All Questions
Tagged with analytic-number-theory riemann-zeta-function
105 questions with no upvoted or accepted answers
2
votes
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173
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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
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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
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
2
votes
0
answers
273
views
Applications of Jensen's Formula to entire functions of finite order
I am trying to understand a frequently omitted technical detail in applications of Jensen's Formula to bound the number of zeros of entire functions of finite order.
We say that an entire function $f(...
- 1,570
2
votes
0
answers
154
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How can collections of rational zeta series that are equal to $\sum_{n=2}^{\infty} (\zeta(n) - 1)^{p} $ be obtained?
It has been discovered long ago that
$$\sum_{n=2}^{\infty} \big(\zeta(n) - 1\big) = 1. \label{1} \tag{1} $$ More recently, a generalization of this result with binomial coefficients has been obtained: ...
- 4,829
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 :...
- 181
2
votes
0
answers
313
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Proving that the Riemann zeta function is zero free on Re=1 using the prime number theorem
Is $\frac{-\zeta'(s)}{\zeta(s)}+\frac{-s}{s-1}$ an analytic continuation, holomorphic for $Re\ s > 0,\ s\neq 1$, of $f(s)=s\int_{1}^{\infty}\frac{\psi(x)-x}{x^{s+1}}\mathrm{d}x$?
If so: Let $s_{0}$ ...
- 129
2
votes
0
answers
537
views
Explicit formula for $n$th prime in terms of Riemann zeros:
We all know there exists an explicit Formula for prime counting function in terms of Riemann zeros.
I'm wondering if similar formula exists for $n$th prime in terms of Riemann zeros?
Or any other ...
- 375
2
votes
0
answers
158
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Subsets of particular values of $\zeta'(k)$ that contain irrational numbers
We consider the set of elements $\zeta'(2),\zeta'(3),\zeta'(4),\zeta'(5),\ldots$ where $\zeta(z)$ is the Riemann zeta function and $\zeta'(z)=\frac{d}{dz}\zeta(z)$ its derivative. Thus we consider ...
2
votes
0
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135
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What is the best known upper bound for $( \gamma_{n+1}-\gamma_{n})\max_{\{T\in(\gamma_{n},\gamma_{n+1})\}}(\vert\zeta(1/2+iT)\vert) $?
For $ n $ a positive integer, denote by $ L(n) : =\gamma_{n+1}-\gamma_{n} $ with $ \gamma_{n} $ the imaginary part of the $ n $-th critical zero of the Riemann zeta function and by $ M(n) : =\...
- 6,984
2
votes
0
answers
176
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Dirichlet series as rational zeta expressions
Let $D(f,s):=\sum_{n=1}^\infty \frac{f(n)}{n^s}$, otherwise known as a Dirichlet series. When $f$ is a multiplicative, number theoretic function, $D(f,s)$ tends to be expressed as a rational product ...
- 4,952
2
votes
0
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147
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Skewes' number and the ratio $\frac{\operatorname{li}(x^{1/2})}{\operatorname{li}(x)-\pi(x)}$
(A complementary post is here.)
Given the prime counting function $\pi(x)$ and the logarithmic integral $\operatorname{li}(x)$, we have Table 1,
$$\begin{array}{|c|l|}
\hline
x&\operatorname{li}...
- 12.6k
2
votes
0
answers
451
views
Analytic continuation of "composite" zeta function
Let us define the Dirichlet series $$\mathcal C(s):=\sum_{n\text{ composite}}\frac{1}{n^s},\quad P(s):=\sum_{p\text{ prime}}\frac{1}{p^s}.$$
They are absolutely convergent in the half-plane $\sigma>...
- 21
2
votes
1
answer
561
views
On infinite sum containing logarithmic derivative of Zeta function and Möbius function:
Consider the following function:
$$F(s)= \sum_m \mu(m) \sum_n \frac{e^{-n/2}\zeta^\prime (mns)}{n \zeta(mns)}$$
Now, we can see, that function has simple poles ${\left[\frac{1}{n}\right]}_{n=1}^\...
- 149
1
vote
0
answers
100
views
Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
- 178
1
vote
0
answers
113
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Are there any known statistics on the sign of the Stieltjes Constants?
The Stieltjes Constants $\gamma_n$ arise from considering the laurent series of the Riemann Zeta function at $s=1$
$$ \zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} (-1)^n \frac{\gamma_n}{n!} (s-1)^n $...
- 2,689
1
vote
0
answers
128
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On Zudilin's linear forms in $1,\zeta(5)$ and $\zeta(7)$
I am reading an article "Well-poised hypergeometric service for diophantine problems of zeta values" by W. Zudilin.
Consider the quantities defined here in pg. $617$
$$\tilde{F_n}:= \frac{1}{...
