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Let \mu (n) be the Möbius function and e(\alpha)=e^{2\pi i\alpha }. In this paper, we study upper bounds of the classical sum \[ S(x,\alpha)≔\sum _{1\leq n\leq x}\mu (n)e(\alpha n). \] We can improve some classical results of Baker and Harman [J. London Math. Soc. (2) 43 (1991), pp. 193–198]. [ABSTRACT FROM AUTHOR]

It is a classic result that two number fields have equal Dedekind zeta functions if and only if the arithmetic type of a prime p is the same in both fields for almost all prime p. Here, almost all means with the possible exception of a set of Dirichlet density zero. One of the results of this paper shows that the condition density zero can be improved to a specific positive density that depends solely on the degree of the fields. More specifically, for every positive n we exhibit a positive constant c_{n} such that any two degree n number fields K and L are arithmetically equivalent if and only if the set of primes p such that the arithmetic type of p in K and L is not the same has Dirichlet density at most c_n. We in fact show that \displaystyle c_n=\frac {1}{4n^2} works and give a heuristic evidence that points to the fact that this value might be improved to \displaystyle \frac {2}{n^2}. We also show that to check whether or not two number fields are arithmetically equivalent it is enough to check equality between finitely many coefficients of their zeta functions, and we give an upper bound for such number. [ABSTRACT FROM AUTHOR]

In this paper we investigate a result of Ueno on the modularity of generating series associated to the zeta functions of binary Hermitian forms previously studied by Elstrodt et al. We improve his result by showing that the generating series are Eisenstein series. As a consequence we obtain an explicit formula for the special values of zeta functions associated with binary Hermitian forms. [ABSTRACT FROM AUTHOR]

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q-analogue of the Euler constant and proved the irrationality of certain numbers involving q-Euler constant. In this paper, we improve their results and prove the linear independence result involving q-analogue of the Euler constant. Further, we derive the closed-form of a q-analogue of the k-th Stieltjes constant \gamma _k(q). These constants are the coefficients in the Laurent series expansion of a q-analogue of the Riemann zeta function around s=1. Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1. Finally, we study the transcendence nature of some infinite series involving \gamma _1(2). [ABSTRACT FROM AUTHOR]

In 1916, Riesz proved that the Riemann Hypothesis is equivalent to the bound \sum _{n=1}^\infty \frac {\mu (n)}{n^2} \exp \left (- \frac {x}{n^2} \right) = O_{\epsilon } \left (x^{-\frac {3}{4} + \epsilon } \right), as x \rightarrow \infty, for any \epsilon >0. Around the same time, Hardy and Littlewood gave another equivalent criterion for the Riemann Hypothesis while correcting an identity of Ramanujan. In the present paper, we establish a one-variable generalization of the identity of Hardy and Littlewood and as an application, we provide Riesz-type criteria for the Riemann Hypothesis. In particular, we obtain the bound given by Riesz as well as the bound of Hardy and Littlewood. [ABSTRACT FROM AUTHOR]

In this paper, it is demonstrated that derived equivalence between smooth, projective varieties that are either surfaces or abelian implies equality of zeta functions. [ABSTRACT FROM AUTHOR]

Let K be a function field over a finite field k of characteristic p and let K∞/K be a geometric extension with Galois group Zp. Let Kn be the corresponding subextension with Galois group Z/pnZ and genus gn. In this paper, we give a simple explicit formula for gn in terms of an explicit Witt vector construction of the Zp-tower. This formula leads to a tight lower bound on gn which is quadratic in pn. Furthermore, we determine all Zp-towers for which the genus sequence is stable, in the sense that there are a, b, c ∈ Q such that gn = ap2n + bpn + c for n large enough. Such genus stable towers are expected to have strong stable arithmetic properties for their zeta functions. A key technical contribution of this work is a new simplified formula for the Schmid-Witt symbol coming from local class field theory. [ABSTRACT FROM AUTHOR]

We compute the local pro-isomorphic zeta functions at all but finitely many primes for a family of class-two-nilpotent Lie lattices of even rank, parametrized by irreducible monic non-linear polynomials f(x) \in \mathbb {Z}[x]. These Lie lattices correspond to a family of groups introduced by Grunewald and Segal. The result is expressed in terms of a combinatorially defined family of rational functions. [ABSTRACT FROM AUTHOR]

