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We also present a brief account of a number of analogous results associated with the (Hurwitz's) generalized zeta function. The estimates of Atle Selberg and Karatsuba can not be improved in respect of the order of growth as . a function of a complex variable s= x+ iyrather than a real variable x. Through connections with random matrix theory and quantum chaos, the appeal is even broader. The principal applications of Weil's conjectures in number theory deal with the study of congruences. In addition to Riemann's zeta-function one … The statistics of the Riemann zeta zeros are a topic of interest to mathematicians because of their connection to big problems like the Riemann hypothesis, distribution of prime numbers, etc. The formula, $$\pi(x)=\mathrm{li}x+O\left(xe^{-B\ln^{3/5}x}\right)$$, is the corresponding statement for prime numbers. Dedekind's zeta-function $\zeta_k(s)$ of the field $k$ is the defined by the series, $$\zeta_k(s)=\sum_{\mathfrak{A}}\frac{1}{N^s_{\mathfrak{A}}},$$. where z and w are complex numbers and the real part of z is greater than zero. The Riemann zeta function is then a particular solution of this equation. The zeta-function in algebraic geometry is an analytic function of a complex variable $s$ describing the arithmetic of algebraic varieties over finite fields and schemes of finite type over $\text{Spec}\,\,\,\mathbb{Z}$. uniformly for $\sigma\geq\alpha+1/\ln T$. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It can therefore be expanded as a Laurent series about s = 1; the series development then is, The constants γn here are called the Stieltjes constants and can be defined by the limit. Karatsuba, A. The properties of the functions $\zeta_k(s;H_j)$ resemble those of $\zeta_k(s)$. It is known that any non-trivial zero lies in the open strip {s ∈ C: 0 < Re(s) < 1}, which is called the critical strip. Hasse, Helmut (1930). An important part in the theory of the zeta-function is played by the problem of estimating the function $N(\sigma,T)$ which denotes the number of zeros $\beta+i\gamma$ of $\zeta(s)$ for $\beta>\sigma$, $0<\gamma\leq T$. It was demonstrated in 1933 by H. Hasse for curves of genus one (for genus zero the situation is trivial), and by A. Weil (1940) for curves of arbitrary genus with the aid of results of the theory of Abelian varieties (cf. No such simple expression is known for odd positive integers. Weil's first conjecture says that the category of algebraic varieties over finite fields admits a cohomology theory which satisfies all the formal properties required to obtain the Lefschetz formula. Titchmarsh, "The theory of the Riemann zeta-function" , Clarendon Press (1986) ((Rev. Corrections? This accounts for the important role played by $\zeta(s)$ in number theory. Orsay (1983), A. Ivic, "The Riemann zeta-function" , Wiley (1985). Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to real s > 1. These estimates were generalized to include sums involving any number of variables. The Riemann zeta function, ζ(s), is a function of a complex variable s that analytically continues the sum of the infinite series which converges when the real part of s is greater than 1. Incidentally, this relation is interesting also because it actually exhibits ζ(s) as a Dirichlet series (of the η-function) which is convergent (albeit non-absolutely) in the larger half-plane σ > 0 (not just σ > 1), up to an elementary factor. To date (1993), the sharpest known zero-free region is given by the following theorem [Iv2]: There is an absolute constant $C>0$ such that $\zeta(s)\neq0$ for, $$\sigma\geq 1-C(\ln t)^{-2/3}(\ln\ln t)^{-1/3}\quad(t\geq t_0).$$. In the theory of the Riemann zeta function, the set {s ∈ C: Re(s) = 1/2} is called the critical line. The hypothetical form of such an equation was proposed in [Se2]. Hardy and J.E. The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers. For $k>2$, all that is known is that if $\sigma>1-1/k$, $$\lim_{T\to\infty}\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2k}\,\mathrm{d}t=\sum_{n=1}^\infty\frac{\tau_k^2(n)}{n^{2\sigma}},$$, where $\tau_k(n)$ is the number of multiplicative representations of $n$ in the form of $k$ positive integers, and that the asymptotic relation, $$\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2k}\,\mathrm{d}t\sim \sum_{n=1}^\infty\frac{\tau_k^2(n)}{n^{2\sigma}}$$. The (Riemann) formula used here for analytic continuation is $$\zeta(s) = 2^s \pi^{s-1} \sin(\pi s/2) \Gamma(1-s) \zeta(1-s).$$ This is actually one of several formulas, but this one was discovered by Riemann himself and is called the functional equation . Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. then, uniformly for $1/2\leq\sigma\leq 1$, $$N(\sigma,T)=O(T^{2(1+2r)(1-\sigma)}\ln^5T).$$. Chebyshev, "Selected mathematical works" , Moscow-Leningrad (1946) (In Russian), N.G. Karatsuba, A. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series. Vol.54, No.1, p. 626 (1996). Let $k$ be an algebraic number field of degree $n=r_1+2r_2>1$, where $r_1$ is the number of real fields and $r_2$ is the number of complex-conjugated pairs of fields in $k$; further, let $\Delta$ be the discriminant, $h$ the number of divisor classes, and $R$ the regulator of the field $k$, and let $g$ be the number of roots of unity contained in $k$. "Continued-fraction expansions for the Riemann zeta function and polylogarithms". Schilling (ed.) There exist general methods by which such results may be obtained not only for the class of zeta-functions, but in general for Dirichlet functions with a Riemann-type functional equation \ref{func}. A. $$\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2}\,\mathrm{d}t=\ln T+2C-1-\ln 2\pi+O\left(\frac{\ln T}{\sqrt{T}}\right),$$, $$\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{4}\,\mathrm{d}t=\frac{\ln^4T}{2\pi^2}+O(\ln^3T).$$, $$\lim_{T\to\infty}\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{2}\,\mathrm{d}t=\zeta(2\sigma)$$, $$\lim_{T\to\infty}\frac{1}{T}\int_1^T\lvert \zeta(\sigma+it)\rvert^{4}\,\mathrm{d}t=\frac{\zeta^4(2\sigma)}{\zeta(4\sigma)}$$.