All Questions
Tagged with analytic-number-theory riemann-zeta-function
319 questions
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
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
$$
\int_2^x (\psi(y)...
- 3,062
1
vote
1
answer
122
views
Best possible unconditional partial sum estimate of $\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$:
Consider the following partial sum:
$$S(x,n)=\sum_{p\leq x}\frac{\ln(p)}{({p})^{n/2}}$$
Here p runs through primes and $n$ is constant
What is the best possible unconditional( using best known version ...
- 149
5
votes
0
answers
260
views
What is the winding behavior of the Riemann zeta function around zero along the line $s=1+it$?
Let $\phi: \mathbb R \setminus \{0\} \to S^1 \subset \mathbb C$ be defined by
$$\phi(t)= \zeta(1+it)/|\zeta(1+it)|$$
(the nonvavishing of the denominator being a bit weaker than the prime number ...
- 63.9k
4
votes
2
answers
649
views
How can one deduce an approximation for the density function of prime numbers from this Euler's theorem?
The author of Riemann's Zeta Function, H.M.Edwards, says:
According to Euler, $\sum_{p<x}\frac{1}{p}\sim \log(\log(x))$ when $x\longrightarrow\infty$.
$\log(\log(x))=\int_{1}^{\log(x)} \frac{du}{...
0
votes
0
answers
101
views
Prime races in two competing arithmetic progressions - error bound
I read an article by Andrew Granville on the subject, there's actually quite a bit of recent literature on the topic. My problem is as follows. I have two sequences of primes: $(p_{1,n})$ and $(p_{3,n}...
- 3,098
4
votes
1
answer
364
views
Zeros of Dirichlet function $L(s,\chi_4)$
I am wondering if there are some know results for the non-trivial roots at ${\rm Re}(s) = \frac{1}{2}$, even maybe a table of the first few roots with $t>0$. This sister function
$$
L_4^* (s,\chi_4)...
- 3,098
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
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
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}+...
- 21
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
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
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
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
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
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 ...
- 20.2k
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 $R= \{\sigma+i t\mid |\sigma -\frac{1}{2}| \leq \Phi(t) \}$...
- 163
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 ...
- 589
-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 ...
user485483
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
$$
|\log\zeta(\...
- 1,315
0
votes
0
answers
131
views
Generalization of the zeta values
Consider the constants $c(s_1,s_2,...)=\sum_{n=1}^\infty \frac{1}{n^{s_n}}$ where the $s_i$'s are fixed positive integers and the number of $i$ with $s_i=1$ is finite (so that the previous sum ...
- 361
1
vote
0
answers
85
views
Why this plot of Siegel Z function is very close to two halve lines?
Let $Z(s)$ be Siegel $Z$ function.
For real $S,T$, define $fu7(S,T)=|Z((S-\frac12)/i + T)|$.
In other words for $s=S + i T$ we have $fu7(s)=|Z((s-\frac12)/i)|$.
For $S_0 \in (0,1)$ and $T_0=\rho_1= 14....
- 25.4k
9
votes
0
answers
414
views
From holes in the image of peculiar functions to new perspective on the Riemann Hypothesis
I am working with the Dirichlet eta function $\eta(z)$, with $z=\sigma+it$, $\sigma > \frac{1}{2}$, and $t>0$. Let us define
$$\eta_n(z,\gamma)= \sum_{k=1}^n (-1)^{k+1}\lambda_k^{-\sigma} e^{-it\...
- 3,098
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}$$
Where $\rho_k$ runs through non trivial zeta zeros with $Im(\...
- 790
1
vote
0
answers
187
views
$\lim_{x\to \infty} \left(\sum_{n\leq x} (\log n)^k/n - \int_1^x (\log t)^k/t\right) = \text{?}$
It is easy to see (by Euler-Maclaurin, say, or just by thinking of a graph) that
$$\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
1
vote
0
answers
213
views
Convergence of zeta Euler product with additional term
Let's consider the following Euler product ($s=\sigma+it)$:
$$ P(s)=\prod_{p \; \text{prime}} \frac{1}{1-p^{-s}} \; e^{-p^{-s}}$$
So for $\sigma>1$, it is clear the product converges and we have:
$$...
- 1,199
4
votes
1
answer
326
views
$\zeta(s) = 1 + X^{1-s}/(s-1) + ...$?
Let $s = \sigma+ i t$ with $0\leq \sigma\leq 1$, $|t|\leq X$, where $1\leq X<2$.
