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Questions tagged [riemann-zeta]

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

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An analytic doubt from Riemann zeta function-Titchmarsh

Doubt form the Book Theory of Riemann Zeta Function by Titchmarsh I've problem on Theorem $5.8$ in the equation $5.8.2$. When $k=l$ the from Lemma $5.7$ we get, $$ \sum_{n=N+1}^bn^{-it}=O\left(N^{1-1/...
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I think I am gonna get answer here faster sorry for bothering but can you answer me this pls? [on hold]

$$\zeta (\frac{1}{2})=\sum_{x=1}^{\infty }\frac{1}{\sqrt{x}}=?$$ Asking this question btw because I am considering the infinite series that looks like this: $$S = \sum_{x=1}^{\infty} \frac{1}{{x^{bi} \...
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Question related to derived formula for zeta-zero counting function

I've been attempting to derive a zeta-zero counting function based on the distributional or Fourier series representation of the second Chebyshev function $\psi(x)$ or its first-order derivative $\psi'...
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Riemann-zeta Function Evaluated at $\zeta(0)$ [duplicate]

WolframAlpha says that $\zeta(0) = - \frac{1}{2}$ but I can't seem to get that result. I found that for $\Re(s) < 1 $, \begin{equation}\label{1} \zeta(s) = 2^s \pi^{s-1}\sin\Bigl(\frac{s\pi}{2}\...
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What would be the integral of the zeta function or $ \sum\limits_{n=1}^{\infty} \frac {1}{n^x} $?

The zeta function is defined as: $$ \zeta (x) = \sum\limits_{n=1}^{\infty} \frac {1}{n^x} $$ Does an integral of this function exist? If it does then what would it be? More information about zeta ...
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Where can I find papers and research on the Alternating Hurwitz Zeta Function?

The function is as follows (I don't know it's name, it could be 'Generalised Dirichlet Eta Function') $$f(s,q)=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(n+q)^{s}}$$
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Asymptotics of the reciprocal Riemann Zeta Function

Assuming Riemanns hypothesis, I would like to obtain an upper bound on $$\left|\frac{1}{\zeta(\sigma+it)}\right|$$ for large $t$ and fixed $\sigma$. I believe it should be easy to show that it ...
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A query regarding the non-trivial zeroes of Zeta function?

The zeta function in its functional form is described by: $$\zeta(s) = 2^s\pi ^{s-1}\sin\left( \frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$$ We know that the zeroes of the zeta function are ...
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The Meaning of R's non-Trivial Zeros

(I have read through the various similar questions on SE listed by the system but not found an answer that helps). Is there an intuitive explanation for why the Riemann zeta Function (rather than ...
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Fourier Transform of the Riemann zeros (Dirac comb)?

Lets assume RH and $\rho_i, i\in\Bbb N$ be the imaginary parts of the non-trivial zeros of the Riemann $\zeta$ function: $\zeta(\frac{1}{2}\pm\imath \rho_i)=0$, $(\forall i)$. Does anonye know if ...
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Fixed point of Riemann Zeta function

I have been looking for fixed points of Riemann Zeta function and find something very interesting, it has two fixed points in $\mathbb{C}\setminus\{1\}$. The first fixed point is in the Right half ...
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Why isn't $\zeta(3) = 2^3\pi ^{3-1}\sin\left( \frac{\pi 3}{2}\right)\Gamma(1-3)\zeta(1-3) = 0$?

The zeta function (for $\Re(s)>1$) is given by the definition: $\zeta(s) = \sum _{n=1}^{\infty} \left [\frac{1}{n^s} \right]$ The zeta function also can be given by the following functional ...
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reference request - moments zeta function/dirichlet polynomials

I would like to study some material on the moments of the Riemann Zeta function and Dirichlet polynomials (mean value theorems). I was looking both for some introductory material and for some more ...
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How do we know the gamma function and Riemann zeta function combine in such a nice way?

Let $\zeta(s) = \prod\limits_p (1 - p^{-s})^{-1}$ be the Riemann zeta function. If we define $L(s) = \pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$, then $L$ admits a meromorphic continuation to the ...
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Integral $\int_{-\infty}^{\infty} \frac{\sinh(x)}{x [a+\cosh(x)]^2}dx$

I have difficulties with calculating the following integral: $$I(a)=\int_{-\infty}^{\infty} \frac{\sinh(x)}{x [a+\cosh(x)]^2} \mathrm dx~~~~~~~,\text{where } a>1$$ For the case with $a=1$ the ...
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Can the Riemann Explicit Formula be used to find prime numbers?

It is well known that there is a strong link between the Riemann Hypothesis and the distribution of primes. The prime number theorem gives the number of primes less than or equal to a given $N$ as: ...
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Need help showing Riemann's Functional equation for negative numbers and complex numbers

Riemann's Functional equation: $\zeta(-z)$=${-2*z!\over(2\pi)^{z+1}}$$sin({\pi z\over2})$$\zeta(z+1)$This formulas expresses $\zeta(-z)$ in terms of $\zeta(z+1)$ Note: I read that the author said, ...
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Is there something like the Busy Beaver Function for uncountable classes? [closed]

Any statement P over a set with a countable number of elements that can be disproven by a counterexample can be proven by testing a finite number of cases. This can be done by encoding P an n-state ...
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Why is it impossible to invert the analytic continuation of a Dirichlet series?

