All Questions
Tagged with analytic-number-theory riemann-zeta-function
319 questions
-6
votes
1
answer
442
views
On gaps between consecutive zeros of the Riemann zeta function
Let $\gamma$ denote the imaginary part of a non-trivial zero of the Riemann zeta function. Do there exist some function $f$ such that $\gamma_{n+1} - \gamma_n > f(n)>0$ for all large $n$? To be ...
- 1,019
10
votes
2
answers
934
views
$\psi(x)-x$ on average
This is a reference question:
Let $\psi(x)$ be the psi-Chebyshev function. Is there any unconditional result in the literature that proves that there exists $0<a<2$ such that
$$
\int_2^x (\psi(y)...
- 63.9k
9
votes
2
answers
1k
views
On the error term of the Riemann explicit formula
Let: $\rho$ be a non-trivial zero of the Riemann zeta function, $\Lambda$ be the von-Mangoldt function and $\psi(x) =\sum_{n \leq x} \Lambda(n)$. What is the best known upper bound for
$$f(x, T) := \...
- 105k
1
vote
0
answers
482
views
Explicit formula for zeta function with special type of weight
Consider the following line of thinking:
$$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$
Here,
$\operatorname{R}(x) = \...
- 790
4
votes
0
answers
280
views
Order of growth of $\left|\frac{1}{\zeta’(\rho)}\right|$ as $\Im(\rho)\rightarrow\infty$?
Let $\zeta$ denote the Riemann zeta function, and let $\rho\in\mathbb{C}$ be a variable that takes its values among the zeros of the zeta function, so that $\zeta(\rho)=0$, and write $\rho=\sigma+it$. ...
- 63.9k
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 ...
- 1,315
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 ...
- 10.5k
9
votes
1
answer
364
views
Large values of $\zeta(1/2+it)$ from sums of short moments
In a now classical paper, Iwaniec proved the following theorem.
Theorem. Let $T \geq 2$, $T^{1/2} < T_0 \leq T$ and $T \leq t_1 < t_2 < \cdots < t_R \leq 2T$, $t_{r+1} - t_r \geq T_0$. ...
- 1,990
7
votes
1
answer
1k
views
The Correlation of the Möbius Function and Dirichlet Characters
Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$
In other words
$$\phi_{\chi}(n)=\sum_{d|...
- 105k
0
votes
1
answer
168
views
Residue calculation for Eulerian expansion of the cotangent
I am looking for ideas on proving the Eulerian expansion of the cotangent using residue calculation: $$\pi\cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\left(\frac{1}{z+n}+\frac{1}{z-n}\right), \ z\in\...
- 463
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. ...
- 1,890
3
votes
2
answers
813
views
Do all rigorous proofs of Euler's product for $\zeta (s)$ use infinitude of primes?
There are two proofs of
$$\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p \text{ prime}}\frac{1}{1-p^{-s}},\quad \Re (s)\gt 1$$
which I'm aware of. I'll call the first one the Sieve proof and the second one ...
- 50.6k
1
vote
0
answers
218
views
Reference for explicit formula used by Ramanujan
In his work on highly composite numbers http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf , Ramanujan used a version of an explicit formula (equation (329) on page 133) relating primes and zeros of ...
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) ...
- 11.1k
2
votes
0
answers
238
views
Possible regularisation for sum of function of primes
Consider the following sum of function of primes:
$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$
Here $p$ runs through all primes and $e$ is Euler's constant.
We can see that the sum ...
- 149
7
votes
1
answer
596
views
What consequences would follow from the density hypothesis?
Let $N(\sigma,T)$ denote the number of zeros $\rho=\beta+\gamma i$ of the Riemann zeta function satisfying $\beta\ge \sigma$ and $0<\gamma\le T$, counted with multiplicity. Then the "Density ...
- 435
1
vote
1
answer
122
views
Best possible unconditional partial sum estimate of $\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$:
Consider the following partial sum:
$$S(x,n)=\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$$
Here p runs through primes and $n$ is constant
What is the best possible unconditional( using best known version ...
- 5,089
5
votes
0
answers
260
views
What is the winding behavior of the Riemann zeta function around zero along the line $s=1+it$?
Let $\phi: \mathbb R \setminus \{0\} \to S^1 \subset \mathbb C$ be defined by
$$\phi(t)= \zeta(1+it)/|\zeta(1+it)|$$
(the nonvavishing of the denominator being a bit weaker than the prime number ...
- 105k
4
votes
2
answers
649
views
How can one deduce an approximation for the density function of prime numbers from this Euler's theorem?
The author of Riemann's Zeta Function, H.M.Edwards, says:
According to Euler, $\sum_{p<x}\frac{1}{p}\sim \log(\log(x))$ when $x\longrightarrow\infty$.
$\log(\log(x))=\int_{1}^{\log(x)} \frac{du}{...
