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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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On the spacing of the zeros of the Riemann zeta function

Suppose the Riemann zeta function has infinitely many zeros $\rho$ with $\Re(\rho)=\sigma$. Does it follow that for every large $T>0$, there exists some $t$ such that $T<t<3T$, where $t=\Im(\...
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Is the following recursion formula for $\zeta(2n)$ known?

I have discovered (and found an elementary proof of) the following $$\zeta(2k)=(-1)^{k-1}\dfrac{\pi^{2k}}{2^{4k}-2^{2k}}\left[k\dfrac{2^{2k}}{(2k)!}+{\displaystyle \sum_{l=1}^{k-1}(-1)^{l}\dfrac{2^{2k-...
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270 views

How often does the Mertens function vanish?

It is well known that the Mertens function $$M(x)=\sum _{n\leq x}\mu(n)$$ has infinitely many zeros, and this seems to be a short proof. Are there known results about how often the Mertens function ...
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1answer
260 views

How large can $|\zeta(\sigma + it)|$ be for $\sigma<1/2$?

Let $\zeta$ be the Riemann zeta function. My question is: For fixed $\sigma<1/2$, how large can $|\zeta(\sigma+it)|$ be for $t\in \mathbb{R}$, even assuming zeta conjectures like the RH or the LH ?...
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1answer
168 views

Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for finite fields?

The Balazard-Saias-Yor criterion for the Riemann Hypothesis states that the latter is equivalent to the statement that $$\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s^2|}|ds|=0$$ where $\zeta$ denotes ...
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199 views

On a certain representation of the Riemann zeta function

Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation $$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
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1answer
544 views

An integral involving the argument of the Gamma function and the Riemann Hypothesis

Evaluate $$I=\int_{0}^{\infty} \frac{t\arg \Gamma(\frac{1}{4}+\frac{it}{2})}{(\frac{1}{4}+t^2)^2}\mathrm{d}t$$ where $\Gamma(s)=\int_{0}^{\infty}e^{-x}x^{s-1}\mathrm{d}x.$ Note that $I$ converges ...
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1answer
167 views

Inquiry on the bound for $\int_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$

Let $\zeta$ be the zeta function of Riemann. Is the bound for $$I_{T}=\int_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$$ known ? It seems to me that $I_{T} \ll T\log T$ since $\log|\zeta(...
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Upper bound of $\zeta$-function on critical strip

How can I determine any upper bound for $|\frac{\zeta^4(s)}{\zeta(2s)}+d\cdot\zeta^2(s)|$ on the critical strip $s=\frac12+it$ for an integer $d$?
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A computation in Conrey's paper on Riemann zeta function

I am reading the Conrey's paper "More than two fifths of the zeros of the Riemann zeta function are on the critical line" (see here). I have question/doubt in a particular step: In P.10, it claimed ...
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A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$

By use of the Riemann functional equation, it can be shown (see corollary 10.5 of Montgomery-Vaughan) that $$|\zeta(\sigma + it)| \asymp |t|^{\sigma-1/2}|\zeta(1-\sigma - it)|$$. where $\zeta$ ...
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3answers
209 views

Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$

Denote by $\zeta$ the Riemann zeta function. It is known that $$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$ But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...
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123 views

On a certain representation for the Riemann zeta function in Montgomery-Vaughan

On page 338 of Montgomery-Vaughan's ''Multiplicative Number Theory'', there is a somewhat nice representation for the Riemann zeta function. That is, let $0<\delta\leq 1/2$. Then one has $$\zeta(1/...
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193 views

Is there an analogue for the Balazard et al criterion for the Generalised Riemann Hypothesis?

A nice result of Balazard et al says the Riemann Hypothesis is equivalent to the statement that $$\int_{-\infty}^{\infty} \frac{\log|\zeta(1/2 + it)|}{\frac{1}{4}+t^2} \mathrm{d}t=0$$ where $\zeta$ ...
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Is there a (Riemann) explicit formula for $\sum_{p\le x}\frac{1}{p}$ involving a sum over the non-trivial zeros ρ of the Riemann zeta function?

