Articles of zeta functions

Properties of Dedekind zeta function

Suppose $K$ is a quadratic field and $a_K(n)$ denotes the number of ideals in the ring of integers of $K$ whose norm is equal to $n$. Then I need to show that $$\sum_{n\leq x} a_K(n)=O(x).$$ Clearly the above claim will imply that the Dedekind zeta function $\zeta_K(z):=\sum_{n\geq 1}\frac{a_K(n)}{n^z}$ converges for $\mathrm{Re}(z)>1$. Is there an analytic […]

Analytic continuation of Dirichlet function

Suppose $\{a_n\}$ is a sequence of complex numbers such that the sums $A_n=a_1+\cdots+a_n$ satisfy $$|A_n-nb|\leq Cn^{\sigma}$$ for all $n$, where $b\in\mathbb{C},C>0,0\leq\sigma<1$. Prove that the function $f(z)$ defined by $$f(z)=\sum_{n=1}^\infty\dfrac{a_n}{n^z}$$ for $\Re{z}>1$ has an analytic continuation to the region $\Re{z}>\sigma$ except for a simple pole of residue $b$ at $z=1$. ($\Re$ denotes the real part of […]

About the number of zeros of the zeta function?

Let $N(T)$ denote the number of zeros $b=α+iβ$ (counted with multiplicity) of the Riemann zeta function $ζ(s)$ for which $0<β<T$. The functional equation and the argument principle implies that $$N(T)=(T/(2π))log(T/(2πe))+(7/8)+(1/π)argζ((1/2)+iT)+O((1/T))$$ If $T=20$, the we get $$N(20)= 0.99967<1$$ which means that there is no zeros in this region. But we know that there exist only one […]

Concerning Hurwitz Zeta function, how to prove the following identity?

It is claimed that $$\zeta'(0,s)=\ln\left(\frac{\Gamma(s)}{\sqrt{2\pi}}\right)$$ where the derivative is meant by the first argument (as usual with Hurwitz Zeta). How to prove this? Wolfram Alpha confirms this:[HurwitzZeta[s%2C+p]%2C+s]+at+s%3D0 but I have no other clue. Is there a more general form of the identity (with a variable instead of 0)?

The multiplication formula for the Hurwitz zeta function

In a textbook I’m reading, the author states without proof that $$ \zeta(s,mz) = \frac{1}{m^{s}} \sum_{k=0}^{m-1} \zeta \left(s,z+\frac{k}{m} \right), \tag{1}$$ where $\zeta(s,z) $ is the Hurwitz zeta function Supposedly, this isn’t hard to prove. But is it possible to prove $(1)$ using simply the series definition of the Hurwitz zeta function, that is, $ \displaystyle\zeta(s,z) […]

Hint on a limit that involves the Hurwitz Zeta function

I will be honest. Some play with a weird integral has gotten me to this formulation: $$\lim\limits_{n\mathop\to\infty}\frac{\zeta(2,n)}{\frac 1n+\frac 1{2n^2}}=1$$ It seems true because of the numerical approximations made and the actual evaluation of this limit at Wolfram Alpha. But i am stuck. Does anyone have an idea of where to start?

Summation of a function 2

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where $c_k = e_k – 1$ Example: $72 = 2^33^2$, so $f(72) = 2^{3-1}3^{2-1} = 2^{2}3^{1}=12$ $144 = 2^43^2$, so $f(144) = 2^{4-1}3^{2-1} = 2^{3}3^{1}=24$ Now let $$F(N) […]

Evaluate $\displaystyle \sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x.$

I have some trouble in evaluating this series $$\sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x$$ I tried to calculate the integral first, but after that I found the series become so complicated. Besides, I found maybe the series equals to $$\sum_{k=0}^{\infty }\frac{\left ( -1 \right )^{k}\zeta \left ( 2k+\dfrac{3}{2} \right )}{2k+\dfrac{3}{2}}$$ this is definitely a monster to me. So I […]

The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann’s $R$ functions over roots of Riemann’s $\zeta$ resp. Dirichlet $\beta$ function: \begin{align*} \pi^*(x;4,3)&=\sum_{k=0}^\infty 2^{-k-1}\left( \operatorname{R}(x^{1/2^{k}})-\sum_{\rho_\zeta} \operatorname{R}(x^{\rho_\zeta/2^k}) +\sum_{\rho_\beta} \operatorname{R}(x^{\rho_\beta/2^k}) \right) \end{align*} Ilya […]

Volume of first cohomology of arithmetic complex

Let $K$ be a number field and consider the Arithmentic complex $\Gamma_{Ar}(1)^\bullet$ be defined by $$\begin{array} A\Bbb R^{r_1+r_2} & \stackrel{\Sigma}{\longrightarrow} & \Bbb R \\ \uparrow{l(\cdot)=\Pi\log |\sigma_i(\cdot)|} & & \uparrow{-\Sigma \log|\mathfrak p_i|^{-z_i}} \\ K^* & \stackrel{\operatorname{val}}{\longrightarrow} & \oplus_\mathfrak p\Bbb Z \end{array} $$ The left arrow going up is just the same map that appears in the […]