All Questions
Tagged with analytic-number-theory riemann-zeta-function
319 questions
0
votes
2
answers
389
views
What is the dirichlet series of $f(n)=\sum_{d | n}(\log d) / d$ function? [closed]
My opinion is ;
We may use id(d)=d arithmetic function and log*id dirichlet convolution in the question.
i thought that ; when we multiply and divide n with $(\log d) / d$ we obtain
$F(S)=\sum_{n=...
- 105
0
votes
0
answers
185
views
On the asymptotics of the Chebyshev psi function
Denote by $\psi(x)$ the Chebyshev psi function over prime powers. Assuming the RH, it can be shown via the Riemann explicit formula that
$$\psi(x)-x \ll √x |\sum_{|\gamma| < T} \frac{1}{\rho} | +...
- 41
1
vote
1
answer
222
views
Understanding a deduction in research paper of Sprang, Fischler and Zudilin ("Many Odd zeta values are irrational")
I am a master's student interested in number theory. I am reading a research paper in Analytic Number theory which is "Many Odd zeta values are irrational" by Stephane Fischler, Johannes Sprang and ...
- 793
3
votes
1
answer
344
views
Unable to deduce an inequality in paper on odd zeta values of Fischler, Sprang and Zudilin
I am a masters student and I am interested in number theory. Due to lockdown I have a lot of time and I thought of reading a research paper in Number theory which is " Many Odd zeta values are ...
- 793
4
votes
0
answers
126
views
What is the closed form of this integral?
Consider the Chebyshev first function $\psi(y):=\sum_{p^j \leq y} \log p$, where $p$ is a prime. Define $$F(s, k) = s\int_{1}^{\infty} \psi(x + x^k)x^{-s-1} \mathrm{d}x$$ for $ \Re(s) >$ max $(1, ...
- 41
2
votes
0
answers
537
views
Explicit formula for $n$th prime in terms of Riemann zeros:
We all know there exists an explicit Formula for prime counting function in terms of Riemann zeros.
I'm wondering if similar formula exists for $n$th prime in terms of Riemann zeros?
Or any other ...
- 375
15
votes
2
answers
728
views
If $\zeta(s)=0$ with $\Re(s)=\frac{1}{2}$, is then $|\hat{\zeta}(s,3)|^2=\frac{1}{2}$?
Helmut Hasse has proved that for $s \in \mathbb{C}-\{1\}$ the Riemann zeta function can be written as:
$$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=0}^\infty\frac{1}{2^{n+1}}\sum_{k=0}^n(-1)^k\ {n \choose k}...
user6671
-2
votes
1
answer
314
views
Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$
(Question is short and straight-forward. )
What is/are "nice and non-trivial" series representation/s of $\log(|\zeta(\frac{1}{2}+it)|)$ ??
By "nice and non-trivial" I mean contains no ...
- 375
-1
votes
1
answer
371
views
An argument involving integratibility of a research paper of Rivoal
I am self studying a research paper in analytic number theory (Ball and Rivoal - Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs) and I am unable to think about an ...
- 793
9
votes
1
answer
940
views
A question on the Riemann zeta function
Yesterday, a certain very talented and passionate young student from Southern Africa asked me the following question about the Riemann zeta function $\zeta(s)$. He says he "thinks" he knows the answer,...
1
vote
1
answer
706
views
How to prove a result related to prime number theorem in research paper of Rivoal and Zudilin
Question is ->I am studying research paper: A note on odd zeta values and I am unable to think how to deduce a result which the authors don't prove. This result has to be proved assuming the prime ...
- 793
3
votes
0
answers
219
views
About generalized binomial theorem and Grünwald-Letnikov fractional derivative
I have run into a problem while computing the fractional derivatives of order $\alpha$ for the Riemann zeta function. My Theorem states
Let $s\in\mathbb{C}$, $\mathfrak{Re}(s)>1$, then the ...
- 169
0
votes
0
answers
104
views
Proportionality constant for the growth rate of $\zeta\left(s\right)$ along the imaginary axis
Going through what it says on page 95 of Titchmarsh's book on the Zeta function, and using his remark about $\mu$ at the bottom of that page, I conclude that there is some constant $C>0$ so that: $$...
