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Using some formulas of S. Ramanujan, we compute in closed form the Fourier transform of functions related to Riemann zeta function $\zeta (s)=\sum \nolimits _{n=1}^{\infty } {1}/{n^{s}}$ and other Dirichlet series.
The multiple T-value, which is a variant of the multiple zeta value of level two, was introduced by Kaneko and Tsumura [‘Zeta functions connecting multiple zeta values and poly-Bernoulli numbers’, in: Various Aspects of Multiple Zeta Functions, Advanced Studies in Pure Mathematics, 84 (Mathematical Society of Japan, Tokyo, 2020), 181–204]. We show that the generating function of a weighted sum of multiple T-values of fixed weight and depth is given in terms of the multiple T-values of depth one by solving a differential equation of Heun type.
We study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over $\mathbb {Z}$ and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp forms for $\mathrm {SL} _2(\mathbb {Z})$, we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo $\zeta (2)$ and modulo lower depth. We formulate a single conjecture about the homology of this Lie algebra which implies conjectures due to Broadhurst and Kreimer, Racinet, Zagier, and Drinfeld on the structure of multiple zeta values and on the Grothendieck–Teichmüller Lie algebra.
We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of $\operatorname{Sp}(n,\mathbb{Z})$ on $\mathbb{C}[(L^{\prime }/L)^{n}]$. By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree $n$, a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard $L$-function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.
Let $\mathbb {F}_q$ be the finite field of q elements. In this paper, we study the vanishing behavior of multizeta values over
$\mathbb {F}_q[t]$ at negative integers. These values are analogs of the classical multizeta values. At negative integers, they are series of products of power sums
$S_d(k)$ which are polynomials in t. By studying the t-valuation of
$S_d(s)$ for
$s < 0$, we show that multizeta values at negative integers vanish only at trivial zeros. The proof is inspired by the idea of Sheats in the proof of a statement of “greedy element” by Carlitz.
We consider the sum
$\sum 1/\gamma $
, where
$\gamma $
ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval
$(0,T]$
, and examine its behaviour as
$T \to \infty $
. We show that, after subtracting a smooth approximation
$({1}/{4\pi }) \log ^2(T/2\pi ),$
the sum tends to a limit
$H \approx -0.0171594$
, which can be expressed as an integral. We calculate H to high accuracy, using a method which has error
$O((\log T)/T^2)$
. Our results improve on earlier results by Hassani [‘Explicit approximation of the sums over the imaginary part of the non-trivial zeros of the Riemann zeta function’, Appl. Math. E-Notes16 (2016), 109–116] and other authors.
Let E(s, Q) be the Epstein zeta function attached to a positive definite quadratic form of discriminant D < 0, such that h(D) ≥ 2, where h(D) is the class number of the imaginary quadratic field
${{\mathbb{Q}}(\sqrt D)}$
. We denote by NE(σ1, σ2, T) the number of zeros of E(s, Q) in the rectangle σ1 < Re(s) ≤ σ2 and T ≤ Im (s) ≤ 2T, where 1/2 < σ1 < σ2 < 1 are fixed real numbers. In this paper, we improve the asymptotic formula of Gonek and Lee for NE(σ1, σ2, T), obtaining a saving of a power of log T in the error term.
As pointed out by Alexandre Bailleul, the paper mentioned in the title contains a mistake in Theorem 2.2. The hypothesis on the linear relation of the almost periods is not sufficient. In this note, we fix the problem and its minor consequences on other results in the same paper.
Let $E/\mathbb {Q}$ be a number field of degree $n$. We show that if $\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central point is $\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$. The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$ associated to the maximal order of $E$, and the escape of mass of the torus orbit associated to the trivial ideal class.
In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least
$(19-\cot (1/4))/16 = 94.27\ldots \%$
of the L-functions under consideration do not vanish at 1/2.
Let
$a_1$
,
$a_2$
, and
$a_3$
be distinct reduced residues modulo q satisfying the congruences
$a_1^2 \equiv a_2^2 \equiv a_3^2 \ (\mathrm{mod}\ q)$
. We conditionally derive an asymptotic formula, with an error term that has a power savings in q, for the logarithmic density of the set of real numbers x for which
$\pi (x;q,a_1)> \pi (x;q,a_2) > \pi (x;q,a_3)$
. The relationship among the
$a_i$
allows us to normalize the error terms for the
$\pi (x;q,a_i)$
in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.
