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On a Mertens-Type Conjecture for Number Fields
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Abstract
We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous Mertens function, expanding upon work of Ng. Finally, we explore properties of the generalized Mertens function of certain dicyclic number fields as consequences of Artin factorization.
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Keating and Snaith modeled the Riemann zeta-function ζ(s) by characteristic polynomials of random N×N unitary matrices, and used this to conjecture the asymptotic main term for the 2k-th moment of ζ(1/2+it) when
k > -1/2. However, an arithmetical factor, widely believed to be part of the leading term coefficient, had to be inserted in an ad hoc manner. Gonek, Hughes and Keating later developed a hybrid formula for ζ(s) that combines a truncation of its Euler product with a product over its zeros. Using it, they recovered the moment conjecture of Keating and Snaith in a way that naturally includes the arithmetical factor. Here we use the hybrid formula to recover a conjecture of Hughes, Keating and O'Connell concerning the discrete moments of the derivative of the Riemann zeta-function averaged over the zeros of ζ(s), incorporating the arithmetical factor in a natural way.
We give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula of the latter is then proved for quadratic extensions in the lowest order case. We are also able to reproduce our moments conjecture in the case of quadratic extensions by using a modified version of the moments recipe of Conrey et al. Generalising our methods, we then provide a conjecture for moments of non-primitive L-functions, which is supported by some calculations based on Selberg's conjectures.
In this article we prove a general theorem which establishes the existence of
limiting distributions for a wide class of error terms from prime number
theory. As a corollary to our main theorem, we deduce previous results of
Wintner (1935), Rubinstein and Sarnak (1994), and of Ng (2004). In addition, we
establish limiting distribution results for the error term in the prime number
theorem for an automorphic L-function, weighted sums of the M\"{o}bius
function, weighted sums of the Liouville function, the sum of the M\"{o}bius
function in an arithmetic progression, and the error term in Chebotarev's
density theorem.
We study the natural analogue of the Mertens conjecture in the setting of global function fields. Building on the work of Cha, we show that most hyperelliptic curves do not satisfy the Mertens conjecture, but that if we modify the Mertens conjecture to have a larger constant, then this modified conjecture is satisfied by a positive proportion of hyperelliptic curves.
We give explicit upper bounds for residues at s=1 of Dedekind zeta functions of number fields, for |L(1, χ)| for nontrivial primitive characters χ on ray class groups, and for relative class numbers of CM fields. We also give explicit lower bounds for relative class numbers of CM fields (which do not contain any imaginary quadratic subfield). Most of these bounds stem from a useful lemma which enables us to easily obtain neat bounds.
This article studies the zeros of Dedekind zeta functions. In particular, we
establish a smooth explicit formula for these zeros and we derive an effective
version of the Deuring-Heilbronn phenomenon. In addition, we obtain an explicit
bound for the number of zeros in a box.
We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function which involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. For the second and fourth moments, we establish all of the steps in our approach rigorously. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory
Random matrix theory (RMT) is used to model the asymptotics of the discrete moments of the derivative of the Riemann zeta function, i(s), evaluated at the complex zeros 2 + ifl n , using the methods introduced by Keating and Snaith in [14]. We also discuss the probability distribution of ln ji (1=2 + ifl n)j, proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations. 1
The Mertens conjecture states that M(x) < x 1 /2 for all x > 1, where M(x) = n x S (n) , and (n) is the Mo .. bius function. This conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the Riemann hypothesis. This paper disproves the Mertens conjecture by showing that x lim sup M(x) x - 1 /2 > 1.06 . The disproof relies on extensive computations with the zeros of the zeta function, and does not provide an explicit counterexample. Disproof of the Mertens Conjecture A. M. Odlyzko AT&T Bell Laboratories Murray Hill, New Jersey 07974 USA and H. J. J. te Riele Centre for Mathematics and Computer Science Kruislaan 413 1098 SJ Amsterdam The Netherlands 1. Introduction Let (n) denote the Mo .. bius function, so that (n) = 0 , p 2 n for some prime p , ( - 1) k , n = i = 1 P k p i , p i distinct primes , 1 , n = 1 , and let M(x) = nx S (n) . (1.1) Then M(x) is the dif...
