The properties of the functions $\zeta_k(s;H_j)$ resemble those of $\zeta_k(s)$. $$\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2k}\,\mathrm{d}t\sim \sum_{n=1}^\infty\frac{\tau_k^2(n)}{n^{2\sigma}}$$ $$\lim_{T\to\infty}\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2}\,\mathrm{d}t=\zeta(2\sigma)$$ which converges absolutely and uniformly in any bounded domain of the complex $s$-plane for which $\sigma\geq1+\delta$, $\delta>0$.
A modified version of this example exists on your system. Modern estimates of $N(\sigma,T)$ are based on convexity theorems of the average values of analytic functions, applied to the function If $\sigma>1$, $\eta(\sigma)=0$, and if $\sigma<0$, then $\eta(\sigma)=(1/2)-\sigma$.
For $a=1$ it becomes identical with Riemann's zeta-function. The results have applications in the study of the zeros of the zeta-function, and in number theory directly. 4) All other zeros of $\zeta_k(s)$ lie in the critical strip $0\leq\sigma\leq1$. The following formula is valid: Web browsers do not support MATLAB commands.Choose a web site to get translated content where available and see local events and offers. Weil noted in 1967 that a consequence of the general hypotheses on the function $\zeta_X^{(1)}(s)$ for an elliptic curve $X$ over $\mathbb{Q}$ is that the curve $X$ is uniformized by modular functions, while the function $\zeta_X^{(1)}(s)$ is the Mellin transform of the modular form corresponding to a differential of the first kind on $X$. More exactly, any function which can be represented by an ordinary Dirichlet series and which satisfies equation \ref{func} coincides, under fairly broad conditions with respect to its regularity, with $\zeta(s)$, up to a constant factor $$ \chi(s)=\pi^{s-1/2}\frac{\Gamma(1-s/2)}{\Gamma(s/2)}$$ $$\frac{1}{\zeta(s)}=\sum_{n=1}^\infty\frac{\mu(n)}{n^s},\quad \zeta^2(s)=\sum_{n=1}^\infty\frac{\tau(n)}{n^s},$$ $$\zeta(s,a)=\frac{e^{-\pi is}\Gamma(1-s)}{2\pi i}\int_L\frac{z^{s-1}e^{-az}}{1-e^{-z}}\,\mathrm{d}z,$$ Lavrik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. where the integral is taken over a contour $L$ which is a path from infinity along the upper boundary of a section of the positive real axis up to some given $0