We study lower bounds of a general family of L-functions on the
$1$
-line. More precisely, we show that for any
$F(s)$
in this family, there exist arbitrarily large t such that
$F(1+it)\geq e^{\gamma _F} (\log _2 t + \log _3 t)^m + O(1)$
, where m is the order of the pole of
$F(s)$
at
$s=1$
. This is a generalisation of the result of Aistleitner, Munsch and Mahatab [‘Extreme values of the Riemann zeta function on the
$1$
-line’, Int. Math. Res. Not. IMRN2019(22) (2019), 6924–6932]. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg L-functions of the type
$L(s,f\times f)$
on the
$1$
-line.
We prove that, for any small $\varepsilon > 0$, the number of irrationals among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\ldots ,\zeta (s)$ is at least $( c_0 - \varepsilon )({s^{1/2}}/{(\log s)^{1/2}})$, provided $s$ is a sufficiently large odd integer with respect to $\varepsilon$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. Our work improves the lower bound $2^{(1-\varepsilon )({\log s}/{\log \log s})}$ of the previous work of Fischler, Sprang and Zudilin. We follow the same strategy of Fischler, Sprang and Zudilin. The main new ingredient is an asymptotically optimal design for the zeros of the auxiliary rational functions, which relates to the inverse totient problem.
In order to study integers with few prime factors, the average of
$\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$
has been a central object of research. One of the more important cases,
$k=2$
, was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For
$k\geq 2$
, it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve.
Let
$\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$
, where
$\unicode[STIX]{x1D707}_{j}$
denotes the Liouville function for
$(j+1)$
-free integers, and
$0$
otherwise. In this paper we evaluate the average value of
$\unicode[STIX]{x1D6EC}_{j,k}$
in a residue class
$n\equiv a\text{ mod }q$
,
$(a,q)=1$
, uniformly on
$q$
. When
$j\geq 2$
, we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for
$\unicode[STIX]{x1D6EC}_{k}(n)$
involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported
${\mathcal{C}}^{2}$
function.
where
$\chi $
is a primitive Dirichlet character and F belongs to a class of L-functions. The class we consider includes L-functions associated with automorphic representations of
$GL(n)$
over
${\mathbb {Q}}$
.
Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic
$L$
-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet
$L$
-functions modulo
$q$
weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length
$q^{1/4}$
. An auxiliary result of independent interest is a sharp upper bound for a certain sixth moment for automorphic
$L$
-functions, which we also use to improve the best known subconvexity bounds for automorphic
$L$
-functions in the level aspect.
We generalize current known distribution results on Shanks–Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let
$\unicode[STIX]{x1D70B}(x;q,a)$
be the number of primes up to
$x$
that are congruent to
$a$
modulo
$q$
. For a fixed integer
$q$
and distinct invertible congruence classes
$a_{0},a_{1},\ldots ,a_{D}$
, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real
$x$
for which the inequalities
$\unicode[STIX]{x1D70B}(x;q,a_{0})>\unicode[STIX]{x1D70B}(x;q,a_{1})>\cdots >\unicode[STIX]{x1D70B}(x;q,a_{D})$
are simultaneously satisfied admits a logarithmic density.
Higher dimensional analogues of the modular group $\mathit{PSL}(2,\mathbb{Z})$ are closely related to hyperbolic reflection groups and Coxeter polyhedra with big symmetry groups. In this context, we develop a theory and dissection properties of ideal hyperbolic $k$-rectified regular polyhedra, which is of independent interest. As an application, we can identify the covolumes of the quaternionic modular groups with certain explicit rational multiples of the Riemann zeta value $\unicode[STIX]{x1D701}(3)$.
Ihara et al. proved the derivation relation for multiple zeta values. The first-named author obtained its counterpart for finite multiple zeta values in ${\mathcal{A}}$. In this paper, we present its generalization in $\widehat{{\mathcal{A}}}$.
We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $\operatorname{SL}_{3}(\mathbf{Z})$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky–Nelson, corresponding to multiplicative Dirichlet characters, and applies, in particular, to trace functions of small conductor modulo primes.