Prime numbers are the multiplicative building blocks of natural numbers. Understanding their overall influence and especially their distribution gives rise to central questions in mathematics and physics. In particular their finer distribution is closely connected with the Riemann hypothesis, the most important unsolved problem in the mathematical world. Assuming only subjects covered in a standard degree in mathematics, the authors comprehensively cover all the topics met in first courses on multiplicative number theory and the distribution of prime numbers. They bring their extensive and distinguished research expertise to bear in preparing the student for intelligent reading of the more advanced research literature. This 2006 text, which is based on courses taught successfully over many years at Michigan, Imperial College and Pennsylvania State, is enriched by comprehensive historical notes and references as well as over 500 exercises.
Under the assumption of the Riemann hypothesis, the Linear Independence hypothesis, and a bound on negative discrete moments of the Riemann zeta function, we prove the existence of a limiting logarithmic distribution of the normalisation of the weighted sum of the Liouville function, Lα(x)=∑n⩽xλ(n)/nαLα(x)=∑n⩽xλ(n)/nα, for 0⩽α<1/20⩽α<1/2. Using this, we conditionally show that these weighted sums have a negative bias, but that for each 0⩽α<1/20⩽α<1/2, the set of all x⩾1x⩾1 for which Lα(x)Lα(x) is positive has positive logarithmic density. For α=0α=0, this gives a conditional proof that the set of counterexamples to Pólyaʼs conjecture has positive logarithmic density. Finally, when α=1/2α=1/2, we conditionally prove that Lα(x)Lα(x) is negative outside a set of logarithmic density zero, thereby lending support to a conjecture of Mossinghoff and Trudgian that this weighted sum is nonpositive for all x⩾17x⩾17.
where runs over the complex zeros of the zeta-function. We assume the Riemann hypothesis (RH) throughout; it then follows that Ik(, T) converges for every k > 0 when > but for no k = when =. We further note that Jk(T) is only defined for all T if all the zeros are simple and, in that case, Ik(, T) converges for all k<.(Received July 26 1988)
The summatory function of the Möbius function is denoted M(x). In this article we deduce conditional results concerning M(x) assuming the Riemann hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function.
Assuming these conjectures, we show that M(x), when appropriately normalized, possesses a limiting distribution, and also that a strong form of the weak Mertens conjecture
is true. Finally, we speculate on the lower order of M(x) by studying the constructed distribution function. 2000 Mathematics Subject Classification 11M26, 11N56.
We study the characteristic polynomials Z(U, θ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of |Z| and Z/Z
*, and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log
Z as N→∞. In the limit, we show that these two distributions are independent and Gaussian. Costin and Lebowitz [15] previously found
the Gaussian limit distribution for Im log Z using a different approach, and our result for the cumulants proves a conjecture made by them in this case. We also calculate
the leading order N→∞ asymptotics of the moments of |Z| and Z/Z
*. These CUE results are then compared with what is known about the Riemann zeta function ζ (s) on its critical line Re s= 1/2, assuming the Riemann hypothesis. Equating the mean density of the non-trivial zeros of the zeta function at a height
T up the critical line with the mean density of the matrix eigenvalues gives a connection between N and T. Invoking this connection, our CUE results coincide with a theorem of Selberg for the value distribution of log ζ(1/2+iT) in the limit T→∞. They are also in close agreement with numerical data computed by Odlyzko [29] for large but finite T. This leads us to a conjecture for the moments of |ζ(1/2+it) |. Finally, we generalize our random matrix results to the Circular Orthogonal (COE) and Circular Symplectic (CSE) Ensembles.
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