All Questions
Tagged with analytic-number-theory riemann-zeta-function
319 questions
3
votes
2
answers
316
views
Explicit formula: explicit work with general smoothing?
The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an ...
- 3,486
6
votes
2
answers
784
views
On some analytic property of the Riemann zeta function
Denote by $\zeta$ the Riemann zeta function. For $\Re(s)=\sigma>0$, it is well known that
$$\sum_{n\leq x} n^{-s} = \zeta(s) + \frac{x^{1-s}}{1-s}+ O(x^{-\sigma}).$$
But do there exist infinitely ...
- 12.8k
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
$$
\log|\zeta(1/2+it))|\leq \Re\left(\sum_{p\leq x}\frac{1}{p^{1/2+1/\log x+it}}\frac{\log(x/p)}{\...
CommunityBot
- 1
1
vote
1
answer
666
views
Complex integral of logarithmic derivative of $\zeta$
I want to prove that for any $x\geq 2$ we have
$$
\begin{split}
-\frac{\zeta^{\prime}}{\zeta}(s)&=\sum_{n\leq x}\frac{\Lambda(n)}{n^s}\frac{\log(x/n)}{\log x}+\frac{1}{\log x}\left(\frac{\zeta^{\...
- 199
2
votes
1
answer
390
views
Zeros of polynomial approximations of the Riemann $\zeta$ function
I know next to nothing about analytic number theory, or the theory of the Riemann $\zeta$ function in particular, so the following might be too elementary to deserve more than derision; even so it ...
- 91.8k
0
votes
0
answers
162
views
How to estimate the integral of Riemann zeta function with error term trends to zero as T trends to infinity?
$\zeta(s)$ denotes the Riemann zeta function. For $T>0$ large enough $$\int_{T}^{2T}|\zeta(\frac{1}{3} + it)
\ \zeta(\frac{2}{3} + it)|dt = \ ? $$
- 29
6
votes
1
answer
383
views
Tight error terms for partial sums $\sum_{n\leq x} 1/n^s$
(a) Let $s>1$, $x>0$ be real. Then it is not hard to see that
$$\sum_{n\leq x} \frac{1}{n^s} \leq \zeta(s) - \frac{1}{(s-1) x^{s-1}} + \frac{1}{2 x^s},$$
basically because $x\mapsto 1/x^s$ is ...
- 105k
2
votes
2
answers
450
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On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta function
Define $\theta(x)=\sum_{p\leq x} \log p $, where $p>1$ denotes a prime.
Nicolas proved that if the Riemann zeta function $\zeta(s)$ vanishes for some $s$ with $\Re(s)\leq 1/2 + b$, where $b\in(0, 1/...
- 1,018
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],...
- 1,018
3
votes
0
answers
171
views
Estimating integral of product of terms $\cos(t\log p)$
I would like to prove the following proposition from A. Harper's paper "Sharp conditional upper bound for moments of the Riemann Zeta Function"
Proposition.
Let $T$ be large and let $n=p_1^{\...
- 199
1
vote
0
answers
178
views
An integral involving $1/\zeta(s)$ and the zeros of $\zeta(s)$
In my thesis, i stumbled across the following problem:
Define $$F(\sigma, T)=\frac{1}{T}\int_{-T}^{T} \frac{1}{|\zeta(\sigma+it)|^2} \mathrm{d}T$$ where $\zeta$ denotes the Riemann zeta function.
Is ...
- 11
1
vote
0
answers
188
views
Questions on Riemann's explicit formula
If we consider this version of the prime-counting function
$$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$
(with $\pi$ being the normal prime-counting function), then we can write $\...
- 749
12
votes
4
answers
916
views
non-trivial zeros of partial zeta functions
Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as
$$
\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}
$$
where $Re(s)>1$. Let $\omega$ be either ...
- 2,821
3
votes
0
answers
97
views
Supremum of certain modified zeta functions at 1
Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by
$$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$
where $(D/m)$ denotes the Jacobi symbol. ...
- 356
16
votes
2
answers
809
views
Multizeta function values
Francis Brown Theorem says that $\zeta(a_{1},\dots ,a_{r})$ the multi-zeta value of weight $N=a_{1}+\dots +a_{r}$ is a $\mathbb{Q}$-linear combination of elements of the set $S=\{\zeta(a_{1},\dots, ...
- 585
1
vote
0
answers
341
views
Riemann Explicit Formula
I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula:
$$\sum_{n \leq x} \Lambda(n) n^{-s} = -\frac{\zeta'}{\zeta}(s)...
- 547
1
vote
0
answers
74
views
Enquiry on bounds for $\frac{1}{(n-1)!}\frac{d^n}{ds^n}((s-1)\zeta(s))$ at $s=1.$
Let $\zeta$ be the Riemann zeta function and $n$ be a positive integer. What are the known (conditional and unconditional) bounds for $f(n)=\frac{1}{(n-1)!}\frac{d^n}{ds^n}((s-1)\zeta(s))$ at $s=1$ ?
