All Questions
41 questions
45
votes
1
answer
5k
views
Is it possible to show that $\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}$ diverges?
Let $\mu(n)$ denote the Mobius function with the well-known Dirichlet series representation
$$
\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}}.
$$
Basic theorems about Dirichlet series ...
- 20.4k
26
votes
5
answers
3k
views
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 ...
- 4,169
29
votes
4
answers
5k
views
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 ...
- 291
24
votes
1
answer
2k
views
How good is "almost all" when it comes to the Riemann Hypothesis?
Let $N(T)$ be the number of zeroes of the Riemann zeta function $\zeta$ having imaginary part strictly between $0$ and $T$, and let $N_0(T)$ be the number of those zeroes that also have real part ...
- 231
8
votes
2
answers
759
views
$L_2$ 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 $$\int_{\sigma-i\infty}^{\sigma-iT} + \...
- 20.2k
50
votes
5
answers
3k
views
Motivated account of the prime number theorem and related topics
Though my own research interests (described below) are pretty far from analytic number theory, I have always wanted to understand the prime number theorem and related topics. In particular, I often ...
- 482
49
votes
4
answers
6k
views
If the Riemann Hypothesis fails, must it fail infinitely often?
That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. ...
- 17.6k
41
votes
6
answers
9k
views
"Long-standing conjectures in analysis ... often turn out to be false"
The title is a quote from a Jim Holt article entitled, "The Riemann zeta conjecture and the laughter of the primes" (p. 47).1
His example of a "long-standing conjecture" is the Riemann hypothesis,...
34
votes
7
answers
8k
views
Explicit formula for Riemann zeros counting function
I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.
Because there are various explicit formulae ...
- 371
19
votes
3
answers
6k
views
Are the nontrivial zeros of the Riemann zeta simple?
A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all ...
- 6,984
15
votes
5
answers
2k
views
Zeros of the derivative of Riemann's $\xi$-function
The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...
- 11.1k
10
votes
5
answers
3k
views
Riemann–Von Mangoldt formula
Let $$N(T) = \#\{\rho \in \mathbb{C}: \zeta(\rho) = 0,\, \operatorname{Im} \rho \in (0,T]\}$$ denote the number of zeros of $\zeta(s)$, counting multiplicities, with imaginary part lying in the ...
- 4,985
6
votes
1
answer
900
views
What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming the SZC but not the RH?
Let $\zeta$ denote the Riemann zeta function and let $\rho$ denote one of its complex zeros. What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming that all zeros are simple (SZC), ...
- 149
5
votes
0
answers
920
views
Zeta function double product
Is it possible to write the following double product in terms of the zeta function?
\begin{align}
&\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}}
\end{align}
Extending the ...
- 1,903
2
votes
1
answer
448
views
Riemann-Von Mangoldt formula, revised question
This is my last question, building off of Riemann-Von Mangoldt formula
and Does $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t)$ have mean value $0$?. I apologize for taking a while to ...
- 4,985
2
votes
1
answer
340
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{...
- 21
45
votes
4
answers
8k
views
Why is so much work done on numerical verification of the Riemann Hypothesis?
I have noticed that there is a huge amount of work which has been done on numerically verifying the Riemann hypothesis for larger and larger non-trivial zeroes.
I don't mean to ask a stupid question, ...
- 5,146
43
votes
3
answers
3k
views
Is this integral representation of $\zeta(2n+1)$ known?
Background: I'm an undergraduate at an institution with no researchers in analytic number theory, and no ties to the analytic number theory community. I believe I have found what is, as far as I can ...
- 383
35
votes
7
answers
6k
views
Heuristic argument for the Riemann Hypothesis
Is there a heuristic argument that supports the validity of the Riemann hypothesis or are we just relying on numerical evidence? Moreover, what is the strongest theorem that supports the validity of ...
- 3,699
32
votes
2
answers
3k
views
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},$$
"proved" by Ramanujan Euler. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = 1}^{\...
- 7,347
15
votes
1
answer
874
views
values of $\zeta$ function are linearly independent?
Are the elements of the set $\{\zeta(2n+1)| n\in \mathbb{N}\}$ $\mathbb{Q}$-linearly independent?
- 641
14
votes
0
answers
831
views
Growth of residues of $1/\zeta(s)$: conjectures?
Let $\rho$ range over the non-trivial zeroes of the Riemann zeta function. Let
$$M(T) = \max_{|\Im \rho|\leq T} \left|\mathrm{Res}_{s=\rho} \frac{1}{\zeta(s)}\right| =
\max_{|\Im \rho|\leq T} \frac{1}...
- 20.2k
12
votes
1
answer
663
views
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 $x\geq |t|$,
$$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O\left(\...
- 20.2k
9
votes
1
answer
733
views
Does this partial sum over primes spike at all zeta zeros?
Below is a plot of $\exp \sum _p^x -\frac{\cos \left(x \log \
p\right)}{\sqrt{p}}$, where $p$ runs over the primes, and the $x$-values of the Riemann $\zeta$ zeros are marked with dashed lines:
Below ...
- 1,903
9
votes
2
answers
705
views
Oscillation of the summatory Möbius function
Let the Mertens function $$M(x) = \sum_{n \le x} \mu(n)$$ I assume (perhaps foolishly) that it is known that $M(x)$ changes sign infinitely often. If that's true, the question is a quantitative ...
