Explicit zero density theorems for dedekind zeta functions. Large gaps between zeros of dedekind zetafunctions of. Ks x i 1 jjijjs where the sum is taken over all nontrivial ideals of the ring of integers of a number eld k. A zero of a function is a point at which that function vanishes. Kudryavtseva 1 filip saidak peter zvengrowski abstract an exposition is given, partly historical and partly mathematical, of the riemann zeta function s and the associated riemann hypothesis. This zeta function is an in nite product over prime ideals pin o. The first family of computations consists in the verification of the riemann hypothesis on all the first 10 non trivial zeros.
Dedekind zeta functions and the complexity of hilberts. Pdf the 10 first zeros of the riemann zeta function. Dynamical zeta functions mark pollicott contents 1 introduction 2. We will also see that an similar property of ls for.
The riemann zeta function david jekel june 6, 20 in 1859, bernhard riemann published an eightpage paper, in which he estimated \the number of prime numbers less than a given magnitude using a certain meromorphic function on c. A survey of results regarding a computational approach to. Introduction and notations the riemann zeta function and its generalization to number fields, the dedekind zeta function, have been for well over a hundred years one of the central tools in number theory. The fact that riemann zeta function doesnt have a zero on res 1 is the most crucial step in the proof of the prime number theorem. Assuming the riemann hypothesis, montgomery and taylor showed that at least 67. Zeros of riemann zeta function university of chicago.
The mathematician thinks that a problem is easy to solve after it has been solved. Zeros of dedekind zeta functions and holomorphy of artin l. Note that this result was announced by chowla and selberg 7 but they have never published a proof. K s, is a generalization of the riemann zeta function which is obtained in the case where k is the rational numbers q. Zeros of dedekind zeta functions and holomorphy of artin lfunctions 255 3. The famous zeros of the riemann zeta function and its generalizations lfunctions, dedekind and selberg zeta functions are analyzed through several zeta functions built over those zeros. Zeta functions for the riemann zeros by andre voros 1. The riemann zeta function plays a pivotal role in analytic number theory and has applications in. T1 the number of zeros of the dedekind zetafunction on the critical line.
It can be defined as a dirichlet series, it has an euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the. On the error term and zeros of the dedekind zeta function. Zeta functions over zeros of zeta functions springerlink. As a consequence, our argument is closer to mccurleys than to rosser and schoenfelds. The signs of the stieltjes constants associated with the dedekind zeta function saad eddin, sumaia, proceedings. This work proposes to investigate certain meromorphic functions defined by dirichlet series over the nontrivial zeros p of the riemann zeta function s, and to thoroughly compile their explicit analytical features. Some results on the location of zeros of a family of approximations of dedekind zeta functions are also derived. Two important concepts used in this proof are those of zeros and poles.
We prove that, if dk is sufficiently large, then the dedekind zeta function. The dedekind zetafunction attached to kcan be expressed by ks sls d for s6 1, where s is the riemann zetafunction, the character. On the real part of nontrivial zeros of riemann zeta function 3 preface. The number of zeros of the dedekind zetafunction on the. These results extend results of gonek and montgomery on families of approximations of the riemann zetafunction. On the density of the zeros of the dedekind zetafunction author. However, a major di erence is that its zeros can lie very close to the the real axis. What are zeta functions of graphs and what are they good for.
Local zetafunction of a characteristic p variety hasseweil l. Turnagebutterbaugh, gaps between zeros of dedekind zetafunctions of quadratic number fields. In number theory, the dedekind zeta function is associated to an algebraic number eld k, such as qp 2, for example. Moreover, in 1859 riemann gave a formula for a unique the socalled holomorphic extension of the function onto the entire complex plane c except s 1. It is also shown, with no hypothesis, that, with the exception of a bounded region where the zeros can be calculated, the closed left. The exposition is selfcontained, and required a preliminary knowledge of only the elements of function theory. The eulermaclaurin formula is a very general technique that applies to many summations. A mathematical proof is only true if the proof can be reproducible, and perhaps by alternative means than that employed in the first proof. It is shown that the riemann hypothesis implies that the derivative of the riemann zeta function has no zeros in the open left half of the critical strip. On the density of the zeros of the dedekind zetafunction. In contrast, the riemannsiegel formula relies on special properties of the zeta function.
