All Questions
Tagged with analytic-number-theory riemann-zeta-function
319 questions
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Compensation by the residue of the zeta function
(Repost of a question from MSE, where it found no success)
Let $F$ be a global number field. Introduce a local quantity at every place
$$x_p = \frac{\zeta_p(1)}{\zeta_p(2)}$$
for instance. The ...
- 105k
2
votes
0
answers
147
views
Skewes' number and the ratio $\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$
(A complementary post is here.)
Given the prime counting function $\pi(x)$ and the logarithmic integral $\operatorname{li}(x)$, we have Table 1,
$$\begin{array}{|c|l|}
\hline
x&\operatorname{li}...
- 12.6k
7
votes
0
answers
179
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When does the function $F(x)=\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$ reach $F(x) > 8$?
We know from Ramanujan and Riemann that,
$$\pi(x) = \operatorname{li}(x) -\tfrac12\operatorname{li}(x^{1/2})-\tfrac13\operatorname{li}(x^{1/3})-\tfrac15\operatorname{li}(x^{1/5}) +\dots$$
with prime ...
- 12.6k
8
votes
1
answer
577
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$\sum_{d\leq x} (\mu(d)/d) \log(x/d)$: is (the analogue of) Mertens' conjecture still false?
It is known to be false that $\sum_{m\leq x} \mu(m) \leq \sqrt{x}$ for all $x$ (Mertens' conjecture), and it is generally believed that $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$. From the ...
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10
votes
1
answer
640
views
Statement of the pair correlation conjecture
In his paper "The pair correlation of zeros and the zeta function",
Montgomery defines a function
$$F(\alpha,T) = \left(\frac{T}{2 \pi} \log T\right)^{-1} \sum_{0 < \gamma, \gamma' < T} T^{i \...
- 26k
8
votes
2
answers
5k
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Inverse of the Riemann zeta function [closed]
I'm wondering if there is any information on the inverse of the Riemann zeta function (not it's reciprocal, but its functional inverse). This would obviously be a multi-valued function.
- 1,228
8
votes
2
answers
759
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$L_2$ bounds for tails of $\zeta(s)$ on a vertical line
Let $0<\sigma\leq 1$. Let $T$ be large. How can we give good explicit $L^2$ bounds on the tails of $\zeta(\sigma+it)$? That is, we want to bound the quantity $$\int_{\sigma-i\infty}^{\sigma-iT} + \...
- 20.2k
9
votes
1
answer
588
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Double sum of negative powers of integers: a direct approach?
Let $\alpha,\beta\in (0,1\rbrack$, $\alpha\ne \beta$. I wish to estimate $$\sum_{m\leq x} \frac{1}{m^\alpha} \sum_{n\leq x/m} \frac{\log(x/mn)}{n^\beta}.$$ There is an obvious approach, namely, to ...
- 20.2k
12
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2
answers
555
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$\zeta^{(k)}(s) < 0$ for $s\in (0,1)$
A bit of plotting suggests that $\zeta^{(k)}(s) < 0$ for all $s\in (0,1)$ and all integers $k\geq 0$. (Or, what is the same: $\zeta^{(k)}(s)$ has no zeroes on $(0,1)$.) Is there a brief, clean ...
- 10.5k
12
votes
1
answer
380
views
Picking a new set of primes
If $S$ is a subset of the set of the positive integers $\mathbb N$, we may consider the set $S^*$ of all products of elements of $S$, allowing for repeated factors —this is a multiset, really, in ...
- 105k
4
votes
2
answers
366
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On an observation which relates to the exponential sum $\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$
This observation is based on the numerical calculation of the exponential sum:
$$\sum_{n=1}^{[\sqrt{t/2\pi}]} n^{-\frac{1}{2}+it}$$
It is known that this sum is related to the famous Riemann–Siegel ...
- 395
3
votes
1
answer
571
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On the series $\sum_{\rho}x^{\rho}\Gamma(\rho)/\Gamma(\rho+k),\,0<k<1$
Let $x>1$ be a real number. For a work I need to find an uniform estimation of the series the series $$\sum_{\rho}x^{\rho}\frac{\Gamma\left(\rho\right)}{\Gamma\left(\rho+k\right)}\tag{1}$$ where $\...
- 219
0
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1
answer
169
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Analytic extension of the Hurwitz ζ function
For the purpose of formalisation in a theorem prover, I am looking for a simple definition of the analytic extension of the Hurwitz ζ function $\zeta(s,q)$ valid for all $s\in\mathbb{C}\setminus\{1\}$ ...
- 3,403
9
votes
2
answers
705
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Oscillation of the summatory Möbius function
Let the Mertens function $$M(x) = \sum_{n \le x} \mu(n)$$ I assume (perhaps foolishly) that it is known that $M(x)$ changes sign infinitely often. If that's true, the question is a quantitative ...
- 12.3k
6
votes
0
answers
233
views
Lindelöf Hypothesis and the Karatsuba conjectures
I'm aware of Shao-Ji Feng's result that Karatsuba's weaker conjecture ("conjecture 1") is true conditionally on the Lindelöf Hypothesis.