- 1,284
1
vote
0
answers
97
views
On some integral involving the Liouville sum
Define $L(x) = \sum_{n\leq x} \lambda(n)$, where $\lambda$ denotes the Liouville function. If $c$ is the supremum of the real parts of the zeros of the Riemann zeta function and $\lfloor x\rfloor$ ...
- 1,019
-8
votes
1
answer
559
views
A question in paper " A note on Odd zeta values " by Tanguy Rivoal and Wadim Zudilin on page 6
I am studying research paper " A note on odd zeta values " by Tanguy Rivoal and Wadim Zudilin .
Note-> This question has been closed 2 times on math.stackexchange . Earlier it was posted ...
- 793
11
votes
1
answer
518
views
Second moment estimates for $\zeta(s)$: different methods?
What are some different ways to achieve the bound $$\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt = T \log \frac{T}{2 \pi} + (2 \gamma - 1) T + E(T)$$
with an error term $E(T) = O(T^{\...
- 20.2k
7
votes
0
answers
173
views
Fully explicit version of Atkinson's formula?
Let $$I(T)=\int_0^T \left|\zeta\left(\frac{1}{2} + i t\right)\right|^2 dt$$
and let $E(T)$ be $I(T)$ minus what turn out to be its main terms:
$$E(T) = I(T)- T \log \frac{T}{2 \pi} - (2 \gamma - 1) T.$...
- 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 $x\geq |t|$,
$$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} + \frac{x^{1-s}}{s-1} + O\left(\...
- 20.2k
10
votes
0
answers
570
views
Bounding $1/\zeta(s)$ given RH
Let $T\geq 0$. Assume RH(T+100), that is, assume that all non-trivial zeros $\rho$ of the Riemann zeta function with $|\Im(\rho)|\leq T+100$ satisfy $\Re(\rho)=1/2$. Can we then give a good upper ...
- 20.2k
2
votes
0
answers
158
views
Subsets of particular values of $\zeta'(k)$ that contain irrational numbers
We consider the set of elements $\zeta'(2),\zeta'(3),\zeta'(4),\zeta'(5),\ldots$ where $\zeta(z)$ is the Riemann zeta function and $\zeta'(z)=\frac{d}{dz}\zeta(z)$ its derivative. Thus we consider ...
-1
votes
1
answer
512
views
Does $\int_{2}^{\infty} (\pi(x)-Li(x))x^{-s-1} \mathrm{d}x$ converge on the real axis for $s>1/2$? [closed]
Consider the prime zeta function, defined for $\Re(s)>1$, by the infinite series
$$\sum_{p} p^{-s} = \sum_{m=1}^{\infty}\frac{\mu(m)}{m}\log \zeta(ms)$$ where $p$ denotes a prime, $\mu$ the Mobius ...
user146617
3
votes
2
answers
1k
views
On the Dirichlet series for $1/\zeta(s)$ for real $s$ and the zeros of zeta
For $\Re(s)>1$, it is well known that
$$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$$ where $\mu$ denotes the Mobius function and $\zeta$ is the Riemann zeta function. I have heard ...
- 39
3
votes
0
answers
239
views
Riemann hypothesis and ternary Goldbach
Is there any result of the following shape: There exists an absolute constant $\delta>0$ such that the Riemann hypothesis for some $L$-functions is equivalent to the following estimate for all ...
- 3,062
4
votes
0
answers
922
views
Guessing of $n$th prime from "super- regularized" product of primes
( I've been thinking about asking this for a long time . Though this is not rigorous; It can be thought of as heuristic or extraction of information from different viewpoint.)
We know "super-...
- 790
4
votes
1
answer
928
views
On a possible equivalent of Riemann hypothesis
I've read in a Bombieri's paper on official problem statement of Riemann hypothesis for Clay Math institute's millennium problems, a statement and what I understood of it is the following :
The ...
- 790
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
2
votes
1
answer
197
views
Interpretation of an equivalence to the Riemann hypothesis due to de Reyna and Toulisse in the spirit of a formula from an article
In [1] the authors present an equivalence to the Riemann hypothesis that is the Theorem 6.2.
