Questions tagged [riemann-zeta-function]

The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.

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74 views

Dynamics of extrema of the Landau Riemann $\xi(1/2+it)$ function

"Relations and positivity results for derivatives of the Riemann xi function" by Coffey characterizes the Riemann xi function and its relation to the Riemann zeta function, and there has been much ado ...
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Are there any extensive treatments on rational zeta series?

I've been trying to find an extensive, in-depth treatment of rational zeta series. Via the Wikipedia article on the topic, I've found two articles on this subject. While they are certainly very ...
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Is regularization of infinite sums by analytic continuation unique?

There are ill-posed summations that we can assign values to, take for concreteness, $$ S = \sum_{k=0}^\infty k $$ to which we can assign $-1/12$ by several methods. Is there a fundamental and rigorous ...
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Not sure about meaning of a term in English in a French research paper

I am self studying a research paper which is in French and i am not a native french speaker so I used Google Translator and Deepl translator . But I am confused over meaning of a term and have no ...
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Efficient boxing for a mean value in the Bombieri Iwaniec method

One of the nice applications of decoupling is Bourgain’s record towards Lindelöf: https://arxiv.org/pdf/1408.5794.pdf Wooley has developed some techniques known as efficient congruencing which allow ...
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Does this series, related to the Hasse/Ser series for $\zeta(s)$, converge for all $s \in \mathbb{C}$?

I have asked this question at math stack exchange, however it did not get any traction. Still curious about the answer though. Numerical evidence suggests that: $$\lim_{N \to +\infty} \sum_{n=1}^N\...
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On the Absolute Value of the Riemann Zeta Function on the Critical Line [closed]

Is there any sort of (closed form preferably, though if not, it's fine) function for $|\zeta(\frac12+it)|$ where $\zeta$ is the Riemann zeta function? Anything is welcome, so I can take it from there. ...
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What is the dirichlet series of $f(n)=\sum_{d | n}(\log d) / d$ function? [closed]

My opinion is ; We may use id(d)=d arithmetic function and log*id dirichlet convolution in the question. i thought that ; when we multiply and divide n with $(\log d) / d$ we obtain $F(S)=\sum_{n=...
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On the asymptotics of the Chebyshev psi function

Denote by $\psi(x)$ the Chebyshev psi function over prime powers. Assuming the RH, it can be shown via the Riemann explicit formula that $$\psi(x)-x \ll √x |\sum_{|\gamma| < T} \frac{1}{\rho} | +...
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Understanding a deduction in research paper of Sprang, Fischler and Zudilin (“Many Odd zeta values are irrational”)

I am a master's student interested in number theory. I am reading a research paper in Analytic Number theory which is "Many Odd zeta values are irrational" by Stephane Fischler, Johannes Sprang and ...
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Unable to deduce an inequality in paper on odd zeta values of Fischler, Sprang and Zudilin

I am a masters student and I am interested in number theory. Due to lockdown I have a lot of time and I thought of reading a research paper in Number theory which is " Many Odd zeta values are ...
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What is the closed form of this integral?

Consider the Chebyshev first function $\psi(y):=\sum_{p^j \leq y} \log p$, where $p$ is a prime. Define $$F(s, k) = s\int_{1}^{\infty} \psi(x + x^k)x^{-s-1} \mathrm{d}x$$ for $ \Re(s) >$ max $(1, ...
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Explicit Formula for $n$th prime in terms of Riemann zeros?

We all know there exists a explicit Formula for prime counting function in terms of Riemann zeros. I'm wondering if similar formula exists for $n$th prime in terms of Riemann zeros?
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Question about proof of irrationality of $\zeta(3)$ [closed]

I'm reading this article of Henri Cohen about Apery's proof of the irrationality of $\zeta(3)$ but I don't really get the details of "THEOREME 1". My first doubt is about the relation $a_n \sim A \...
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Unable to think the reasoning behind an argument in one of Lemmas in research paper of T.Rivoal and K.Ball

I am reading this research paper in analytic number theory by myself "Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs Keith Ball & Tanguy Rivoal ". In lemma 4 on ...
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If $\zeta(s)=0$ with $\Re(s)=\frac{1}{2}$, is then $|\hat{\zeta}(s,3)|^2=\frac{1}{2}$?

Helmut Hasse has proved that for $s \in \mathbb{C}-\{1\}$ the Riemann zeta function can be written as: $$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=0}^\infty\frac{1}{2^{n+1}}\sum_{k=0}^n(-1)^k\ {n \choose k}...
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Can this quantity be expressed as $x\cdot \zeta(k)+y, x,y \in \mathbb{Q}$?

