All Questions
Tagged with analytic-number-theory riemann-zeta-function
319 questions
2
votes
1
answer
277
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 ...
- 105
-1
votes
1
answer
243
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
$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
- 105
-2
votes
1
answer
270
views
A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$
By use of the Riemann functional equation, it can be shown (see corollary 10.5 of Montgomery-Vaughan) that
$$|\zeta(\sigma + it)| \asymp |t|^{\sigma-1/2}|\zeta(1-\sigma - it)|$$.
where $\zeta$ ...
- 35
3
votes
3
answers
273
views
Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$
Denote by $\zeta$ the Riemann zeta function.
It is known that
$$\int_{0}^{T} \zeta(1/2 + it) \mathrm{d}t = T + O(T^{1/2}).$$
But is a similar result for $\int_{0}^{T} \zeta(\sigma + it) \mathrm{d}...
- 35
-1
votes
1
answer
239
views
Enquiry on an equality involving the Riemann zeta function
Let $\zeta$ denote the Riemann zeta function. Does there exist a $t\geq 0$ such that
$$\Re(1/4 + t^2)\zeta(1/2 + it)=2t\arg \zeta(1/2 + it) + 2(1/4 + t^2)\ ?$$
- 29
3
votes
1
answer
312
views
Closed form for an integral involving the Riemann zeta function at the critical line
After seeing this question $L_2$ bounds for $\zeta(1/2 + it)$ and a related integral i became curious if/how the approach in the answer by reuns can be applied to evaluate
$$I_{a,b}=\int_{-\infty}^{\...
- 29
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
3
votes
0
answers
194
views
On some inequality involving the Riemann zeta function integral at $\Re(s)=1/2$ [closed]
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 $T\geq 2$, ...
user123305
1
vote
0
answers
242
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, is:...
user129745
2
votes
0
answers
135
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) : =\...
- 6,984
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
12
votes
1
answer
283
views
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 $T$ ...
user1688
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,...
2
votes
1
answer
397
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 ?
- 1,019
0
votes
4
answers
716
views
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 $\frac{1}{2}<\Re(s)\leq 1$ ?
- 1,019
7
votes
1
answer
1k
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 ...
- 171
1
vote
1
answer
419
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 ...
- 1,019
4
votes
2
answers
448
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 $0\leq a<b \leq 1$?
- 395
2
votes
0
answers
176
views
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 ...
- 4,952
1
vote
0
answers
152
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 ...
52
votes
3
answers
6k
views
Is the Riemann zeta function surjective?
Is the Riemann zeta function surjective or does it miss one value?
- 2,375
12
votes
0
answers
392
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 ...
- 20.2k
0
votes
2
answers
339
views
Error term in França-LeClair approximation of zeta zeros
The imaginary part of the $n$th critical zero of the Riemann zeta function with positive imaginary part (in increasing order) is asymptotically
$$
t_n \sim 2\pi\frac{n}{\log n}
$$
and has been ...
- 9,114
3
votes
2
answers
861
views
On the growth of the Riemann zeta function on the critical line
Is it true that
$$|\zeta(\frac{1}{2}+it)|< \frac{1}{4}+t^2$$ for all $t\geq 0$, where $\zeta$ denotes the Riemann zeta function ?
It is known that the left hand-side is $O(t^{0.25})$, and on the ...
- 31
2
votes
1
answer
150
views
Residue of the following variant of Dirichlet function [closed]
I am working with the Piltz divisor problem where the number of ways in which a number $n$ can be written as a product of $k$ is of the form
$$D_{k}(x)=xP_{k}(\log x)+\Delta _{k}(x)$$
where $P_k$ is ...
5
votes
0
answers
97
views
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 ...
- 5,424
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,
$$\begin{array}{|c|l|}
\hline
x&\operatorname{li}...
- 12.6k
7
votes
0
answers
179
views
When does the function $F(x)=\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$ reach $F(x) > 8$?