On the other hand I know a statement from [2], in English this is the article Andrew Granville and Greg ...
2
votes
1
answer
2k
views
Books on complex analysis for self learning that includes the Riemann zeta function?
I am searching for an introductory book in the field of complex analysis for self learning, that would contain the following:
Analytic number theory : the connection between complex analysis and
...
- 29
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 ...
- 20.2k
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}).$$
But do there exist infinitely ...
user140392
3
votes
1
answer
709
views
Conditional bound on RH for $\Re\left(\sum_{p\leq\sqrt{x}}\frac{(1/2)}{p^{1+2it}}\right)$
I would like to prove that
Assume RH. Let $T$ large, $2\leq x \leq T^2$ and $T\leq t \leq 2T$, then
$$
\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
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
$$
\begin{split}
-\frac{\zeta^{\prime}}{\zeta}(s)&=\sum_{n\leq x}\frac{\Lambda(n)}{n^s}\frac{\log(x/n)}{\log x}+\frac{1}{\log x}\left(\frac{\zeta^{\...
- 199
2
votes
1
answer
390
views
Zeros of polynomial approximations of the Riemann $\zeta$ function
I know next to nothing about analytic number theory, or the theory of the Riemann $\zeta$ function in particular, so the following might be too elementary to deserve more than derision; even so it ...
- 21
0
votes
0
answers
162
views
How to estimate the integral of Riemann zeta function with error term trends to zero as T trends to infinity?
$\zeta(s)$ denotes the Riemann zeta function. For $T>0$ large enough $$\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},$$
basically because $x\mapsto 1/x^s$ is ...
- 20.2k
2
votes
2
answers
412
views
Robin's inequality and the zeros of the Riemann zeta function
Robin showed that if $a\in(1/2, 1]$ is the supremum of the real parts of the zeros of the Riemann zeta function $\zeta(s)$, then $f(x)=\Omega_{\pm} (x^{-b})$, where $b$ is some number on $(a-1/2, 1/2],...
- 1,019
3
votes
0
answers
171
views
Estimating integral of product of terms $\cos(t\log p)$
I would like to prove the following proposition from A. Harper's paper "Sharp conditional upper bound for moments of the Riemann Zeta Function"
Proposition.
Let $T$ be large and let $n=p_1^{\...
- 199
1
vote
0
answers
178
views
An integral involving $1/\zeta(s)$ and the zeros of $\zeta(s)$
In my thesis, i stumbled across the following problem:
Define $$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))$$
(with $\pi$ being the normal prime-counting function), then we can write $\...
- 749
9
votes
2
answers
2k
views
Dynamics of Riemann zeta function
Has the dynamics of the Riemann zeta function been studied? By dynamics I mean the limiting behavior of the sequence of iterates $s, \zeta (s), \zeta (\zeta (s)), \zeta (\zeta (\zeta (s)))\dots $ for ...
- 91
3
votes
0
answers
97
views
Supremum of certain modified zeta functions at 1
Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by
$$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$
where $(D/m)$ denotes the Jacobi symbol. ...
45
votes
4
answers
8k
views
Why is so much work done on numerical verification of the Riemann Hypothesis?
I have noticed that there is a huge amount of work which has been done on numerically verifying the Riemann hypothesis for larger and larger non-trivial zeroes.
I don't mean to ask a stupid question, ...
- 5,146
2
votes
2
answers
450
views
On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta function
Define $\theta(x)=\sum_{p\leq x} \log p $, where $p>1$ denotes a prime.
Nicolas proved that if the Riemann zeta function $\zeta(s)$ vanishes for some $s$ with $\Re(s)\leq 1/2 + b$, where $b\in(0, 1/...
user136836
1
vote
0
answers
341
views
Riemann Explicit Formula
I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula:
$$\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
5
votes
2
answers
850
views
Local phase statistics of the nontrivial Riemann zeros
(The question is inspired by Owen Maresh's post)
The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$.
Numerical results on the first 10000 zeros suggest ...
- 9,501
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>...
- 1,019
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 ...
- 587
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 ?...
- 13