Shao-Ji Feng, "On Karatsuba conjecture and the Lindelöf ...
- 17.6k
1
vote
0
answers
130
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On the asymptotic form of a sum over the nontrivial zeros of the Riemann zeta function
What is the asymptotic form of the sum
$$\sum_{\rho} \dfrac{x^{\rho}-(x-2)^{\rho}}{\rho}$$
where the summation is over the nontrivial zeros of the Riemann zeta function?
By the Prime Number Theorem,...
-3
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1
answer
651
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A question about Yitang Zhang's paper "On the zeros of ζ’(s) near the critical line"
We conclude that, in the case $\sigma = 1/2$ and $\zeta’\left(s\right) \neq 0$,
$$\mathrm{Re}\frac{\eta’}{\eta}\left(s\right) = \sum_{\beta’ \gt 0}{\mathrm{Re}\frac{1}{s - \rho’}} + O(1)$$
It is easy ...
CommunityBot
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7
votes
4
answers
4k
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Is this sum of reciprocals of zeta zeros correct?
I am trying to find or get a numerical approximation of
$$ \sum_{\rho \text{ non-trivial zeros of } \zeta} \frac{1}{\rho} $$
In The Riemann Hypothesis: Arithmetic and Geometry Lagarias gives the ...
- 105k
19
votes
4
answers
2k
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What are the obstructions to showing that $\zeta$ doesn't vanish on the strip $1- \varepsilon < {\rm Re}(s) \leq 1$
Most (if not all) of the proofs of the Prime Number Theorem that I have seen in the
literature rely on the fact that the Riemann zeta function, $\zeta(s)$, does not vanish
on the line ${\rm Re}(s) = 1$...
- 105k
1
vote
2
answers
1k
views
Intuition behind the Riemann $\zeta$ functional equation
Let $\Lambda(s)=\pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s).$ Then $\Lambda\left(\frac{1}{2} + s\right) = \Lambda\left(\frac{1}{2} - s\right)$ constitutes the functional equation for the Riemann ...
- 17.6k
1
vote
1
answer
343
views
Question on the zeta and sigma functions
EDIT:
The observation described here was due to a bug in the code used to generate the data (as commented by Lucia), which renders this question irrelevant.
The answer, however, is worth reading.
The ...
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1
vote
0
answers
202
views
Estimation of the $k$-th derivative zeta function
When I was doing some task in number theory which involves bounds for the Riemann zeta function, I was stuck on the following question:
Let $\zeta$ be the Riemann zeta function and let $\zeta^{(k)}$ ...
- 1,257
4
votes
1
answer
585
views
$\sum_{d\leq x} (\mu(d)/d) \log x/d$: elementary estimates?
Let
$$F(x) = \sum_{d\leq x} \frac{\mu(d)}{d} \log \frac{x}{d}.$$
s it possible/feasible to give an elementary proof of the fact that $F(x)= 1 + o(1)$ (and, ideally, $1+O(1/\log x)$, or better)? By "...
- 4,713
0
votes
1
answer
249
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Can there be more than two zeta zeros in between a Gram point and a França-LeClair point?
According to formula 163 at page 47 in the paper A theory for the zeros of Riemann Zeta and other L-functions by Guilherme França and André LeClair, the Gram points can be approximated with the ...
15
votes
1
answer
874
views
values of $\zeta$ function are linearly independent?
Are the elements of the set $\{\zeta(2n+1)| n\in \mathbb{N}\}$ $\mathbb{Q}$-linearly independent?
- 47.4k
-6
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2
answers
357
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Are the zeros of $\zeta'$ exactly the zeros of $\zeta$? [closed]
The Riemann Hypothesis is known to be equivalent to the statement that $\zeta'$ (the derivative of the Riemann zeta function) has no zeros in the region $0< \Re(s) < 1/2$. By the functional ...
- 11.1k
6
votes
1
answer
2k
views
How to understand the explicit formula for zeta function?
The explicit formula for the zeta function, e.g.
$$\psi_0(x) = \dfrac{1}{2\pi i}\int_{\sigma-i \infty}^{\sigma-i \infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}ds=x-\sum_\rho\frac{x^\...
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15
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5
answers
2k
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Zeros of the derivative of Riemann's $\xi$-function
The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...
CommunityBot
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2
votes
1
answer
396
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Is $|\zeta(e^{ni})|\leq \log(n)$ true for $n > 19$ and how do i can show it if it is?
I performed some computations in wolfram alpha looking at the behavior of the values of $|\zeta(e^{ni})|$ trying to predict a lower bound. I have got the following result:
For $n > 19 :|\zeta(e^{...
- 105k
3
votes
2
answers
597
views
lower bound for $\Re\zeta(1+it)$
Hi
is there any lower bound for $\Re\zeta(1+it)$.
I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.
If it is true, is there any reference to prove it.
thanks
CommunityBot
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0
votes
1
answer
1k
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Analytical continuation of the reciprocal of the Zeta function [closed]
Is the reciprocal of the Zeta function analytically continuable?
