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Questions tagged [dirichlet-series]

21 questions from the last 365 days
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1 answer
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when does $h$ exist?

Let $\zeta(s)$ denote the Reimann zeta function in the critical strip. It is easy to see that $$ \zeta(s) = 0 \Longleftrightarrow \Re(\zeta(s))+\Im(\zeta(s)) = 0 ~~~~ \text{and} ~~~~~~ \Re(\zeta(s)) \...
Roy Burson's user avatar
-2 votes
0 answers
48 views

Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs?

To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$ $$ \eta(s) = \sum_{n=...
Roy Burson's user avatar
5 votes
2 answers
237 views

Residue of Dirichlet series at $s = 1$

Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, and suppose that the sequence has a well-defined "average", in the sense that $$ \lim_{N \to \infty} \frac{1}{N}\sum_{i = 1}^N a_i = R$...
David Loeffler's user avatar
1 vote
1 answer
188 views

Can one show $h(x)=|2(\zeta'(x))^2-\zeta''(x)\zeta(x)|$ is a decreasing function for $x\in\mathbb{R}\cap [1,\infty)$?

This question is related to This question. When I tried to approach it I couldn't even proof that the LHS is a decreasing function on the given domain using regular methods. I have tried to write the ...
Haidara's user avatar
  • 178
0 votes
2 answers
364 views

Can one show $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for $x\in\mathbb{R}\cap [1,\infty)$?

I have found that $\left|\frac{2(\zeta'(x))^2-\zeta''(x)\zeta(x)}{\zeta^3(x)}\right|\leq \frac{2}{(x-\frac{1}{2})^2}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the ...
Haidara's user avatar
  • 178
6 votes
1 answer
568 views

Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?

I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
Haidara's user avatar
  • 178
0 votes
0 answers
103 views

Validity of a Tauberian theorem for Dirichlet series

I encountered this statement about Dirichlet series but couldn't find a similar result in Korevaar's "Tauberian Theory". Is this statement valid? Statement: Let $f(s) = \sum_{n=1}^{\infty} \...
 Babar's user avatar
  • 611
1 vote
0 answers
191 views

Prerequisites for Chen's theorem?

I am an undergraduate theoretical physics student, and I am trying to understand Chen's theorem. But when I tried to read Chen Jingrun's 1973 paper (https://www.sciengine.com/Math%20A0/doi/10.1360/...
Ben's user avatar
  • 11
1 vote
2 answers
310 views

Dirichlet Series that fail to be L-functions

For $\sigma \in \mathbb{R}$, let each $\mathbb{C}_\sigma = \{s \in \mathbb{C} : \Re(s) > \sigma\}$. For a sequence $a_n \in \mathbb{C}$, consider the Dirichlet series $D(s) = \sum_{n\ge 0} a_n n^{-...
Greg Zitelli's user avatar
  • 1,094
0 votes
0 answers
88 views

Closed form formula

Let $m$ and $n$ are positive integers, then find the sum of the infinite series defined as $$\sum_{k=0}^\infty \frac{(-1)^k\Gamma(m+k) }{\Gamma(m)k!(k+m)^n}.$$ I was managed to the sum with $m=2$ and $...
user90533's user avatar
  • 109
3 votes
0 answers
139 views

Do the denominators of A006571(n)/A366450(n) have a Dirichlet generating function? Because they partially match A071974(n) and A056622(n)?

Consider the expansion: $$A006571(n) = q \prod _{k=1}^{\infty } \left(1-q^k\right)^2 \left(1-q^{11 k}\right)^2 \label{1}\tag{1}$$ A006571 and the triple sum: $$A366450(n)=\sum _{k=1}^n \left(\sum _{y=...
Mats Granvik's user avatar
  • 1,183
5 votes
0 answers
95 views

Possible extension of Ikehara's theorem for Dirichlet series with not necessarily positive coefficients?

I'm wondering about a possible extension of Ikehara's theorem for Dirichlet series with coefficients that are not necessarily positive. Consider the following: Let $D(s) = \sum_{n \geq 1} \frac{a(n)}{...
 Babar's user avatar
  • 611
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 \...
Mats Granvik's user avatar
  • 1,183
6 votes
1 answer
247 views

Convergence and meromorphic continuation of a Dirichlet series under RH

Consider a Dirichlet series $F(s) = \sum_{n\geq1} \frac{a_{n}}{n^{s}}$ such that $a_{n} = O(1)$. Suppose the Dirichlet series $$\zeta(s)F(s)=\sum_{n\geq1} \frac{(a\star1)(n)}{n^{s}}$$ converges ...
 Babar's user avatar
  • 611
4 votes
0 answers
265 views

Ramanujan-like series for $1/\pi^m$ and Dirichlet L-values

A rational Ramanujan-like series for $\pi^{-m}$ and character $\chi$ is a series with rational parameters which is of the following form: $$ \sum_{n=0}^{\infty} \left(\prod_{i=0}^{2m} \frac{(s_i )_{n}...
Jesús Guillera's user avatar
3 votes
0 answers
167 views

A sharper estimate for a generalization of the sum-of-divisors function

I am interested in the function $f_n(m)$ which can be defined by the Dirichlet generating function $$\zeta(s)\zeta(s - 1) \cdots\zeta(s - n + 1) = \sum\limits_{m = 1}^\infty \frac{f_n(m)}{m^s} $$ This ...
Bear's user avatar
  • 31
-1 votes
1 answer
109 views

Analyzing a Dirichlet series with log-oscillating terms via Fourier methods

I am investigating the series $S(z)$ defined as follows: $$ S(z) = \sum_{n=1}^{\infty} n^{-a}\cos(b\ln(n)), $$ where $z = a + bi \in \mathbb{C}$, with $0 < a < 1$, and $b \in \mathbb{R}$. I want ...
swami's user avatar
  • 375
1 vote
1 answer
258 views

Classification of L functions and Dirichlet series by poles

I am interested in the connection between particular Dirichlet series' abscissa of convergence and the poles of L-functions. Let $D(z) = \sum_{n=1}^\infty\frac{a_n}{n^z}$ be a Dirichlet series ...
user avatar
2 votes
2 answers
231 views

Conditional convergence of exponential sums related to a Hecke modular form

Definition Consider the Fourier coefficients $\psi(n)$ of the modular form $\eta^4(6\tau)$, which are defined in terms of $q=\exp(i2\pi\tau)$ by the identity: $$\eta^4(6\tau) = q \prod_1^\infty (1-q^{...
Christopher-Lloyd Simon's user avatar
0 votes
1 answer
179 views

A question about the setup of zero density estimates for Dirichlet $L$-functions

For $L(s,\chi)= \sum_{n \geq 1}\frac{\chi(n)}{n^s}$, where $s = \sigma + it$, we define the function $N(\sigma, T, \chi)$ which counts the zeros $\rho = \beta + i\gamma$ for which $L(\rho, \chi) =0$ ...
Josh's user avatar
  • 1
6 votes
0 answers
286 views

Approximating $\zeta'/\zeta$ (and its derivatives) by a finite sum

Let $A(s) = (-\zeta'/\zeta)^{(r)}(s) = \sum_n a_n n^{-s}$, where $r\geq 0$. (We can consider $r=0$ first for simplicity.) Say I want to approximate $A(s)$ for $s=1+it$ by a finite sum - preferably a ...
H A Helfgott's user avatar
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