Let $f(x)=\sum_{p\le x}\frac{1}{p}$ and $f_0(x)=\frac{1}{2}(f(x+0)+f(x-0))$. Then is there a (Riemann) explicit formula for $f_0(x)$ involving a sum over the non-trivial zeros ρ of the Riemann zeta ...
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271 views

Witten zeta function v.s. Riemann zeta function

From a talk, we learned that The symplectic volume of the space $M$ of gauge equivalence classes of flat G connections is given by the “Witten zeta function”: where we sum over irreducible ...
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1answer
221 views

Enquiry on an equality involving the Riemann zeta function

Let $\zeta$ denote the Riemann zeta function. Does there exist a $t\geq 0$ such that $$\Re(1/4 + t^2)\zeta(1/2 + it)=2t\arg \zeta(1/2 + it) + 2(1/4 + t^2)\ ?$$
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1answer
140 views

Closed form for an integral involving the Riemann zeta function at the critical line

After seeing this question $L_2$ bounds for $\zeta(1/2 + it)$ and a related integral i became curious if/how the approach in the answer by reuns can be applied to evaluate $$I_{a,b}=\int_{-\infty}^{\...
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1answer
226 views

$L_2$ bounds for $\zeta(1/2 + it)$ and a related integral

Denote by $\zeta$ the Riemann zeta function. I just learnt from this question $L_2$ bounds for tails of $\zeta(s)$ on a vertical line that $\int_{T}^{\infty} \frac{\zeta(1/2 + it)}{1/4 + t^2}\mathrm{...
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108 views

On some inequality involving the Riemann zeta function integral at $\Re(s)=1/2$

I recently saw on p.$458$ of Montgomery-Vaugahn's ''Multiplicative number theory'' that the inequality $$\int_{1}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2})$$ holds uniformly for $T\geq 2$, ...
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188 views

Does the Riemann Xi function possess the universality property?

Here is the question.   Does the Riemann Xi function possess the universality property,  or something similar to Voronin 's universality property?  This is the reason why the answer to this question ...
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90 views

Remainder term in an integral linked to the Riemann zeta function

Sorry if this is not research level, but the following problem occurs in my own research: it is trivial to show that for $k\ge2$ integral we have $\zeta(k)=(1/(k-1)!)\int_0^\infty t^{k-1}/(e^t-1)\,dt$ ...
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194 views

Hardy-Littlewood vs heuristics on the zeta zeros

The first Hardy-Littlewood Conjecture asserts: Conjecture 1: Fix integers $0< a_1 < \ldots < a_k$. The density of those primes $p\le x$ such that $p + 2a_1,\ldots, p+2a_k$ are also primes, ...
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261 views

$\sum_{n=1}^\infty \Lambda(n) e^{-nz}$ and $L(s,\chi)$

Let $$W(z)=\sum_{n=1}^\infty \frac{\Lambda(n)}{n^{1/2}} e^{-nz}, \qquad\Re(z) > 0$$ For $\frac{y}{2\pi}=\frac{a}{q} \in \mathbb{Q}$, as $x \to 0^+$ we have $$W(x-iy) -{\scriptstyle \underset{(n,q) &...
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248 views

Strange inequality with $\zeta(5)$ [closed]

$$\frac{\pi^2}{1+\exp(-1/\pi^2)}<\sum\limits_{k=1}^{\infty}\frac{5}{k^5}<\frac{\pi^2}{1+\exp(-\pi/31)}$$ How can I prove it (not only with computation)?
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Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?

Apéry's proof of the irrationality of $\zeta(3)$ astounded contemporary mathematicians for its wealth of new ideas and techniques in proving the irrationality of a known constant. It is often the case ...
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0answers
116 views

What is the best known upper bound for $( \gamma_{n+1}-\gamma_{n})\max_{\{T\in(\gamma_{n},\gamma_{n+1})\}}(\vert\zeta(1/2+iT)\vert) $?

For $ n $ a positive integer, denote by $ L(n) : =\gamma_{n+1}-\gamma_{n} $ with $ \gamma_{n} $ the imaginary part of the $ n $-th critical zero of the Riemann zeta function and by $ M(n) : =\...
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Analogues of the Riemann zeta function that are more computationally tractable?

Many years ago, I was surprised to learn that Andrew Odlyzko does not consider the existing computational evidence for the Riemann hypothesis to be overwhelming. As I understand it, one reason is as ...
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5answers
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Motivated account of the prime number theorem and related topics

Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...
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230 views

Order of magnitude of extremely abundant numbers and RH

I have always been intrigued by the fact that Riemann's hypothesis is equivalent to the assertion (you can find the scanned paper here) that the inequality $$\frac{\sigma(n)}n<e^\gamma \log\log n \...
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1answer
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Partial product of Euler factors

Let $\mathbb P$ denote the set of prime numbers and for a subset $T\subset \mathbb P$ let $$ \zeta_T(s)=\prod_{p\in T}\frac1{1-p^{-s}}, $$ where $\mathrm{Re}(s)>1$. Is there any $T$ such that $T$ ...
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109 views