As $1/\zeta(n) = \sum_{n=1}^\infty \mu(n)/{n^s}$, this does not look obvious.
CommunityBot
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2
votes
0
answers
452
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Analytic continuation of "composite" zeta function
Let us define the Dirichlet series $$\mathcal C(s):=\sum_{n\text{ composite}}\frac{1}{n^s},\quad P(s):=\sum_{p\text{ prime}}\frac{1}{p^s}.$$
They are absolutely convergent in the half-plane $\sigma>...
- 17.6k
10
votes
0
answers
740
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Implications of divergence of $1/\zeta(s) $ at 1/2
$1/\zeta(s)=\sum_{n>0}\frac{\mu(n)}{n^s}$ where $\mu$ is the Moebius function.
This series is known to converge for $s\ge 1$ and diverge for $s\le 1/2$.
Its convergence is unknown if $1/2< s&...
- 17.6k
5
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0
answers
504
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An explicit formula for $\zeta(2m+1)$ with good convergence
The question: Is the following formula known?
$$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!}\cdot \sum\limits_{v=k}^m \frac{(2^{2v-2k+...
- 293
1
vote
1
answer
2k
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Upper bound for real part of Riemann Zeta function zeros
I have been reading about Riemann Zeta function $\zeta(s)$ and have been thinking about it for some time. I did some calculations, and reached a conclusion where $\Re(\rho) \le \log_2(3) - 1$ as $\Im(\...
- 109k
1
vote
1
answer
243
views
Do we know an upper bound for the number of possible real parts of the non trivial zeroes of $\zeta$?
Let $n_{\zeta}$ denote the number of possible real parts for the non trivial zeroes of the Riemann Zeta function. RH is equivalent to $n_{\zeta}=1$, and the symmetry arising from the functional ...
CommunityBot
- 1
3
votes
0
answers
196
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Relation between the sign of the Stieltjes constants and some zero-free region of $\zeta$
One may recall that the Stieltjes constants $\gamma_{k}$ appear as the scaled coefficients in the regular part of the Laurent series expansion of the Riemann zeta function about $s = 1$:
$$
\begin{...
- 243
3
votes
1
answer
592
views
Some identities with the Riemann-Hurwitz zeta function
The only definition that I have ever seen of this Riemann-Hurtwitz zeta-function is this,
For $0 < a \leq 1$ we have the identity
$$ \zeta(z, a) = \frac{2 \Gamma(1 - z)}{(2 \pi)^{1-z}} \left[\sin ...
CommunityBot
- 1
15
votes
1
answer
901
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Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?
Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be ...
- 43.7k
1
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0
answers
223
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Is the difference of these two real-rooted functions real-rooted?
During our on-going search of approximations to the Riemann $\Xi(z)$ function, we discovered a family of functions $W_n(z)$ as shown in (1).
Our final goal is to prove that:
Proposition 1: $W_{n}(z)...
CommunityBot
- 1
3
votes
1
answer
1k
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Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function
Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum_{k|n}^{n}1.
$$
One can observe the following pattern in the values of $...
- 105k
19
votes
3
answers
6k
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Are the nontrivial zeros of the Riemann zeta simple?
A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all ...
- 1
26
votes
5
answers
3k
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Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?
The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here:
Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real ...
CommunityBot
- 1
6
votes
1
answer
4k
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About the logarithmic derivative of the Riemann zeta function
Let $\rho=\beta+i\gamma$ a non-trivial zeros of the Riemann zeta function and $s=\sigma+it$ a complex number. It is possible to prove that $$\frac{\zeta'}{\zeta}\left(s\right)=\sum_{\left|t-\gamma\...
- 105k
32
votes
2
answers
3k
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Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?
Consider the equation
$$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$
"proved" by Ramanujan Euler. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = 1}^{\...
- 10.1k
29
votes
4
answers
5k
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Good uses of Siegel zeros?
The short version of my question goes: What is known to follow from the existence of Siegel zeros?
A longer version to give an idea of what I have in mind: The "exceptional zeros" of course first ...
- 50.6k
8
votes
2
answers
297
views
How to formalize the *loci of equal arg($\zeta(s)$)* ("isogones") in the near of a nontrivial root
(This is an extension and specification of a question which I initially asked in MSE having now one comment (which I could not yet digest completely) and which I also detailed further (after working ...
CommunityBot
- 1
9
votes
0
answers
265
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Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?
Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
- 13.4k
1
vote
1
answer
190
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Values of the completed Riemann $\xi(1+it)$ for small t?
I'm editing this question heavily for clarity:
I am looking for methods to compute $\zeta(1+it)$, or the (partially) completed Riemann zeta function
$$\pi^{-s/2}\Gamma(s/2)\zeta(s)$$
along the line ...
- 3,799
5
votes
0
answers
195
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Moments of completed L-functions?
This is a follow up question to this one.
It seems that results on moments of L-functions, that is, estimates for integrals of the form
$$\int^{T}_1|\zeta(\sigma+it)|^{2k}dt$$
are typically for the ...
CommunityBot
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