Chatterjee and Garg [Proc. Amer. Math. Soc. 151 (2023), pp. 2011-2022] established closed form for a q-analogue of the Euler-Stieltjes constants. In this article, we aim to build upon their work by extending it to a q-analogue of the double zeta function. Specifically, we derive a closed form expression for \gamma _{0,0}(q) which is a q-analogue of Euler's constant of height 2 and appear as the constant term in the Laurent series expansion of a q-analogue of the double zeta function around s_1 = 1 and s_2=1. Moreover, we examine the linear independence of a set of numbers involving the constant \gamma _0^{\prime *}(q^i), where 1 \leq i \leq r for any integer r \geq 1, that appears in the Laurent series expansion of a q-double zeta function. Finally, we discuss the irrationality of certain numbers involving a 2-double Euler-Stieltjes constant (\gamma _{0,0}(2)). [ABSTRACT FROM AUTHOR]

Abstract. Let Ps ∊ Ds[X0,X1, . . . ,Xl] be a polynomial whose coefficients are the ring of all general Dirichlet series which converge absolutely in the half-plane ʀ(s) > 1/2. In the present paper, we show that the function Ps(L(s), L(1)(s), . . . , L(l)(s)) has infinitely many zeros in the vertical strip D := {s ∊ C : 1/2 < ʀ(s) < 1} if L(s) is hybridly universal and Ps ∊ Ds[X0,X1, . . .,Xl] is a polynomial such that at least one of the degrees of X1, . . . ,Xl is greater than zero. As a corollary, we prove that the function (dk/dsk)Ps(L(s)) with k ∊ N has infinitely many zeros in the strip D when L(s) is hybridly universal and Ps ∊ Ds[X] is a polynomial with degree greater than zero. The upper bounds for the numbers of zeros of Ps(L(s), L(1)(s), . . . , L(l)(s)) and (dk/dsk)Ps(L(s)) are studied as well. [ABSTRACT FROM AUTHOR]

In 2000, Keating and Snaith suggested that the value distribution of the Riemann zeta function ζ(1/2+it) is related to that of the characteristic polynomials of random unitary matrices, ΛU(θ) = ∏n=1N(1 - ei(θn+θ)), with respect to the circular unitary ensemble. They derived the conjecture for the moment of the Riemann zeta function through computing the exact formula for the moments of the characteristic polynomials. In this paper, we compute the shifted moments of the characteristic polynomials of random unitary matrices and express them in a determinant form. When shifts are the roots of unity, we can obtain a precise formula, and this also leads to a new formula analogous to the Selberg's identity applied in Keating and Snaith's computation. [ABSTRACT FROM AUTHOR]

Borel's rank theorem identifies the ranks of algebraic K-groups of the ring of integers of a number field with the orders of vanishing of the Dedekind zeta function attached to the field. Following the work of Gross, we establish a version of this theorem for Artin L-functions by considering equivariant algebraic K-groups of number fields with coefficients in rational Galois representations. This construction involves twisting algebraic K-theory spectra with rational equivariant Moore spectra. We further discuss integral equivariant Moore spectra attached to Galois representations and their potential applications in L-functions. [ABSTRACT FROM AUTHOR]

Let \digamma (n) denote a multiplicative function with range \{-1,0,1\}, and let F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \digamma (n). Then F(x)/\sqrt {x} = a\sqrt {x} + b + E(x), where a and b are constants and E(x) is an error term that either tends to 0 in the limit or is expected to oscillate about 0 in a roughly balanced manner. We say F(x) has persistent bias b (at the scale of \sqrt {x}) in the first case, and apparent bias b in the latter. For example, if \digamma (n)=\mu (n), the Möbius function, then F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \mu (n) has apparent bias 0, while if \digamma (n)=\lambda (n), the Liouville function, then F(x) = \sum _{n=1}^{\left \lfloor x\right \rfloor } \lambda (n) has apparent bias 1/\zeta (1/2). We study the bias when \digamma (p^k) is independent of the prime p, and call such functions fake \mu 's. We investigate the conditions required for such a function to exhibit a persistent or apparent bias, determine the functions in this family with maximal and minimal bias of each type, and characterize the functions with no bias. For such a function F(x) with apparent bias b, we also show that F(x)/\sqrt {x}-a\sqrt {x}-b changes sign infinitely often. [ABSTRACT FROM AUTHOR]

Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field L has infinitely many nontrivial zeros of multiplicity at least 2 if L has a subfield K for which L/K is a nonabelian Galois extension. We also extend this to zeros of order 3 when Gal(L/K) has an irreducible representation of degree at least 3, as predicted by the Artin holomorphy conjecture. [ABSTRACT FROM AUTHOR]

We consider two sequences a(n) and b(n), 1\leq n<\infty, generated by Dirichlet series of the forms \begin{equation*} \sum _{n=1}^{\infty }\dfrac {a(n)}{\lambda _n^{s}}\qquad \text {and}\qquad \sum _{n=1}^{\infty }\dfrac {b(n)}{\mu _n^{s}}, \end{equation*} satisfying a familiar functional equation involving the gamma function \Gamma (s). A general identity is established. Appearing on one side is an infinite series involving a(n) and modified Bessel functions K_{\nu }, wherein on the other side is an infinite series involving b(n) that is an analogue of the Hurwitz zeta function. Six special cases, including a(n)=\tau (n) and a(n)=r_k(n), are examined, where \tau (n) is Ramanujan's arithmetical function and r_k(n) denotes the number of representations of n as a sum of k squares. All but one of the examples appear to be new. [ABSTRACT FROM AUTHOR]