We know from Ramanujan and Riemann that,
$$\pi(x) = \operatorname{li}(x) -\tfrac12\operatorname{li}(x^{1/2})-\tfrac13\operatorname{li}(x^{1/3})-\tfrac15\operatorname{li}(x^{1/5}) +\dots$$
with prime ...
- 12.6k
25
votes
2
answers
2k
views
$P(s)=1-\sqrt{\frac{2}{\zeta(s)}-\sqrt{\frac{2}{\zeta(2s)}-\sqrt{\frac{2}{\zeta(4s)}-\sqrt{\frac{2}{\zeta(8s)}-...}}}}$
Vassilev-Missana - A note on prime zeta function and Riemann zeta function¹ claims the following remarkable identity:
$$
P(s)=1-\sqrt{\frac{2}{\zeta(s)}-\sqrt{\frac{2}{\zeta(2s)}-\sqrt{\frac{2}{\zeta(...
- 1,962
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
$$F(\alpha,T) = \left(\frac{T}{2 \pi} \log T\right)^{-1} \sum_{0 < \gamma, \gamma' < T} T^{i \...
- 26k
8
votes
2
answers
759
views
$L_2$ bounds for tails of $\zeta(s)$ on a vertical line
Let $0<\sigma\leq 1$. Let $T$ be large. How can we give good explicit $L^2$ bounds on the tails of $\zeta(\sigma+it)$? That is, we want to bound the quantity $$\int_{\sigma-i\infty}^{\sigma-iT} + \...
- 20.2k
9
votes
1
answer
588
views
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
votes
2
answers
555
views
$\zeta^{(k)}(s) < 0$ 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 ...
- 20.2k
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 ...
- 47.3k
12
votes
2
answers
2k
views
What are the implications of a zero of zeta off the critical line
So what happens if there is a non-trivial zero of the Riemann zeta function off the critical line? Has there been any work in the following direction: We know from Landaus theorem that there is a ...
- 3,699
4
votes
2
answers
366
views
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
0
votes
1
answer
169
views
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 $s\in\mathbb{C}\setminus\{1\}$ ...
- 1,241
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
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
3
votes
1
answer
571
views
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
1
vote
0
answers
130
views
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
votes
1
answer
651
views
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 ...
- 5
3
votes
1
answer
330
views
Turan Inequalities
A real entire function
$$\psi(x)=\sum_{k=0}^{\infty} \gamma_k\frac{x^k}{k!}$$
is said to be in the Laguerre-Polya class, denoted $\psi(x) \in \mathcal{LP}$, if it can be represented in the form
\...
- 73
5
votes
1
answer
627
views
Do Riemann-Weil formulas exist for functions other than the Mangoldt function $ \Lambda (n) $
Are there formulas similar to the Riemann-Weil formula for other arithmetical functions like $ \mu (n) $ or $ \lambda (n) $, for example a sum of the form $ \sum_{n=1}^{\infty}a(n) f(n) $ with this ...
- 113
4
votes
1
answer
621
views
Seek a reference for Theorem 1.2 on p. 6 of the Riemann Hypothesis sourcebook of Borwein et. al
The book "The Riemann Hypothesis, A Resource for the Afficionado and Virtuoso Alike", Borwein, Choi, Rooney, Weirathmueller, Eds., states on its page six the following theorem (Theorem 1.2):
...
- 51
1
vote
2
answers
999
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 ...
- 655
5
votes
2
answers
1k
views
Riemann Hypothesis and Euler product
It is conjecture that under certain conditions a L-function satisfies RH.
Among these conditions there is the necessity for the L-function to have an Euler product. (Some L-functions with a functional ...
- 1,199
17
votes
3
answers
3k
views
Largest known zero of the Riemann zeta function
Numerical calculations on the zeroes of the Riemann zeta function have reached a very high degree of refinement and sophistication and I think that the first $10^{20}$ (with positive imaginary part) ...
- 16.2k
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 ...
- 37
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