Pdf using stechkins lemma we derive explicit regions of the half complex plane s. The riemann zeta function and its generalization to number fields, the dedekind zeta function. In mathematics, the dedekind zeta function of an algebraic number field k, generally denoted. Gaps between zeros of dedekind zetafunctions of quadratic number elds. In particular, we establish a smooth explicit formula for these zeros and we derive an effective version of. From this perspective, it behaves similarly to a dirichlet lfunction. The following notes summarize five years of effort. N2 erich hecke first showed that the dedekind zetafunction for an ideal class in an imaginary quadratic field has an infinite number of zeros on the critical line. A proof of the riemann hypothesis should be generalizable because there exists zeta functions such as the dedekind zeta function, dirichlet series, generalized zeta functions, and lfunctions.
Erich hecke first showed that the dedekind zetafunction for an ideal class in an imaginary quadratic field has an infinite number of zeros on the critical line. When conrey proved that at least 25 of the zeros of the riemann zeta function are on the critical line, he proved the asymptotic formula for the mean square of. Pdf simple zeros of dedekind zeta functions researchgate. Using techniques similar to those of riemann, it is shown how to locate and count nontrivial zeros of s. Ii article pdf available in the quarterly journal of mathematics 673 october 2014 with 33 reads. Dedekind zeta function of a number eld epstein zeta function of a quadratic form goss zeta function of a function eld padic zeta function from algebraic geometry. Zeros of high derivatives of the riemann zeta function binder, thomas, pauli, sebastian, and saidak, filip, rocky mountain journal of mathematics, 2015. The riemann zeta function or eulerriemann zeta function. Riemann, dedekind, selberg, and ihara zetas audrey terras crm montreal 2009 more details can be found in. Dedekind explored the distribution of primes in di erent number elds. This article studies the zeros of dedekind zeta functions. Let kbe a quadratic number eld with discriminant d. Pdf real zeros of dedekind zeta functions researchgate.
The dedekind zeta function measures the properties of the ring of integers of a number eld which is a dedekind ring. Chandrasekharan and raghavan narasimhan proved the result for both real and imaginary quadratic fields. Gaps between zeros of dedekind zetafunctions of quadratic. Jensen polynomials for the riemann zeta function and other. Assuming grh, we prove an explicit upper bound for the number of zeros of a dedekind zeta function having imaginary part in. On simple zeros of the dedekind zetafunction of a quadratic number field volume 65 issue 4 xiaosheng wu, lilu zhao. A proof of the riemann hypothesis should be generalizable because there exists zeta functions such as the dedekind zeta function, dirichlet series, generalized zeta functions, and l functions. In this article, we study the zeros of the partial sums of the dedekind zeta function of a cyclotomic field k defined. Large gaps between zeros of dedekind zetafunctions of quadratic number elds. The aim of these lectures is to provide an intorduc tion to the theory of the riemann zetafunction for stu dents who might later want to do research on the subject.
The prime number theorem, hardys theorem on the zeros of. Let k be a number field, nk be its degree, and dk be the absolute value of its discriminant. Using montgomery and taylors argument together with an elementary. Next, we investigate estimate for the number of zeros of dedekind zeta function. Deuring 9 and stark 31 showed that all zeros of these epstein zetafunctions in the. Simple zeros of dedekind zeta functions 4 3 stechkins tric k and proof of theorem 3 since z k.
Zeros on rez1 we are almost ready to prove an important theorem about the zeros of the zeta function in the critical strip. This explicit formula involves the zetafunction and, in particular, its nontrivial zeros. Dedekind zeta functions the roman number theory association. Zeros of the riemann zetafunction on the critical line.
Dyson has drawn my attention to the fact that the eigenvalues of a random complex hermitian or unitary matrix of large order have precisely the same pair correlation. The dedekind zeta function is similar to the riemann zeta function as it also has a simple pole at s 1. On simple zeros of the dedekind zeta function of a quadratic number field volume 65 issue 4 xiaosheng wu, lilu zhao skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Zeros of dedekind zeta functions in the critical strip. Turnagebutterbaugh3 1school of mathematics, university of manchester, manchester m 9pl, uk, hung. To make this precise, recall that dyson 8, montgomery 9, and odlyzko 10 conjecture that the nontrivial zeros of the riemann zeta function are distributed like the eigenvalues of random hermitian matrices.
Pdf building on stechkin and kadiris ideas we derive an explicit zerofree region of the real axis for dedekind zeta functions of number fields. We also prove a bound for the multiplicity of the zeros. Simple zeros of dedekind zeta functions project euclid. Zeros of partial sums of the dedekind zeta function of a cyclotomic field. As a consequence of their papers, we study some aspects of zeros of the derivatives of the riemann zeta function. However, the formula 2 cannot be applied anymore if the real part. On zeros and cvalues of epstein zetafunctions 183 1 2 and 1 if k. Why does the dedekind zeta function of a number field have.