- 156k
0
votes
1
answer
186
views
On the explicit upper bound of $|\log\zeta(s)|$ near $\Re(s)=1$
I learn from Montgomery & Vaughan's Multiplicative Number Theory I: Classical theory that there exists $c_1,c_2>0$ such that whenever $\sigma\ge1-c_1/\log|t|,|t|\ge4$ there is
$$
|\log\zeta(\...
- 5,089
0
votes
0
answers
101
views
Prime races in two competing arithmetic progressions - error bound
I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: $(p_{1,n})$ and $(p_{3,n}...
- 3,098
4
votes
1
answer
364
views
Zeros of Dirichlet function $L(s,\chi_4)$
I am wondering if there are some know results for the non-trivial roots at ${\rm Re}(s) = \frac{1}{2}$, even maybe a table of the first few roots with $t>0$. This sister function
$$
L_4^* (s,\chi_4)...
- 50.6k
2
votes
0
answers
173
views
An identity among values of the logarithmic derivative of $\zeta(s)$
From some known special values of the Riemann zeta function and its derivative, one can show that
$$\gamma =1+ \frac{\zeta'(2)}{\zeta(2)} -\frac{\zeta'(0)}{\zeta(0)}+ \frac{\zeta'(-1)}{\zeta(-1)}.$$
...
- 4,985
2
votes
0
answers
158
views
Error or gap in "Modular Functions and Dirichlet Series", by Apostol
My question concerns Apostol's Chapter 7, Kronecker's Theorem with Applications. It's Theorem 7.11, page 156.
I’m attaching the proof in question. There is a lot going on, but I’ve highlighted the ...
- 579
22
votes
2
answers
3k
views
What are the consequences of an ineffective proof of the Riemann Hypothesis?
Suppose a proof came out (and was verified by credible peer review) of the following statement:
There is a $T_0$ such that for all $t>T_0$, all zeros $\zeta(\beta+it)=0$ have $\beta=1/2.$
where $...
- 9,114
2
votes
1
answer
623
views
The nontrivial zeros of the zeta function and the prime counting function
The truncated explicit formula has the shape
\begin{equation}\label{1}
\psi(x) =x-\frac{\zeta^{\prime}(0)}{\zeta(0)}-\sum_{|\rho|\leq T}\frac{x^{\rho}}{\rho}+\sum_{n=1}^{\infty}\frac{x^{-2n}}{2n}+...
- 5
4
votes
1
answer
925
views
A question on the use of fractional derivatives in Riemann Hypothesis
We already know that Riemann-zeta function on the critical band is defined as follows:
$$(1-2^{1-\alpha})\zeta(\alpha) = \sum_{k=1}^{\infty} (-1)^{k+1}k^{-\alpha},\quad \Re(\alpha) \in ]0, 1[ $$
Is ...
- 71
1
vote
0
answers
155
views
Function involving argument of the Riemann zeta function
When $t$ is an ordinate of a zero of Riemann zeta function, we define \begin{equation}
f(t):=\frac{t}{2\pi}\log\left(\frac{t}{2\pi e}\right)+S(t)-\frac{1}{8}+\frac{1}{48 \pi t}+\frac{7}{5760 t^3}+...
- 19
4
votes
0
answers
450
views
Question about a paper by Franca and LeClair in analytic number theory
I am reading an article "Transcendental equations satisfied by the individual
zeros of Riemann $\zeta$, Dirichlet and modular
L-functions" by G. Franca and A. LeClair (2015) see here. The ...
- 41
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
29
votes
4
answers
5k
views
What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differences of zeta zeros?
The number theory community here at University of Michigan is abuzz with talk of this paper recently posted to the arxiv. If you haven't seen it already, the punch line is that the global differences ...
- 2,821
10
votes
1
answer
1k
views
Approximation for the $n$th nontrivial zero of $\zeta(s)$
For all positive integers $n$, let $$t_n = \frac{1}{2\pi} \operatorname{Im} \rho_n,$$ where $\rho_n$ donates the $n$th nontrivial zero of the Riemann zeta function in the upper half plane (listed in ...
CommunityBot
- 1
0
votes
0
answers
169
views
On $\sum_{\rho\in D} \text{dist}(\rho)=\frac{1}{2\pi i} \int_{\partial{D}}\log \zeta(s)\ ds$
Let $D$ denote a closed two dimensional figure as: $D=2+iT\to 2\to 2-\delta\to 2-\delta+i(T-\delta)\to \frac{1}{2}+\epsilon+i(T-\delta)\to\frac{1}{2}+\epsilon\to\frac{1}{2}-\epsilon\to \frac{1}{2}-\...
- 11
-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 ...
- 17k
20
votes
1
answer
745
views
On the equation $\zeta(s) = F(s)+F(s+1)$
Define the function $F(s)$ as the Dirichlet series
$$
F(s) = \sum_{n=1}^\infty \frac{1}{(n+1)n^{s-1}},
$$
which converges for $\operatorname{Re}(s)>1$.
Has anyone seen/studied this function before? ...
