# Questions tagged [dirichlet-series]

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### Exceptional zeros of a convolutional inverse

Let $\kappa:\lbrace 1,2,3,\ldots\rbrace\longrightarrow \mathbb Z$ be the convolutional inverse in the Dirichlet ring of $n\longmapsto {n+1\choose 2}$. It is defined by $\kappa(1)=1$ and by the ...
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### Is this a valid method of extending convergence of the Maclaurin series for $\frac{x}{x+1}$ from $|x|<1$ to $\Re(x)>-1$?

I originally asked this question on Math StackExchange a few months ago and no answers or even comments have yet been posted, so I'm asking this question again here on Math OverFlow. This Math ...
582 views

### An inequality related to Catalan's constant and $\zeta(3)$

Problem : Show that : $$\frac{1}{\zeta(3)}<2C-1$$ Where we can see the zeta function and the Catalan's constant . After a bounty on Maths Stack Exchange there is no satisfying answer . See https://...
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### Positivity of partial Dirichlet series for a quadratic character?

Let $\chi\colon(\mathbb{Z}/N\mathbb{Z})^\times\rightarrow\{\pm1\}$ be a primitive quadratic Dirichlet character of conductor $N$. For any integer $m=1,2,\cdots,\infty$, consider the partial Dirichlet ...
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1 vote
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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 ...
154 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 ...
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### Mean value theorem for Dirichlet series of prime support?

Let $\{a_n\}_{1\leq n\geq N}$, $a_n\in \mathbb{C}$. Let $F(s) = \sum_{n=1}^N a_n n^{-s}$. By a mean-value theorem (Montgomery-Vaughan, 1973), $$\int_0^T |F(i t)|^2 = \sum_{n=1}^N |a_n|^2 (T + O(n)).$$ ...
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### Converse theorem for zeta universality

Voronin's Universality Theorem for $\zeta(s)$ is that the zeta function can uniformly approximate any non-vanishing holomorphic function to any degree of accuracy in the right-half of the critical ...
1 vote
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### If the Dirichlet series $L(z,\chi)$ diverges for $\sigma< 1$, does its alternating version converge for some $\sigma_0 < 1$, and conversely?

Here $\chi$ is a completely multiplicative function with $\chi(p)\in\{-1,+1\}$ for any prime $p$, but not necessarily a character. Also $s=\sigma + it$ as usual. The series corresponding to $L(s,\chi)$...
1 vote
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### Dirichlet series associated to divisor function

How can I express the following Dirichlet series $$\sum_{n=1}^{\infty}\frac{d_{k}^{2}(n)}{n^{s}}$$ (where $d(n)$-divisor function;$k\geq 1$) in terms of the zeta function?
1 vote
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1 vote
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### Multiplicity of zeros of partial sums of the Dirichlet Eta function

I am studying ways to approach the problem of the multiplicity of zeros of the partial sums of the Dirichlet Eta functions: $$\sum_{n=1}^{K}\frac{(-1)^{n-1}}{n^{s_o}} = 0$$ more in particular, ...
1 vote
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### What is the approximation of $\log(|\zeta'(\frac{1}{2}+it)|)$ in Dirichlet polynomial if it is exists?

I have done some search many times on web to find any approximation of $\log|(\zeta'(s))|$ in Dirichlet polynomial but I didn't got it, Probably that $\log(|\zeta'(s)|$ dosn't have a Dirichlet ...
My opinion is ; We may use id(d)=d arithmetic function and log*id dirichlet convolution in the question. i thought that ; when we multiply and divide n with $(\log d) / d$ we obtain $F(S)=\sum_{n=... 15 votes 2 answers 667 views ### If$\zeta(s)=0$with$\Re(s)=\frac{1}{2}$, is then$|\hat{\zeta}(s,3)|^2=\frac{1}{2}$? Helmut Hasse has proved that for$s \in \mathbb{C}-\{1\}$the Riemann zeta function can be written as:$\$\zeta(s)=\frac{1}{1-2^{1-s}}\sum_{n=0}^\infty\frac{1}{2^{n+1}}\sum_{k=0}^n(-1)^k\ {n \choose k}... 