- 11
1
vote
0
answers
211
views
Understanding the effect of PDE solution on critical strip?
I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
- 383
1
vote
0
answers
218
views
Reference for explicit formula used by Ramanujan
In his work on highly composite numbers http://math.univ-lyon1.fr/~nicolas/ramanujanNR.pdf , Ramanujan used a version of an explicit formula (equation (329) on page 133) relating primes and zeros of ...
- 63
1
vote
0
answers
155
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Function involving argument of the Riemann zeta function
When $t$ is an ordinate of a zero of Riemann zeta function, we define \begin{equation}
f(t):=\frac{t}{2\pi}\log\left(\frac{t}{2\pi e}\right)+S(t)-\frac{1}{8}+\frac{1}{48 \pi t}+\frac{7}{5760 t^3}+...
- 19
1
vote
0
answers
482
views
Explicit formula for zeta function with special type of weight
Consider the following line of thinking:
$$\pi(x) = \operatorname{R}(x) - \sum_{\rho}\operatorname{R}(x^{\rho}) - \frac1{\ln x} + \frac1\pi \arctan \frac\pi{\ln x} $$
Here,
$\operatorname{R}(x) = \...
- 790
1
vote
0
answers
85
views
Why this plot of Siegel Z function is very close to two halve lines?
Let $Z(s)$ be Siegel $Z$ function.
For real $S,T$, define $fu7(S,T)=|Z((S-\frac12)/i + T)|$.
In other words for $s=S + i T$ we have $fu7(s)=|Z((s-\frac12)/i)|$.
For $S_0 \in (0,1)$ and $T_0=\rho_1= 14....
- 25.4k
1
vote
0
answers
97
views
Bounds and repulsion domains for the Dirichlet eta function $\eta(\sigma+it)$, for fixed $\sigma$
Let $\eta(\sigma+it)$ be the Dirichlet eta function, with $t>0$ (the variable) and $\sigma$ be fixed, with $\frac{1}{2}\leq \sigma <2$. I define the hole $\Omega_T
=\Omega_T(\sigma)$ as the ...
- 3,098
1
vote
0
answers
187
views
$\lim_{x\to \infty} \left(\sum_{n\leq x} (\log n)^k/n - \int_1^x (\log t)^k/t\right) = \text{?}$
It is easy to see (by Euler-Maclaurin, say, or just by thinking of a graph) that
$$\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:
$$...
- 1,199
1
vote
0
answers
138
views
Integral of $|1/\zeta(\sigma+i T)|$ (or $|(1/\zeta(\sigma+i T))^{(k)}|$) on a horizontal half-line in the left upper quadrant
Let $T_0\geq 20$. Let $L$ be the half-line from $-\infty + i T$ to (say) $-1/2 + i T$. Since $|\zeta(s)|$ is roughly proportional to $(T/2 \pi e)^\sigma$ for $s=\sigma+ i T$ on $L$, it is clear that ...
- 20.2k
1
vote
0
answers
360
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Incredibly accurate recursions for the Riemann Zeta function
Last update as of Jan 27, 2021: I posted this as an article for laymen, here. It is very light mathematically speaking, but section 3 is a little more accurate than my post here.
During some ...
- 3,098
1
vote
0
answers
253
views
Who formulated the conjecture that the set of real parts of zeros of the Riemann zeta function is dense in $[0,1]$?
Does anyone know who formulated this conjecture related to Riemann's zeta function?
Conjecture. The set $$\{ x : \exists y \space \space \zeta (x+iy) = 0\}$$ is dense in $[0, 1]$.
In ...
1
vote
0
answers
97
views
On some integral involving the Liouville sum
Define $L(x) = \sum_{n\leq x} \lambda(n)$, where $\lambda$ denotes the Liouville function. If $c$ is the supremum of the real parts of the zeros of the Riemann zeta function and $\lfloor x\rfloor$ ...
- 1,019
1
vote
0
answers
178
views
An integral involving $1/\zeta(s)$ and the zeros of $\zeta(s)$
In my thesis, i stumbled across the following problem:
Define $$F(\sigma, T)=\frac{1}{T}\int_{-T}^{T} \frac{1}{|\zeta(\sigma+it)|^2} \mathrm{d}T$$ where $\zeta$ denotes the Riemann zeta function.
Is ...
- 11
1
vote
0
answers
188
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Questions on Riemann's explicit formula
If we consider this version of the prime-counting function
$$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$
(with $\pi$ being the normal prime-counting function), then we can write $\...
- 749
1
vote
0
answers
341
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Riemann Explicit Formula
I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula:
$$\sum_{n \leq x} \Lambda(n) n^{-s} = -\frac{\zeta'}{\zeta}(s)...
