All Questions
13 questions with no upvoted or accepted answers
11
votes
0
answers
530
views
Cesaro summation of a particular Dirichlet series associated with $\zeta(s)$
If you've investigated the error in Perron's formula in general, you've probably noticed that Cesaro summation $$\lim_{x\rightarrow\infty}\sum_{n\leq x} \left(1-\frac{\log n}{\log x}\right)\frac{a_n}{...
- 2,480
9
votes
0
answers
265
views
Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?
Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...
- 2,480
5
votes
0
answers
321
views
Approximating $\zeta^{(r)}(s)$ by a sum
Let $\eta:[0,\infty)\to [0,\infty)$ be compactly supported, continuous and piecewise $C^1$, with its derivative $\eta'$ being of bounded variation. It is completely unsurprising that one can prove (...
- 20.2k
5
votes
0
answers
260
views
What is the winding behavior of the Riemann zeta function around zero along the line $s=1+it$?
Let $\phi: \mathbb R \setminus \{0\} \to S^1 \subset \mathbb C$ be defined by
$$\phi(t)= \zeta(1+it)/|\zeta(1+it)|$$
(the nonvavishing of the denominator being a bit weaker than the prime number ...
- 63.9k
5
votes
0
answers
161
views
On the asymptotics of some sum involving the Mertens function
Let $a_n$ be a sequence of nonnegative real numbers such that $\sum_{n\leq x} a_n \gg \frac{\sqrt x}{\log x}$ for large enough $x$. Denote by $\mu$ the Mobius function, and let $M(N)=\sum_{n\leq N} \...
- 1,019
5
votes
0
answers
694
views
Has the Weierstass transform been used to give Hermite series representations of the Riemann zeta function?
The inverse of the Weierstrass transform
expands a function as a series of Hermite polynomials $H_{n}$. There are several ways to invert the Weierstrass transform which led me to the following ...
4
votes
0
answers
168
views
Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
- 11.1k
4
votes
0
answers
821
views
One of the numbers $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational
I am reading an interesting paper One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive ...
- 11
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
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
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
1
vote
0
answers
381
views
Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$
I am actually interested in the analytic continuation of $\phi_w(s)=\prod_p(1+w\cdot p^{-s})^{-1}$. Here $w$ is rational, or the imaginary unit multiplied by a rational. Consider for now that $w=1$. ...
- 3,098
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