- 2,549
3
votes
0
answers
159
views
Is there a general relationship between definite integrals over functions involving the complete elliptic integral of the first kind and zeta values?
Background
Let $\textbf{K}(k)$ be the complete elliptic integral of the first kind, where $k$ is its elliptic modulus [1]. Moreover, define $k' := \sqrt{1-k^{2}} $ as its complementary modulus. I've ...
- 4,829
3
votes
2
answers
386
views
Explicit bound on $\zeta(s)$ inside a zero-free region?
Does anybody know of a place in the literature where one can find an explicit result of the form $|\zeta(\sigma+it)|\leq C \log t$ for $t$ within a zero-free region (assuming $t$ is larger than an ...
- 5,089
2
votes
1
answer
429
views
'Almost all' zeros of the Dirichlet L function lies 'near' the critical line?
Is there a well known result that states that as $t \to \infty$, 'almost all' zeros of any Dirichlet L function $L(s,\chi)$ lie in the region $R= \{\sigma+i t\mid |\sigma -\frac{1}{2}| \leq \Phi(t) \}$...
2
votes
1
answer
301
views
Averages of Möbius function in arithmetic progressions
It is mentioned in multiple occasions here that the bound
$$
\mathop{\sum_{n=1}^{N}}_{n\equiv a\mod l} \mu(n) = o(N)
$$
is equivalent to the prime number theorem in arithmetic progressions. But I am ...
- 50.6k
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 ...
- 105k
-1
votes
1
answer
250
views
Significance of $N_0(T+1)-N_0(T)\sim \frac{1}{2\pi}\log \frac{T}{2\pi}$
Let $N(T)$ be the number of zeros of Riemann zeta function upto height $T$ in the critical strip and $N_0(T)$ be the number of zeros on the critical line.
What will be the significance of proving ...
- 105k
0
votes
0
answers
131
views
Generalization of the zeta values
Consider the constants $c(s_1,s_2,...)=\sum_{n=1}^\infty \frac{1}{n^{s_n}}$ where the $s_i$'s are fixed positive integers and the number of $i$ with $s_i=1$ is finite (so that the previous sum ...
- 361
1
vote
0
answers
85
views
Why this plot of Siegel Z function is very close to two halve lines?
Let $Z(s)$ be Siegel $Z$ function.
For real $S,T$, define $fu7(S,T)=|Z((S-\frac12)/i + T)|$.
In other words for $s=S + i T$ we have $fu7(s)=|Z((s-\frac12)/i)|$.
For $S_0 \in (0,1)$ and $T_0=\rho_1= 14....
- 25.4k
9
votes
0
answers
414
views
From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis
I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
- 3,098
8
votes
1
answer
868
views
A question on an equivalence of RH
In page 6, RH Equivalence 5.3. An equivalence of the Riemann Hypothesis says that
$$\sum_{\rho} \frac{1}{|\rho|^2} =\sum_{\rho} \frac{1}{\rho (1{-}\rho)}= 2 + \gamma - \log 4\pi$$
where $\rho$ is ...
CommunityBot
- 1
1
vote
0
answers
97
views
Bounds and repulsion domains for the Dirichlet eta function $\eta(\sigma+it)$, for fixed $\sigma$
Let $\eta(\sigma+it)$ be the Dirichlet eta function, with $t>0$ (the variable) and $\sigma$ be fixed, with $\frac{1}{2}\leq \sigma <2$. I define the hole $\Omega_T
=\Omega_T(\sigma)$ as the ...
- 3,098
0
votes
0
answers
157
views
Does it make sense to express upper bounds on arithmetic sequences with Dirichlet generating functions?
In order to see what happens when taking the functional equation in this form:
$$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$
$$\xi(s) = \xi(1 - s)$$
$$\pi^{-s/2}\ \Gamma\left(\...
- 1,183
3
votes
0
answers
101
views
Regularised value of cardinality of non trivial Zeta zeros:
This is a straight forward question so apologies in advance
Consider the following sums:
$$\sum_k1_{\rho_k}$$
$$\sum_k{\rho_k}$$
(i.e. first sum counts non trivial zeros of Zeta function)
I want ...
- 790
6
votes
1
answer
546
views
On Cramér's theorem about roots of Zeta function
Cramér proved the following theorem (see the announcement in [1] and [2]):
Consider the following function:
$$V(z)=\sum_k e^{\rho_kz}$$
Where $\rho_k$ runs through non trivial zeta zeros with $Im(\...
1
vote
0
answers
187
views
$\lim_{x\to \infty} \left(\sum_{n\leq x} (\log n)^k/n - \int_1^x (\log t)^k/t\right) = \text{?}$
It is easy to see (by Euler-Maclaurin, say, or just by thinking of a graph) that
$$\lim_{x\to \infty} \sum_{n\leq x} \frac{(\log n)^k}{n} = \int_1^x \frac{(\log t)^k}{t} + C + O\left(\frac{(\log x)^k}{...
- 20.2k