By Mathematica (and the truncated Euler MacLaurin formula) I know that: $$\zeta(s)=\lim_{k\to \infty } \, \left(\sum _{n=1}^k \frac{1}{n^s}+\frac{1}{(s-1) k^{s-1}}\right) \tag{1}$$ when the real part ...
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First order approximation of $\zeta$(s) at s=1

I was playing around with Wolfram Alpha. I found one interesting thing when I asked it to evaluate this particular summation.$$\Sigma_{n=1}^\infty\frac{1}{n^{1+10^{- 10}}}$$ It returned this$$ \...
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Big Oh of values of Riemann zeta function

There is a equality in a proof in Apostol's Analytic Number Theory as follows: $O(x^{\alpha} \zeta(\alpha)) = O(x^{\alpha})$ for arbitrary real number $\alpha \ge 0$. How do we say that? Does ...
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Symmetry of zeros in the critical strip for Riemann Hypothesis

If it can be proven that there are no zeros for real values greater than $0.5$ in the critical strip, does this prove that there are no zeros in the critical strip having a real value of less than $0....
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Line of Symmetry of the Zeta Function

I heard once that 0.5 is the line of symmetry of the Riemann Zeta Function. What does that mean? A graph illustrating would be helpful.
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Limit of an alternating series

Given some fixed integer $n$ (but one may take $n = 0$ for simplicity), I would like to compute the following limit: $$\lim_{N \to \infty}\sum_{\substack{|u_1|, |u_2|, |u_3|, |u_4| \leq N\\|u_1 + u_2 ...
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Questions related to the Dirichlet series for $\frac{\zeta'(s)}{\zeta(s)^2}$

This question is related to the following two functions evaluated with the coefficient function $a(n)=\mu(n)\log(n)$. (1) $\quad f(x)=\sum\limits_{n=1}^x a(n)$ (2) $\quad\frac{\zeta'(s)}{\zeta(s)^2}=...
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Proof claimed that almost all zeroes of the Riemann zeta function lie on the critical line

There have been many results in recent years on the natural density of zeta zeroes on the critical line, with the best bound commonly accepted (as far as I'm aware) that $$\liminf_{t\to\infty} \frac{...
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What is the relationship between simple prime-power counting function and $\log\zeta(s)$?

This question assumes the following definitions of prime-counting functions where $p$ denotes a prime number, $k$ denotes a positive integer, and $\theta(y)$ is the Heaviside step function which takes ...
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Why should the series representation of the zeta function know about its analytic continuation?

In physics, when we calculate the renormalized sum of $S=\sum_{n=1}^\infty n$, we usually use an exponential regulator and instead first calculate $$S_\epsilon = \sum_{n=1}^\infty ne^{-\epsilon n} = ...
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Bound for the number of roots $\rho$ of $\xi(\rho)$

I was reading the book Riemann's Zeta Function, by H. M. Edwards, page 42, where is a theorem that estimates the number of roots of the $\xi$ function $$\xi(s)=\Gamma\Big(\frac{s}{2}+1\Big)(s-1)\pi^{-...
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If I have a polynom $P(x)$, which zeros have the absolute value $q^{-(\frac{n-1}{2})}$. Why is this an accord to the Riemann hypothesis?

You can read the question above. So I'm really " new in terms of Riemann hypothesis". I have read about the hypothesis in wikipedia. So I know the statement of the Hypothesis : The Riemann Zeta ...
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Since $\zeta(1) \neq 1$, does this mean that $\zeta$ is not multiplicative?

Let $\zeta(s)$ be the Riemann zeta function, that is, $$\zeta(s) = \sum_{n=1}^{\infty}{\frac{1}{n^s}}.$$ A function $g$ is said to be multiplicative if, whenever $\gcd(x,y)=1$, we have $$g(xy) = g(x)...
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What makes Riemann hypothesis so much harder to prove than its analogue for curves over finite fields

The analogue of the Riemann hypothesis for curves over finite fields has been shown by André Weil (see also Roadmap to Riemann hypothesis for curves over finite fields) and further deep results (Weil ...
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General formula for $\sum_{n=0}^\infty \frac{1}{(an+b)^c}, a> 0, b>0, c>1$

Is there a general formula for this sum? $$I(a,b,c)=\sum_{n=0}^\infty \frac{1}{(an+b)^c}, a> 0, b>0, c>1$$ I noticed that the first few a, b, and c values yielded seemingly different results, ...
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A question on the approximation of $\ln(x!)$ for small x

In a question of Arfken and Weber's Mathematical Methods for Physicists, For small values of $x$, $\ln(x!)=-\gamma x+\sum\limits_{n=2}^{\infty}(-1)^n\frac{\zeta(n)}{n}x^n$, where the symbols used ...
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Condition on convergence of an integral

I have solved the question. But I have problem with the condition on n. This is what I think. RHS converges if $n>1$(because of Zeta function). For LHS to converge, the given function must tend ...
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Trying to understand a certain form of zeta function