For each natural number $a$ consider the sequence $l(a):=\left(\frac{\gcd(a,b)}{a+b}\right)_{b \in \mathbb{N}}$. Then I have computed that for $k\ge 2, k \in \mathbb{R}$ and $p$ prime, we have: $$|l(...
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Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$

(Question is short and straight-forward. ) What is/are "nice and non-trivial" series representation/s of $\log(|\zeta(\frac{1}{2}+it)|)$ ?? By "nice and non-trivial" I mean contains no ...
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Unable to understand an argument involving integratibility of a research paper of T. Rivoal

I am self studying a research paper in analytic number theory (Ball and Rivoal - Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs) and I am unable to think about an ...
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A question on the Riemann zeta function

Yesterday, a certain very talented and passionate young student from Southern Africa asked me the following question about the Riemann zeta function $\zeta(s)$. He says he "thinks" he knows the answer,...
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Any simplification of this inequality if it is true? :For $t\geq 1.22$: $|\zeta(0.5+it)|\leq 0.5 \frac{|\Gamma(0.5+it)|}{|\Gamma(-0.5+it)|}$

When I tried to give bounds for $\zeta(0.5+it)$ using some transformations over Gamma function using the function $f(x)=\exp(-n x)$ over the range $(0,+\infty)$ , For $ Re(s)=\frac12 $ and $t >0$ ...
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if $\max_{z \in K} |\zeta(z+it)-f(z)|<\epsilon.$ then is this $\lim_{t\to \infty} \inf \frac {|\zeta'(z+it)|}{|f'(z)|} $ a finit limit?

Universality theorem of Riemann zeta function states that :Let $K$ be a compact subset with connected complement lying in the strip $\{1/2 < \operatorname{Re}(z)<1\}$, and let $f : K \rightarrow ...
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Does there exist a rational polynomial $P(x)\in{\mathbb Q}[x]{}$ such that $P(\zeta(s))=\zeta(P(s))$?

let $P(x)\in{\mathbb Q}[x]{}$ be a rational polynomial with $P(1) >1$ and $\zeta $ be the Riemann zeta function , I want to know if there exist a rational polynomial such that $P(\zeta(s))=\zeta(P(...
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Unable to think how to prove a result related to prime number theorem in research paper of T.Rivoal and W.Zudilin

I am asking this question on MO as earlier it was asked on MSE (link below). I even put a bounty on it and waited but noone answered. So I am posting it here as I have no option and I am unable to ...
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A line integral involving $\zeta(s)$

Let $x>1$ and $\zeta$ denote the Riemann zeta function. Is the equality $$\int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\zeta(s)x^{s}}{s(s+1)(s+2)\cdots (s+k)} \mathrm{d}s = 2\pi i$$ possible ...
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About generalized binomial theorem and Grünwald-Letnikov fractional derivative

I have run into a problem while computing the fractional derivatives of order $\alpha$ for the Riemann zeta function. My Theorem states Let $s\in\mathbb{C}$, $\mathfrak{Re}(s)>1$, then the ...
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What are the best unconditional bounds for the discrete moments of $\zeta'$?

I heard this morning through the app Researcher about the following article by Scott Kirila: arXiv:1804.08826v2 in which the author proves under RH that $J_{k}(T)\asymp_{k}(\log T)^{k(k+2)}$, where ...
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Proportionality constant for the growth rate of $\zeta\left(s\right)$ along the imaginary axis

Going through what it says on page 95 of Titchmarsh's book on the Zeta function, and using his remark about $\mu$ at the bottom of that page, I conclude that there is some constant $C>0$ so that: $$...
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On some integral involving the Liouville sum

Define $L(x) = \sum_{n\leq x} \lambda(n)$, where $\lambda$ denotes the Liouville function. If $c$ is the supremum of the real parts of the zeros of the Riemann zeta function and $\lfloor x\rfloor$ ...
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Better trigonometrical inequalities for $\zeta(s)$?

The inequality $$3 + 4 \cos \theta + \cos 2 \theta \geq 0$$ plays a key role in the proof of the classical zero-free region of the Riemann zeta function. Are there other inequalities of the form $$\...
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A zeta function for the Klein Four group?

Let $G$ be a finite group, $S \subset G$ a generating set, $|g|:=|g|_S=$ word-length with respect to $S$. Let $\phi(g,h)=|g|+|h|-|gh| \ge 0$ be the "defect-function" of $S$. The set $\mathbb{Z}\times ...
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Is finding positive integer solutions of $\zeta(a/b) = c$ equivalent to deciding the rationality of $\gamma$?

This question requires little bit of explanation of the background hence it is a bit lengthy. Note: The question was initially posted in MSE but did not get answers hence posting in MO. For every ...
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Second moment estimates for $\zeta(s)$: different methods?

What are some different ways to achieve the bound $$\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt = T \log \frac{T}{2 \pi} + (2 \gamma - 1) T + E(T)$$ with an error term $E(T) = O(T^{\...
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Fully explicit version of Atkinson's formula?

