n) \pi^{2n}$$ where $f(n)$ is an function in rationals: (the denominator = OEIS ...
33
votes
5answers
2k views

Does $\zeta(3)$ have a connection with $\pi$?

The problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)? Details Several ...
24
votes
2answers
619 views

A double series yielding Riemann's $\zeta$

Can you give me some hints to prove equality: $$\sum_{m,n=1}^{\infty} \frac1{(m^2+n^2)^2} =\zeta (2)\ G-\zeta(4)=\frac{\pi^2}{6}\ G-\frac{\pi^4}{90}$$ where $\zeta (t):= \sum\limits_{n=1}^{+\infty} ...
48
votes
8answers
8k views

Nice proofs of $\zeta(4) = \pi^4/90$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute ...
10
votes
3answers
441 views

Erroneous numerical approximations of $\zeta\left(\frac{1}{2}\right)$?

By definition of the Riemann Zeta Function, $$\zeta\left(\frac{1}{2}\right) = \sum_{n=1}^\infty \frac{1}{\sqrt{n}}.$$ Since $\forall n \geq 1 : \frac{1}{\sqrt{n}} \geq \frac{1}{n}$, we have that for ...
5
votes
1answer
329 views

Square-free zeta function zeros

It is a well known fact that the geometric series $$1+x+x^2+x^3+\ldots$$ has the following form $$\frac{1}{1-x}$$ Another possible representation is $$\prod_{k=0}^{\infty}\left(1+x^{2^{k}}\right)$$ ...
8
votes
2answers
568 views

Trig identity to zeta function identity?

The inequality $$\zeta(s)^3 | \zeta(s + it)^4 \zeta(s + 2it)| \ge 1$$ follows from $$3 + 4 \cos(\theta) + \cos(2 \theta) \ge 0$$ How is that done? What is the relationship between zeta and the ...
37
votes
2answers
1k views

Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$

Some time ago when decomponsing the natural numbers, $\mathbb{N}$, in prime powes I noticed a pattern in their powers. Taking, for example, the numbers $\lbrace 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 ...
3
votes
2answers
400 views

Is it problem of Mathematica or my own?

The following is a plot comparing Exp[Derivative[1,0][Zeta][0,x]+1/2Log[2 Pi]] and Gamma[x]: In theory the blue and the red ...
20
votes
4answers
3k views

Riemann zeta function at odd positive integers

Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the ...
5
votes
2answers
1k views

Riemann Zeta Function and Analytic Continuation

The Riemann Zeta Function is defined as $ \displaystyle \zeta(s) = \sum\limits_{n=1}^{\infty} \frac{1}{n^s}$. It is not absolutely convergent or conditionally convergent for $\text{Re}(s) \leq 1$. ...
9
votes
3answers
724 views

An inverse for Euler's zeta function product formula

Of course, Euler proved that the Riemann zeta function can be defined as the analytic continuation of a product over all primes. $$\zeta(s) = \prod_{p \in \mathbb{P}}\frac1{1-p^{-s}}$$ It is well ...