Assuming the Riemann hypothesis, we obtain a formula for the mean value of the k-derivative of \zeta '/\zeta, depending on the pair correlation of zeros of the Riemann zeta-function. This formula allows us to obtain new equivalences to Montgomery's pair correlation conjecture. This extends a result of Goldston, Gonek, and Montgomery where the mean value of \zeta '/\zeta was considered. [ABSTRACT FROM AUTHOR]

We express the quotients of the Riemann zeta function at consecutive positive integers in terms of the limit of an average sum of quotients of special values of Dirichlet L-functions. [ABSTRACT FROM AUTHOR]

In this article, we study the Piltz divisor problem, which is sometimes called the generalized Dirichlet divisor problem, over number fields. We establish an identity akin to Voronoï's formula concerning the error term in the Dirichlet divisor problem. [ABSTRACT FROM AUTHOR]

We explicitly compute the archimedean parts of the Rankin-Selberg integrals on GL2 × GSp4 and GL4 discovered by Pollack and Shah when the archimedean components are the class one principal series representations. [ABSTRACT FROM AUTHOR]

We show that any derived equivalent smooth, projective varieties of dimension 3 over a finite field Fq have equal zeta functions. This result is an application of the extension to smooth, projective varieties over any field of Popa and Schnell's proof that derived equivalent smooth, projective varieties over C have isogenous Albanese torsors; this result is proven in an appendix by Achter, Casalaina-Martin, Honigs and Vial. [ABSTRACT FROM AUTHOR]

Let Λ be the von Mangoldt function and R(n) = ... Λ(h)Λ(k). Let further N,H be two integers, N ≥ 2, 1 ≤ H ≤ N, and assume that the Riemann Hypothesis holds. Then ... where ρ = 1/2+iγ runs over the non-trivial zeros of the Riemann zeta function ζ(s). [ABSTRACT FROM AUTHOR]

Abstract. For a region Ω ⊂ C denote by Ψ(Q;Ω) the number of complex algebraic numbers in Ω of degree ≤ n and naive height ≤ Q. We show that....where ν is the Lebesgue measure on the complex plane and the function ψ will be given explicitly. [ABSTRACT FROM AUTHOR]

We prove a conjecture due to Kimoto and Wakayama from 2006 concerning Apéry-like numbers associated to a special value of a spectral zeta function. Our proof uses hypergeometric series and p-adic analysis. [ABSTRACT FROM AUTHOR]

For an action α of Zd by homeomorphisms of a compact metric space, D. Lind introduced a dynamical zeta function and conjectured that this function has a natural boundary when d ≥ 2. In this note, under the assumption that α is a mixing action by continuous automorphisms of a compact connected abelian group of finite topological dimension, it is shown that the upper growth rate of periodic points is zero and that the unit circle is a natural boundary for the dynamical zeta function. [ABSTRACT FROM AUTHOR]

Let &963;, t ∈ R, s = &963; + it and (s) be the Riemann zeta function. Put f&963;(t) := (&963; - it)/(a - it) and F&963; (t) := f&963;(t)/f&963;(0). We show that f&963;(t) is a characteristic function of a probability measure for any 0 < &963; ≠ 1 by giving the probability density function. By using this fact, we show that for any C ∈ C satisfying C > 10 and -19/2 ≤ C ≤ 17/2, the function (s)+Cs does not vanish in the half-plane &963; > 1/18. Moreover, we prove that F&963;(t) is an infinitely divisible characteristic function for any &963; > 1. Furthermore, we show that the Riemann hypothesis is true if each F&963;(t) is an infinitely divisible characteristic function for each 1/2 < &963; < 1. [ABSTRACT FROM AUTHOR]

We discuss the stabilization of symmetric products Symn (X) of a smooth projective variety X in the Grothendieck ring of varieties. For smooth projective surfaces X with non-zero h0(X, ωX ), these products do not stabilize; we conditionally show that they do not stabilize in another related sense, in response to a question of R. Vakil and M. Wood. There are analogies between such stabilization, the Dold-Thom theorem, and the analytic class number formula. Finally, we discuss conjectural Hodge-theoretic obstructions to the stabilization of symmetric products. We provide evidence for these obstructions by showing that the Newton polygon of the motivic zeta function associated to a curve equals the Hodge polygon of the curve. [ABSTRACT FROM AUTHOR]