- 96.4k
8
votes
1
answer
577
views
$\sum_{d\leq x} (\mu(d)/d) \log(x/d)$: is (the analogue of) Mertens' conjecture still false?
It is known to be false that $\sum_{m\leq x} \mu(m) \leq \sqrt{x}$ for all $x$ (Mertens' conjecture), and it is generally believed that $\lim \sup_{x\to \infty} |M(x)|/\sqrt{x} = \infty$. From the ...
- 20.2k
8
votes
1
answer
373
views
Behaviour of $\zeta(1-it)/\zeta(1+it)$?
I am trying to understand the behaviour of
$$\int^\infty_{-\infty}\frac{\xi(1-it)}{\xi(1+it)}h(t)\frac{dt}{t}$$
where $h$ is a Schwartz function on $\mathbb R$, and $\xi(s)$ the completed Riemann zeta ...
- 3,799
7
votes
0
answers
179
views
When does the function $F(x)=\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$ reach $F(x) > 8$?
We know from Ramanujan and Riemann that,
$$\pi(x) = \operatorname{li}(x) -\tfrac12\operatorname{li}(x^{1/2})-\tfrac13\operatorname{li}(x^{1/3})-\tfrac15\operatorname{li}(x^{1/5}) +\dots$$
with prime ...
- 12.6k
7
votes
2
answers
1k
views
Consequences of a bound on possible counterexamples to Riemann hypothesis
The Riemann hypothesis has many strong consequences in number theory. The question is: would a bound on the number of zeros of Riemann zeta-function in the critical strip with real part not equal 1/2 ...
- 2,205
7
votes
2
answers
906
views
Positivity of the coefficients of Taylor series associated to the Riemann hypothesis
The question below relates to the paper "Jensen Polynomials for the Riemann Zeta Function and Other Sequences" of Griffin, Ono, Rolen and Zagier. I'm asking it here because I am sure the ...
- 7,057
5
votes
0
answers
680
views
The Basel problem revisited?
In the Basel problem, the $sinc$ function is considered at the Wikipedia page.
Let me try to make an alternative function definition:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
- 3,064
5
votes
1
answer
291
views
Asymptotics of the Liouville sum at the primes
Let $\lambda$ denote the Liouville function, and let $L(x):=\sum_{n \leq x} \lambda(n)$ be the Liouville sum. Define $c$ to be the supremum of the real parts of the zeros of the Riemann zeta function. ...
- 103
5
votes
1
answer
766
views
Reference for Lindelöf Hypothesis implying finitely many zeros off critical line?
Can anyone give me a reference for the following theorem on the Riemann zeta function?
If the Lindelöf Hypothesis is true (that is $\zeta(\sigma+it)=O(t^\epsilon)$ as $t\rightarrow\infty$), then ...
- 195
3
votes
1
answer
730
views
what would be the consequences on the distribution of primes of $\Lambda=\infty$?
It is widely believed that the quantity $\Lambda:=\lim\sup\dfrac{t_{n+1}-t_{n}}{2\pi/\log t_{n}}$, where $t_{n}$ is the imaginary part of the $n$-th non-trivial zero on the critical line of the ...
- 6,984
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
- 163
3
votes
1
answer
709
views
Conditional bound on RH for $\Re\left(\sum_{p\leq\sqrt{x}}\frac{(1/2)}{p^{1+2it}}\right)$
I would like to prove that
Assume RH. Let $T$ large, $2\leq x \leq T^2$ and $T\leq t \leq 2T$, then
$$
\log|\zeta(1/2+it))|\leq \Re\left(\sum_{p\leq x}\frac{1}{p^{1/2+1/\log x+it}}\frac{\log(x/p)}{\...
- 199
3
votes
3
answers
273
views
Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$
Denote by $\zeta$ the Riemann zeta function.
It is known that
$$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$
But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...
- 35
2
votes
0
answers
188
views
How to best approximate $1/\zeta(s)$ by a finite sum
I would like to approximate $1/\zeta(s)$ for $s=1+it$ by a finite sum:
$$\frac{1}{\zeta(1+it)} = \sum_n \frac{\mu(n)}{n} \eta\left(\frac{n}{x}\right) +
\epsilon(t)$$
with $\eta$ a function of compact ...
- 20.2k
2
votes
2
answers
412
views
Robin's inequality and the zeros of the Riemann zeta function
Robin showed that if $a\in(1/2, 1]$ is the supremum of the real parts of the zeros of the Riemann zeta function $\zeta(s)$, then $f(x)=\Omega_{\pm} (x^{-b})$, where $b$ is some number on $(a-1/2, 1/2],...
- 1,019
2
votes
1
answer
240
views
On the nearest integer to $\zeta^{(k)}(1-1/B),B \ge 2$
Let $k \ge 1,B \ge 2$ integers and $\zeta^{(k)}(s)$
denote the $k$-th derivative of Riemann zeta function.
For real $x$, let $[x]$ denote the nearest integer to $x$.
Conjecture 1: For all $n \ge 1,[\...
- 25.4k
2
votes
1
answer
1k
views
On the convergence of Dirichlet series over the Mobius Mu function
It is known that if $\sum_{k=1}^{\infty} \frac{\mu(k)}{k^s} = \frac{1}{\zeta(s)}$ for $\Re(s) > 1/2$ then RH holds. My question is:
Under RH why is it not $\sum_{k=1}^{\infty} \frac{\mu(k)}{k^s} = ...
- 731