Let $$I(T)=\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt$$ and let $E(T)$ be $I(T)$ minus what turn out to be its main terms: $$E(T) = I(T)- T \log \frac{T}{2 \pi} - (2 \gamma - 1) T.$...
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Error term when truncating series for $1/\zeta(s)$

Let $s=\sigma+it$, $0\leq \sigma\leq 1$, $|t|\geq 1$, say. Using Euler-Maclaurin, one can easily show that, for $x\geq |t|$, $$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O\left(\...
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Why was the factor $\frac12$ introduced in the Riemann $\xi$ function?

The factor $\frac12$ in the Riemann $\xi$ function: $$\xi(s)=\frac12 s(s-1)\,\pi^{-s/2}\,\Gamma(s/2)\,\zeta(s)$$ was introduced by Riemann, however appears to be redundant. Once he had arrived at: ...
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A new way of approaching the pole of the Riemann zeta function - and a new conjectured formula

On the Wolfram page about the Euler-Mascheroni Constant $\gamma $, the following amazing limit is given without proof (referring to "personal communication"): $$\lim_{z\to\infty}\left[\zeta(\zeta(z))-...
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sum involving Riemann zeta function

In my work arose the following series: $$g(s) = \sum_{n = 0}^\infty \frac{\log(\zeta(n+2))}{n+2}s^{n}.$$ It has radius of convergence $2$ and converges for $|s| \leq 2$ except at $s = 2$. I'm ...
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Looking for a fine exposition of a result of Littlewood

I recently asked for the original journal citation on Littlewood's result $$ \lim\limits_{n\to\infty}|\gamma_n-\gamma_{n-1}| =0 ~~,$$ wherein $\gamma_n$ is an increasing sequence of the imaginary ...
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Looking for a citation for a result of Littlewood

I see that it is proven by Hardy in 1914 that there are an infinite number of zeros on the critical line. I also see that the Hardy and Littlewood conjectures appear in some papers they wrote ...
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Largest observed value of $S(t)$

Let $S(t)$ be the deviation of the number of zeros of the Riemann zeta function up to height $t$ from the expectation. What is the largest observed value of $S(t)$ today? Here is a quote from a ...
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Bounding $1/\zeta(s)$ given RH

Let $T\geq 0$. Assume RH(T+100), that is, assume that all non-trivial zeros $\rho$ of the Riemann zeta function with $|\Im(\rho)|\leq T+100$ satisfy $\Re(\rho)=1/2$. Can we then give a good upper ...
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Subsets of particular values of $\zeta'(k)$ that contain irrational numbers

We consider the set of elements $\zeta'(2),\zeta'(3),\zeta'(4),\zeta'(5),\ldots$ where $\zeta(z)$ is the Riemann zeta function and $\zeta'(z)=\frac{d}{dz}\zeta(z)$ its derivative. Thus we consider ...
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Multiple zeta values related to fractional calculus and an Appell polynomial sequence

There is an Appell sequence of polynomials $p_n(z)$ related to an infinitesimal generator for one rep of the fractional calculus that have coefficients involving the Riemann zeta function values at ...
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Can this infinite sum for the Riemann zeta function be generalised?

I recently derived the following identity (which is probably a rediscovery of something well-known to experts). $$\sum_{k=1}^\infty{\frac{(-1)^{k+1}k^{4n+1}}{e^{k \pi}-(-1)^k}}=\frac{1}{2}\zeta\left(-...
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Does $\int_{2}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$ converge on the real axis for $s>1/2$? [closed]

Consider the prime zeta function, defined for $\Re(s)>1$, by the infinite series $$\sum_{p} p^{-s} = \sum_{m=1}^{\infty}\frac{\mu(m)}{m}\log \zeta(ms)$$ where $p$ denotes a prime, $\mu$ the Mobius ...
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The complex roots of $\left(\Gamma(\kappa s)+2^{1-s} \pi^{-s}\cos\left(\frac{\pi s}{2} \right)\Gamma(s)\,\Gamma(\kappa(1-s))\right)$

Question: With $\kappa \in \mathbb{R}, \kappa \ge 1$, the function: $$\Upsilon(s,\kappa) = \Gamma(\kappa s)\zeta(s)+\Gamma\left(\kappa(1-s)\right) \zeta(1-s)$$ could be rewritten as: $$\Upsilon(s,\...
2
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1answer
375 views

Fourier transform of the von Mangoldt function?

Wikipedia states under the entry for the von Mangoldt function: The Fourier transform of the von Mangoldt function gives a spectrum with spikes at ordinates equal to imaginary part of the Riemann ...
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2answers
376 views

On the Dirichlet series for $1/\zeta(s)$ for real $s$ and the zeros of zeta

For $\Re(s)>1$, it is well known that $$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$$ where $\mu$ denotes the Mobius function and $\zeta$ is the Riemann zeta function. I have heard ...
3
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210 views

Riemann hypothesis and ternary Goldbach

Is there any result of the following shape: There exists an absolute constant $\delta>0$ such that the Riemann hypothesis for some $L$-functions is equivalent to the following estimate for all ...

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