It is easy to use the Euler-Maclaurin formula to prove a result of the form
$$\left|\zeta(s) - 1 - \frac{X^{1-s}}{s-...
- 20.2k
3
votes
0
answers
161
views
$\zeta(s) = \sum_{n\leq x} n^{-s} - x^{1-s}/(1-s) + ...$ through bounded-order Euler-Maclaurin?
It is a basic classical result (Titchmarsh Thm 4.11; credited to Hardy-Littlewood) that,
uniformly for $\Re s \geq \sigma_0>0$, $t\leq 6 x$ (say),
$$\zeta(s) = \sum_{n\leq x} \frac{1}{n^s} - \frac{...
- 20.2k
1
vote
0
answers
138
views
Integral of $|1/\zeta(\sigma+i T)|$ (or $|(1/\zeta(\sigma+i T))^{(k)}|$) on a horizontal half-line in the left upper quadrant
Let $T_0\geq 20$. Let $L$ be the half-line from $-\infty + i T$ to (say) $-1/2 + i T$. Since $|\zeta(s)|$ is roughly proportional to $(T/2 \pi e)^\sigma$ for $s=\sigma+ i T$ on $L$, it is clear that ...
- 20.2k
3
votes
0
answers
4k
views
Intuition for the bias of the partial sums of the Liouville function
It's a well known result that the Dirichlet series of the Liouville function $ \lambda(n) $ is given by
$$ \sum_{k=1}^{\infty} \frac{\lambda(k)}{k^s} = \frac{\zeta(2s)}{\zeta(s)} $$
If we use Perron's ...
- 291
9
votes
1
answer
585
views
Scattering amplitudes and the Riemann zeta function
I'm reading Amplitudes and the Riemann Zeta Function, which recently appeared in Physical Review Letters. It's received some publicity, including my own campus' PR operation. From the abstract (...
- 11.1k
3
votes
0
answers
128
views
Laplace transform of power of zeta function
Let $s$ is the complex variable. I would like to figure out the region of absolutely convergency of the following integral
$$
e^{\frac{is}{2}}\int\limits_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}\...
- 51
9
votes
1
answer
853
views
Moments of the Riemann zeta function
Is it possible to get an upper bound better than $\ll_\sigma T^{3/2-\sigma}$ for $$\int_{0}^{T}|\zeta (\sigma +it)|\,dt,\qquad 0<\sigma<1/2\,?$$
- 150
5
votes
2
answers
1k
views
Density of fake zeros of Zeta
I am investigating whether or not there exist $\epsilon > 0$ such that $\zeta(s) \neq 0$ on the strip $1-\epsilon < \Re(s) \leq 1$.
Suppose not. Then given $\delta > 0$ there exists a zero ...
- 3,699
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
$$M(T) = \max_{|\Im \rho|\leq T} \left|\mathrm{Res}_{s=\rho} \frac{1}{\zeta(s)}\right| =
\max_{|\Im \rho|\leq T} \frac{1}...
- 20.2k
4
votes
1
answer
333
views
Double sum over zeros of Riemann zeta-function
In a paper by Saffari and Vaughan there appears a complicated-looking double sum
$$\Sigma_1=\sum_{\rho_1}\sum_{\rho_2}\frac{(1+\theta)^{\rho_1}-1}{\rho_1}\cdot \frac{(1+\theta)^{\bar{\rho_2}}-1}{\bar{\...
- 465
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:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
- 3,064
6
votes
0
answers
265
views
Best explicit bound on $\zeta'(1+it)/\zeta(1+it)$
Assume the Riemann hypothesis. We know that
$$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| \leq 2 \log \log t + O(1)$$
(see, e.g., Thm. 13.13 in Montgomery-Vaughan). What is the best explicit bound ...
- 20.2k
6
votes
0
answers
225
views
Conditional results on average size of Mertens' function
Let $M(x) = \sum_{n \le x} \mu(n)$ where $\mu$ is the Möbius function. Titchmarsh, in his book on the Riemann zeta function, considers consequences of the hypothesis that
$$\int_{1}^{X} \left( \frac{M(...
- 14.6k
0
votes
1
answer
195
views
Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?
Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1} $$ for all positive integers $n$.
Also $\lambda_n$ is given as a sum over the non ...
user469908
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 ...
- 4,985
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
7
votes
0
answers
328
views
Does $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t)$ have mean value $0$?
I'm curious about what is known about the distribution of the function $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t) \in (-1,1]$, on a linear or logarithmic scale, where $\operatorname{Arg}$ ...
- 4,985