A month ago I have asked a question about a certain form of the zeta function. I will now try to be more accurate. facts: Let $N_s$ be the number of points on the projective hypersurface $\bar{H}_f(...
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Closed form for $\sum\limits_{n=1}^{\infty}\frac{O_{n}^{(p)}}{(2n-1)^{q}}$, with $O_{n}^{(s)}=\sum\limits_{k=1}^n\frac1{(2k-1)^{s}}$

Consider the sum $$\sum_{n=1}^{\infty}\frac{O_{n}^{(p)}}{(2n-1)^{q}}\text{, with }O_{n}^{(s)}=1+\frac{1}{3^{s}}+\dots+\frac{1}{(2n-1)^{s}}$$ My question is: if there exists some general theorems ...
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Generalizing $\sum\limits_{m\geq1}\sum\limits_{n\geq1}\frac{(-1)^n}{n^3}\sin(n/m^{2k})=\frac1{12}\zeta(6k)-\frac{\pi^2}{12}\zeta(2k)$

I am trying to generalize the fact that, for $k>\frac12$, $$\sum_{m\geq1}\sum_{n\geq1}\frac{(-1)^n}{n^3}\sin(n/m^{2k})=\frac1{12}\zeta(6k)-\frac{\pi^2}{12}\zeta(2k)$$ To reach this I start off ...
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A strange type of complex root

This function is related to the zeta function for $-7.$ Here is a function which has $2$ complex roots at $\frac{1}{2}$ which I find very strange. https://www.wolframalpha.com/input/?i=Roots+n%5E2%...
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Extending Riemann's zeta function to $0<s<1$

This is in reference to baby Rudin's Ch 6 exer 16: He defines there $\zeta(s)=\sum_1^\infty\frac1{n^s}$ for $1<s<\infty$, and then asks to show that $$\zeta(s)=\frac s{s-1}-s\int_1^\infty \frac{...
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Uniform convergence of $ \zeta'(x) = \sum \frac{\ln(k)}{k^x}$

As a part of a proof that the $\zeta$ function is differentiable I want to check that the series: $$-\sum_{k=1}^\infty \frac{\ln(k)}{k^x}$$ Converges uniformly for $x \in (a, \infty)$ whenever $a>1$...
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Are there known zeros of the Zeta function off the line 1/2?

I've been doing research in information theory that had some superficial connections to the Zeta function, since I was making use of logarithms of complex numbers. In short, we'd be looking at the ...
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Why $\zeta(\exp(s)+\exp(1-s))$ for $s=0.5+it $ should be reals?

I have got a simple complex valued function behave more like zeta function for $(s)=0.5$ which is defined as :$f(s)=\exp(s)+\exp(1-s))$ vanish at $s=0.5+i (2n+1)\frac{\pi}{2}$ with $n$ is integer and ...
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Rearrangement of summations in series expansion of $-\log(\zeta(s))$

This is a step in Titchmarsh's The theory of Riemann Zeta functions pg 2 of Chpt 1 eq 1.13 which I do not find obvious, though I can prove it directly. Assume $\sigma=Re(s)>1$. Denote by $\pi(x)$ ...
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Is this equivalent to the Riemann Hypothesis?

By a result of Spira, we know that the Riemann Hypothesis (RH) is equivalent to the statement that $|\zeta(1-s)|$ increases as $\Re(s)$ varies on $(\frac{1}{2}, \infty)$ with $|t|=|\Im(s)|\geq 165$ ...
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Evaluating $\frac1{3^k}+\frac1{6^k}+\frac1{10^k}+\cdots$

The series $\frac1{3^k}+\frac1{6^k}+\frac1{10^k}+\cdots$ for integers $k>1$ at the triangular numbers can be written as $$\sum_{i=2}^\infty\frac1{\left(\frac{i(i+1)}2\right)^k}=2^k\sum_{i=2}^\infty\...
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1answer
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Why is the Euler product expected to play a role in a solution of the Riemann Hypothesis?

The Riemann Hypothesis is the statement that the Riemann zeta function $\zeta(s)$ does not vanish for $1/2<\Re(s)<1$. $\zeta(s)$ can also be expressed by the Euler product over primes $$\zeta(s)=...
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Is there a connection between $\zeta(-1)$ and Ramanujan's calculation of the sum over $\mathbb{N}$?

Let me elaborate a little on the matter that I've been mulling over for a little while. This essentially concerns the summation of $1+2+3+...$, how it equals $-1/12$ (in a certain sense, obviously not ...
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An observation on Non-Trivial Zeros of Riemann Zeta Function.

I observed this property in month of July this year but unable to design a mathematical proof or mathematical way to state my observation. I need help to state this property. We know that for certain ...
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Using Euler's equation to calculate integral of $\zeta$ over the curve $Re(\zeta(t)=0)$ between first two zeros?

Is Euler's equation the right equation to use if I want to calculate a numerical value of $\int_{\{ {Re} (\zeta (t)) = 0 : 14.134 \ldots < {Im} (t) < 21.022 \ldots . \}} \zeta (t) d t$ It ...