Dirichlet eta function and Stirling Permutations

The Stirling permutations of order $k$ is a permutation of the multiset $1, 1, 2, 2, ..., k, k$. The Dirichlet $\eta$-function is a function closely related to the Riemann $\zeta$-function. According ...
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2answers
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The Riemann hypothesis as a problem in analysis

The recent post("Long-standing conjectures in analysis ... often turn out to be false") prompted me to think about a question which I have not given much though before: to what extent the ...
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2answers
329 views

A recurrence relation for $\zeta(2n)$ - reference request

In the textbook https://www.springer.com/gp/book/9783034851688 (Klassische elementare Analysis, by M. Koecher) the following elegant recurrence relation is proved for $\zeta(2n)$ (on p. 157): $$\left(...
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6answers
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“Long-standing conjectures in analysis … often turn out to be false”

The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1 His example of a "long-standing conjecture" is the Riemann hypothesis,...
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1answer
258 views

On the Dirichlet series for $1/\zeta(s)$ at $\Re(s)=1/2$

Suppose that $1/2+it$ is not a zero of the Riemann zeta function $\zeta$, where $t \in \mathbb{R}$. Can $1/\zeta(1/2+it)$ be expressed as a Dirichlet series ?
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An interesting phenomenon of the analytic continuation of Riemann zeta function [closed]

It is known that $$\Gamma (s) \zeta (s)=\int_0^{\infty} \frac{x^{s-1}}{e^x-1}dx$$ this function is valid only for $\Re{s}>1$. However, if we ignore this restriction, and integrate by using $$\frac{...
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2answers
203 views

On a certain integral representation for Hurwitz zeta functions

A recent question On a certain integral representation for Dirichlet L-functions referenced an integral representation of $\zeta(s)$ due to Jensen that was new to me: $$ (s-1)\zeta(s)=\frac{\pi}{2(s-1)...
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4answers
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On the real part of the Riemann zeta function inside the critical strip

Denote by $\zeta$ the Riemann zeta function. Does $\Re\zeta(s)$ ever vanish for $\frac{1}{2}<\Re(s)\leq 1$ ?
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1answer
254 views

Confusion about Montgomery's Pair Correlation Conjecture

This question will be based roughly on the Bourgade Keating review on Zeta function and eigenvalue asymptotics (BK): https://link.springer.com/chapter/10.1007/978-3-0348-0697-8_4 To set up the ...
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1answer
208 views

On a certain integral representation for Dirichlet L-functions

It is an ancient result of Jensen that $$(s-1)\zeta(s)=\frac{\pi}{2} \int_{-\infty}^{\infty} \frac{(1/2+it)^{1-s}}{\cosh^{2}\pi t} \mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function. Is ...
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2answers
315 views

Is Riemann zeta function injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?

Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
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1answer
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Why did Euler consider the zeta function?

Many zeta functions and L-functions which are generalizations of the Riemann zeta function play very important roles in modern mathematics (Kummer criterion, class number formula, Weil conjecture, BSD ...
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Dirichlet series as rational zeta expressions

Let $D(f,s):=\sum_{n=1}^\infty \frac{f(n)}{n^s}$, otherwise known as a Dirichlet series. When $f$ is a multiplicative, number theoretic function, $D(f,s)$ tends to be expressed as a rational product ...
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115 views

A mixed of the Dedekind zeta function and the L-function

I have recently come across the following function, which seems like a "mix" between the Dedekind zeta function and the L-function: $\sum_I\frac{\chi_k(N(I))}{N(I)^s}$ where $\chi_k(n)$ is the ...
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2answers
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Is the Riemann zeta function surjective?

Is the Riemann zeta function surjective or does it miss one value?
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127 views

On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function

Let $$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function. What are the reasonable asymptotic estimates for $I(T)...
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290 views

Computing Mertens' function in time O(sqrt(x)) - in practice

As far as I know, there is one way currently known to -- in principle -- compute the Mertens function $M(x) = \sum_{n\leq x} \mu(n)$ in time essentially $O\left(x^{1/2}\right)$, namely, a modification ...
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1answer
142 views

Product representations of the Riemann zeta function

The Riemann zeta function, initially defined as $$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s} $$ for $\Re(s)>1$ also has infinite sum representations on larger domains. One such sum is $$\zeta(s)=\...
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206 views

Does the Hilbert-Polya conjecture implies The Riemann hypothesis?

I wanted to know if I have a direct involvement, I have been reading the article by Alan Connes "Trace formula in noncommutative Geometry and the zeros of the Riemann zeta function". I wanted ...