Skip to main content

All Questions

36 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
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}{...
Kevin Smith's user avatar
  • 2,480
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\...
Vincent Granville's user avatar
9 votes
0 answers
264 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 ...
Kevin Smith's user avatar
  • 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 (...
H A Helfgott's user avatar
  • 20.1k
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 ...
Tim Campion's user avatar
  • 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} \...
Q_p's user avatar
  • 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 ...
Craig Calcaterra's user avatar
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 ...
Stopple's user avatar
  • 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 ...
Max's user avatar
  • 11
4 votes
0 answers
279 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 $\rho=\sigma+it$. ...
EGME's user avatar
  • 1,018
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 ...
Williams's user avatar
4 votes
0 answers
268 views

Four infinite series involving Riemann zeta function

Can you provide a proof for at least one of the claims given below? It is known that $\pi=\displaystyle\sum_{n=1}^{\infty}\frac{3^n-1}{4^n} \cdot \zeta(n+1)$ where $\zeta$ denotes Riemann zeta ...
Pedja's user avatar
  • 2,661
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, ...
user156584's user avatar
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}\...
Mark's user avatar
  • 51
3 votes
0 answers
481 views

Characterizing essential singularities

In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ...
Victor Liu's user avatar
2 votes
0 answers
217 views

Zeta function associated with a function $f$

Let the function $f(t) = \cos(at)$, where ($0 < a < 1$). Let us define $$\zeta(z, f) = \frac{1}{\Gamma(z)} \int_0^{+\infty} \frac{t^{z-1}\cos(at)}{e^t-1}\, dt. $$ Is there a general formula that ...
L.L's user avatar
  • 463
2 votes
0 answers
210 views

Binomial transform of Dirchlet series (2)

Referring to this MO question, i managed to do the following : We denote by $J(k+1,z)$ the sum : $$J(n+1,z)=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\frac{\theta(k+1)}{(k+1)^{z}}$$ and by $S(k+1,z)$ the sum :...
mohammad-83's user avatar
2 votes
0 answers
180 views

Multiple zeta values related to fractional calculus and an Appell polynomial sequence

There is an Appell sequence of polynomials $p_n(z)$ related to an infinitesimal generator for one rep of the fractional calculus that have coefficients involving the Riemann zeta function values at ...
Tom Copeland's user avatar
  • 10.5k
2 votes
0 answers
131 views

multiplicative sequences $b(n)$ such that $F(s)$ is meromorphic

Is it possible to describe the set of sequences $a(n) = \mathcal{O}(1)$ such that $$\frac{F'}{F}(s) = \sum_{p^k} a(p^k) p^{-sk} \ln p \qquad (Re(s) > 1)$$ is the logarithmic derivative of a ...
reuns's user avatar
  • 3,403
2 votes
0 answers
484 views

Mellin inverse of the Hadamard product rep. of the Riemann zeta function?

The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely : $$\left \lfloor x \right \rfloor=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta(s)\frac{x^{s}}{...
mohammad-83's user avatar
2 votes
1 answer
561 views

On infinite sum containing logarithmic derivative of Zeta function and Möbius function:

Consider the following function: $$F(s)= \sum_m \mu(m) \sum_n \frac{e^{-n/2}\zeta^\prime (mns)}{n \zeta(mns)}$$ Now, we can see, that function has simple poles ${\left[\frac{1}{n}\right]}_{n=1}^\...
Zaza's user avatar
  • 149
1 vote
0 answers
113 views

Are there any known statistics on the sign of the Stieltjes Constants?

The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$ $$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
Sidharth Ghoshal's user avatar
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}+...
Steve's user avatar
  • 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: $$...
Bertrand's user avatar
  • 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 $w=1$. ...
Vincent Granville's user avatar
1 vote
0 answers
105 views

sum involving Riemann zeta function

In my work arose the following series: $$g(s) = \sum_{n = 0}^\infty \frac{\log(\zeta(n+2))}{n+2}s^{n}.$$ It has radius of convergence $2$ and converges for $|s| \leq 2$ except at $s = 2$. I'm ...
Jesse Elliott's user avatar
1 vote
0 answers
131 views

Looking for a citation for a result of Littlewood

I see that it is proven by Hardy in 1914 that there are an infinite number of zeros on the critical line. I also see that the Hardy and Littlewood conjectures appear in some papers they wrote ...
Jimmy Rankerman's user avatar
1 vote
0 answers
62 views

Example of an integer $n_0$ such that $1+\sum_{k=2}^{n_0} \zeta(k)^s=0$ has repeated roots

After I was studying the exercise Problem 4.20 from [1] I was inspired to ask about next problem, where $\zeta(k)$ denotes, for integers $k>1$, particular values of the Riemann zeta function. And $...
user142929's user avatar
0 votes
0 answers
74 views

Singular behavior of zeros of incomplete zeta function

I've been looking at the zeros of the incomplete zeta function $\zeta_{lower}(s, z)$ recently. $$ \zeta_{\mathrm{lower}}(s,z)=-\frac{{\Gamma(1-s)}}{2\pi i}\int_{z}^{\infty}\frac{{(-t)^{s-1}}}{e^{t}-1}...
Zhobbyist's user avatar
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}-\...
Honor's user avatar
  • 11
0 votes
0 answers
151 views

Abscissa of convergence of transformed Dirichlet series

Let $$F(s)=\sum_{k=1}^\infty \frac{f(k)}{k^s} \mbox{ and }F^*(s)=\sum_{k=2}^\infty \frac{f(k)g(k)}{k^s},$$ where the infinite sum $\sum f(k)$ diverges, $f(k)$ and $g(k)$ are real numbers, $s$ a ...
Vincent Granville's user avatar
0 votes
2 answers
682 views

On integral relating logarithm of absolute value of Zeta function

Sorry for such a direct question: Consider the following integral: $$I(t)=\int_{1/2}^{1} {\log|\zeta(a+it)|}da.$$ How to find the nature of $I(t)$ as $t\rightarrow\infty$?
TPC's user avatar
  • 774
0 votes
0 answers
108 views

A line integral involving $\zeta(s)$

Let $x>1$ and $\zeta$ denote the Riemann zeta function. Is the equality $$\int_{\sigma - i \infty}^{\sigma + i \infty} \frac{\zeta(s)x^{s}}{s(s+1)(s+2)\cdots (s+k)} \mathrm{d}s = 2\pi i$$ possible ...
user152442's user avatar
0 votes
0 answers
126 views

On an integral involving $\zeta(1/2 + i\tau)$

Denote by $\zeta$ the Riemann zeta function. Define $$F_{y}(x)= \int_{-\infty}^{\infty} \frac{x^{iu}\zeta(1/2 + it + iu)}{u^2 + y^2} \mathrm{d}u.$$ For some fixed real number $t$, is there any $y&...
user avatar
0 votes
0 answers
158 views

On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function

Let $$I(T) = \int_{-T}^{T} \frac{\log|\zeta(\frac{1}{2} + it|)|}{\frac{1}{4}+t^2}\mathrm{d}t$$ where $\zeta$ denotes the Riemann zeta function. What are the reasonable asymptotic estimates for $I(T)...
user avatar
0 votes
0 answers
194 views

Simple proof for Riemann/Hurwitz ζ functional equation

For the purpose of formalisation, I am looking for a simple proof of the function equation of the Riemann ζ function or the generalisation thereof, the Hurwitz's formula for the Hurwitz ζ function. ...
Manuel Eberl's user avatar
  • 1,241