...
- 11
11
votes
1
answer
624
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Riemann zeta function: pair correlations vs. neighbor spacings
Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the ...
- 188k
1
vote
0
answers
150
views
Do we believe that the distribution of spacings of successive critical zeros of zeta is log-symmetric?
Let $\gamma^{+}(T)$ be the imaginary part of the critical zero of $\zeta$ closest to $1/2+iT$ with $\gamma^{+}(T)\ge T$ and define similarly $\gamma^{-}(T)$ with a reversed inequality. Let $g(T)$ be ...
- 6,984
5
votes
1
answer
273
views
Where can I find this result of Ingham?
Sometime ago, I read somewhere (should be in Titschmarsh) that, if $N(\sigma, T)$ denotes the number of zeros of the Riemann zeta function $\zeta(s)$ with $\Re(s)\geq \sigma>1/2$ up to height $T>...
- 3,107
6
votes
2
answers
788
views
How often does the Mertens function vanish?
It is well known that the Mertens function
$$M(x)=\sum _{n\leq x}\mu(n)$$
has infinitely many zeros, and this seems to be a short proof.
Are there known results about how often the Mertens function ...
- 4,713
1
vote
1
answer
416
views
How large can $|\zeta(\sigma + it)|$ be for $\sigma<1/2$?
Let $\zeta$ be the Riemann zeta function.
My question is: For fixed $\sigma<1/2$, how large can $|\zeta(\sigma+it)|$ be for $t\in \mathbb{R}$, even assuming zeta conjectures like the RH or the LH ?...
- 5,424
2
votes
1
answer
277
views
Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for finite fields?
The Balazard-Saias-Yor criterion for the Riemann Hypothesis states that the latter is equivalent to the statement that
$$\int_{\Re(s)=1/2} \frac{\log|\zeta(s)|}{|s^2|}|ds|=0$$ where $\zeta$ denotes ...
- 2,060
-1
votes
1
answer
243
views
On a certain representation of the Riemann zeta function
Let $\zeta$ denote the Riemann zeta function. In this answer: https://mathoverflow.net/a/314066/133634, @Paul Garret considers the representation
$$\frac{\zeta(s)}{s} = \int_1^\infty (\sum_{1 \le n \...
- 105k
-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 $\zeta$ ...
CommunityBot
- 1
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}...
- 105k
-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
$$\Re(1/4 + t^2)\zeta(1/2 + it)=2t\arg \zeta(1/2 + it) + 2(1/4 + t^2)\ ?$$
- 105k
5
votes
1
answer
999
views
Generalization of Mertens' theorem
One classical Mertens' theorem tells us that $$\prod_{p \leq n} (1-\frac{1}{p})^{-1} = e^\gamma \log n + \mathcal{O}(1).$$
It is now very natural to ask, whether we have some good estimate to $$\prod_{...
- 113
3
votes
1
answer
312
views
Closed form for an integral involving the Riemann zeta function at the critical line
After seeing this question $L_2$ bounds for $\zeta(1/2 + it)$ and a related integral i became curious if/how the approach in the answer by reuns can be applied to evaluate
$$I_{a,b}=\int_{-\infty}^{\...
- 23k
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{...
- 4,713
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 $T\geq 2$, ...
CommunityBot
- 1
14
votes
1
answer
1k
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On meromorphic continuation of zeta function(s) and special values at negative integers
Euler developped (at least) two different approaches in order to calculate the values $\zeta(-m)$ of the zeta function $$\zeta(s) = \sum_{n\geq 1} \frac{1}{n^s}$$ at non-positive integers.
In one ...
- 13.4k
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,...
- 3,209
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:...
CommunityBot
- 1
11
votes
2
answers
2k
views
Is there an explicit expression for the imaginary part of some non-trivial zero of zeta,
Is there an explicit expression for the imaginary part of some non-trivial zero of zeta, in terms of well-known constants, such as say $\gamma$ or $\pi$ say ?
- 1,183
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 ...
CommunityBot
- 1
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
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 $T$ ...
- 12.8k
17
votes
1
answer
3k
views
What is the relation between Quasicrystals, Riemann Hypothesis, and PV numbers?
Could somebody explain to me, from a mathematical stand-point, what is a quasi-crystal, and how it relates to the set of Pisot numbers, and the Riemann Hypothesis?
I've heard Freeman Dyson say that ...
- 5,407
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 ...
- 11.1k
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 ?
- 105k
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 $0\leq a<b \leq 1$?
- 105k
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 $\frac{1}{2}<\Re(s)\leq 1$ ?
- 105k
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 ...
- 105k
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 ...
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
37
votes
2
answers
3k
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The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function?
Yesterday Bourgain, Demeter and Guth released a preprint proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only ...
- 8,842
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 ...
- 18.4k
2
votes
1
answer
150
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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 ...
- 131