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-3 votes
1 answer
193 views

Bounding a number-theoretic integral

Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH. My attempt here is ...
charlie_beck's user avatar
2 votes
2 answers
361 views

Size of $\zeta'(s)$ at its zeros

How large can the derivative of the Riemann zeta function be at its zeros? More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
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
3 votes
1 answer
458 views

Limit of an infinite series with quadratic arguments

I have encountered a limiting process on some infinite series. So, I would like to ask: QUESTION. Assume $n$ is an even positive integer. Is this true? $$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
T. Amdeberhan's user avatar
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
1 vote
0 answers
127 views

an eigenvalue problem for Jacobi Forms

Assume $G(q,z)$ is a Jacobi form of a certain index k. It is known that $G$ can be expanded in a Taylor series with coefficients in the ring of quasi-modular forms (generators $E_2, E_4$ and $E_6$). $\...
T. Amdeberhan's user avatar
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
1 vote
1 answer
344 views

Is there a way to tie up even and "newly suggested odd" Riemann zeta values?

Define the sequence $$a_s=(-1)^{\binom{s-1}2}\left(\frac{\pi}2\right)^s\frac1{2\cdot s!}\begin{cases} s\,E_{s-1}, \qquad \text{if $s$ is odd} \\ 2^{2s}B_s, \qquad \,\,\text{if $s$ is even};\end{cases}$...
T. Amdeberhan's user avatar
7 votes
2 answers
788 views

Reference request for the explicit formula for $\sum_{n\leq x} \Lambda(n)n^{-s}$

Denote by $\Lambda(n)$ the von Mangoldt function, which is equal to $\log p$ if $p\geq 2$ is a prime, and $0$ otherwise. Let $\rho$ denote a complex zero of the Riemann $\zeta$-function. If I recall ...
Q_p's user avatar
  • 1,019
0 votes
1 answer
116 views

Logarithms of $L$-functions of irreducible characters of Galois group

We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...
asrxiiviii's user avatar
2 votes
1 answer
186 views

Existence of analytic continuation of Dirichlet series corresponding to the indicator sequence of a complement of a special multiplicative set

Let $K/ \mathbb Q $ be a finite Galois extension and let $X$ be a proper non-empty subset of the Galois group $G=Gal(K/ \mathbb Q)$ that is closed under conjugation. Consider a set of integer primes $...
asrxiiviii's user avatar
7 votes
1 answer
244 views

Volume of solution sets for polynomials in $\mathbb{C}[x]$

Denote $\pmb{a}=(a_1,\dots,a_d)\in\mathbb{R}^d$ and consider the set $$\mathcal{E}_d=\{\pmb{a}\in\mathbb{R}^d: \text{each root $\xi$ of $x^d+a_dx^{d-1}+\cdots+a_2x+a_1=0$ lies in $\vert\xi\vert<1$}\...
T. Amdeberhan's user avatar
4 votes
0 answers
261 views

Reference request for some result of de Bruijn on zeros of some holomorphic function

In a video lecture on Youtube, ''Vaporizing and freezing the Riemann zeta function'', Terry Tao states that ''de Bruijn proved that if for some $t_0$ the zeros of $H_{t_0}$ are contained in the strip $...
Merton's user avatar
  • 57
5 votes
0 answers
268 views

Reference Request on logarithm derivative of L-functions

I'm looking for references that show almost all Dirichlet characters $\chi \mod q$ satisfy $$|\frac{L'}{L}(1+it, \chi)|=o(\log q)$$ where $t\in \mathbb{R}$ is fixed. I have been able to adapt a method ...
J. Pan's user avatar
  • 51
15 votes
5 answers
2k views

Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as $$ \xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s). $$ It is an entire function whose zeros are precisely those of $\zeta(s)$. Since $\xi$ is real ...
Stopple's user avatar
  • 11.1k
13 votes
2 answers
725 views

Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (...
Myshkin's user avatar
  • 17.6k
9 votes
2 answers
2k views

References on Taylor series expansion of Riemann xi function

I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$. $$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$ where $$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
mike's user avatar
  • 603
8 votes
2 answers
2k views

Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$

Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example ...
Stopple's user avatar
  • 11.1k
2 votes
3 answers
632 views

How to find the almost period of an exponential polynomial

Let $u(t) = \Sigma_{k=1}^n c_k e^{i \lambda_k t} (c_k \in \mathbb C, \lambda_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the ...
Vagabond's user avatar
  • 1,795