- 547
1
vote
0
answers
74
views
Enquiry on bounds for $\frac{1}{(n-1)!}\frac{d^n}{ds^n}((s-1)\zeta(s))$ at $s=1.$
Let $\zeta$ be the Riemann zeta function and $n$ be a positive integer. What are the known (conditional and unconditional) bounds for $f(n)=\frac{1}{(n-1)!}\frac{d^n}{ds^n}((s-1)\zeta(s))$ at $s=1$ ?
...
- 11
1
vote
0
answers
150
views
Do we believe that the distribution of spacings of successive critical zeros of zeta is log-symmetric?
Let $\gamma^{+}(T)$ be the imaginary part of the critical zero of $\zeta$ closest to $1/2+iT$ with $\gamma^{+}(T)\ge T$ and define similarly $\gamma^{-}(T)$ with a reversed inequality. Let $g(T)$ be ...
- 6,984
1
vote
0
answers
242
views
Hardy-Littlewood vs heuristics on the zeta zeros
The first Hardy-Littlewood Conjecture asserts:
Conjecture 1: Fix integers $0< a_1 < \ldots < a_k$. The density of those primes $p\le x$ such that $p + 2a_1,\ldots, p+2a_k$ are also primes, is:...
user129745
1
vote
0
answers
152
views
A mixed of the Dedekind zeta function and the L-function
I have recently come across the following function, which seems like a "mix" between the Dedekind zeta function and the L-function:
$\sum_I\frac{\chi_k(N(I))}{N(I)^s}$
where $\chi_k(n)$ is the ...
1
vote
0
answers
130
views
On the asymptotic form of a sum over the nontrivial zeros of the Riemann zeta function
What is the asymptotic form of the sum
$$\sum_{\rho} \dfrac{x^{\rho}-(x-2)^{\rho}}{\rho}$$
where the summation is over the nontrivial zeros of the Riemann zeta function?
By the Prime Number Theorem,...
1
vote
0
answers
202
views
Estimation of the $k$-th derivative zeta function
When I was doing some task in number theory which involves bounds for the Riemann zeta function, I was stuck on the following question:
Let $\zeta$ be the Riemann zeta function and let $\zeta^{(k)}$ ...
- 1,257
1
vote
0
answers
223
views
Is the difference of these two real-rooted functions real-rooted?
During our on-going search of approximations to the Riemann $\Xi(z)$ function, we discovered a family of functions $W_n(z)$ as shown in (1).
Our final goal is to prove that:
Proposition 1: $W_{n}(z)...
- 603
0
votes
0
answers
88
views
Accentuating the appearance of convergence of the Möbius function Dirichlet series on the line $\sigma = \frac{2}{3}$ in the critical strip
Set the constant $c$ to:
$$c = -\frac{3}{4}$$
which is in the interval: $$-1 < c < 0$$
and let the matrix $A$ be:
$$A(n,k)=[k|n] - [n=k](1+c)$$
Then form the matrix power series:
$$M=\sum _{n \...
- 1,183
0
votes
0
answers
268
views
Do plots $(5)$ and $(6)$ go to infinity not at the same rate but at similar rates?
The following has been proven by joriki and GH from MO (see here): assuming that $n>1$, then the von Mangoldt function
$$
\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\...
- 1,183
0
votes
0
answers
393
views
Spiralling cycles surrounding zeros
The following came up, as a vague idea, in dialogue with a bright, female, 20 year old student of mine. It is a bit vague, but it seems that conjecture 1 is not present in the literature, which seems ...
- 232
0
votes
0
answers
159
views
The argument of Riemann zeta function and the number of zeros on the critical line
Back ground
I studied the proof of "$KT$ zero theorem" and "$KT\log T$" theorem in Edwards book.
And I'm looking for other kind of evaluation of the number of zeros on the line.
...
- 328
0
votes
0
answers
351
views
On a Duality between Riemann-weil explicit formula and Abel- Plana summation of trigonometric prime counting function:
Consider the analytic function $g(x)$
Now define
$f(x)=g(x)\frac{\sin^2\left(\frac{π\Gamma(x)}{2x}\right)}{\cos^2\left(\frac{π}{2x}\right)}$
Such that
$|f(x+it)|=o(e^{2πt})$
uniformly for every $x$...
- 790
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
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
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
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:
$$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$
$$\xi(s) = \xi(1 - s)$$
$$\pi^{-s/2}\ \Gamma\left(\...
- 1,183
0
votes
0
answers
220
views
Mertens function via Perron's formula without assuming the simplicity of the Riemann zeros
Let $\mu$ denote the Möbius function, and define the the Mertens function $M(x) = \sum_{n \leq x} \mu(n)$. By Person's formula, one can express $M(x)$ as a sum over the nontrivial zeros of the ...
- 149
0
votes
0
answers
151
views
Abscissa of convergence of transformed Dirichlet series
Let
$$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 ...
- 3,098