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We study the moments
$M_k(T;\,\alpha) = \int_T^{2T} |\zeta(s,\alpha)|^{2k}\,dt$
of the Hurwitz zeta function
$\zeta(s,\alpha)$
on the critical line,
$s = 1/2 + it$
with a rational shift
$\alpha \in \mathbb{Q}$
. We conjecture, in analogy with the Riemann zeta function, that
$M_k(T;\,\alpha) \sim c_k(\alpha) T (\!\log T)^{k^2}$
. Using heuristics from analytic number theory and random matrix theory, we conjecturally compute
$c_k(\alpha)$
. In the process, we investigate moments of products of Dirichlet L-functions on the critical line. We prove some of our conjectures for the cases
$k = 1,2$
.
We study Ohno–Zagier type relations for multiple t-values and multiple t-star values. We represent the generating function of sums of multiple t-(star) values with fixed weight, depth and height in terms of the generalised hypergeometric function
$\,_3F_2$
. As applications, we get a formula for the generating function of sums of multiple t-(star) values of maximal height and a weighted sum formula for sums of multiple t-(star) values with fixed weight and depth.
We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.
satisfying a familiar functional equation involving the gamma function $\Gamma (s)$. Two general identities are established. The first involves the modified Bessel function $K_{\mu }(z)$, and can be thought of as a ‘modular’ or ‘theta’ relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are $K_{\mu }(z)$, the Bessel functions of imaginary argument $I_{\mu }(z)$, and ordinary hypergeometric functions ${_2F_1}(a,b;c;z)$. Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan’s arithmetical function $\tau (n)$, the number of representations of n as a sum of k squares $r_k(n)$, and primitive Dirichlet characters $\chi (n)$.
We establish the optimal order of Malliavin-type remainders in the asymptotic density approximation formula for Beurling generalized integers. Given
$\alpha \in (0,1]$
and
$c>0$
(with
$c\leq 1$
if
$\alpha =1$
), a generalized number system is constructed with Riemann prime counting function
$ \Pi (x)= \operatorname {\mathrm {Li}}(x)+ O(x\exp (-c \log ^{\alpha } x ) +\log _{2}x), $
and whose integer counting function satisfies the extremal oscillation estimate
$N(x)=\rho x + \Omega _{\pm }(x\exp (- c'(\log x\log _{2} x)^{\frac {\alpha }{\alpha +1}})$
for any
$c'>(c(\alpha +1))^{\frac {1}{\alpha +1}}$
, where
$\rho>0$
is its asymptotic density. In particular, this improves and extends upon the earlier work [Adv. Math. 370 (2020), Article 107240].
The Euler–Mascheroni constant
$\gamma =0.5772\ldots \!$
is the
$K={\mathbb Q}$
example of an Euler–Kronecker constant
$\gamma _K$
of a number field
$K.$
In this note, we consider the size of the
$\gamma _q=\gamma _{K_q}$
for cyclotomic fields
$K_q:={\mathbb Q}(\zeta _q).$
Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that
In other words, under EH, the
$\gamma _q /\!\log q$
in these ranges converge to the one point distribution at
$1$
. This theorem refines and extends a previous result of Ford, Luca and Moree for prime
$q.$
The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.
Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric
$\{\pm 1\}$
-matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random
$\{\pm 1\}$
-matrices over
$\mathbb{F}_p$
for primes
$2 < p \leq \exp(O(n^{1/4}))$
. Previously, such estimates were available only for
$p = o(n^{1/8})$
. At the heart of our proof is a way to combine multiple inverse Littlewood–Offord-type results to control the contribution to singularity-type events of vectors in
$\mathbb{F}_p^{n}$
with anticoncentration at least
$1/p + \Omega(1/p^2)$
. Previously, inverse Littlewood–Offord-type results only allowed control over vectors with anticoncentration at least
$C/p$
for some large constant
$C > 1$
.
We obtain some improved results for the exponential sum
$\sum _{x<n\leq 2x}\Lambda (n)e(\alpha k n^{\theta })$
with
$\theta \in (0,5/12),$
where
$\Lambda (n)$
is the von Mangoldt function. Such exponential sums have relations with the so-called quasi-Riemann hypothesis and were considered by Murty and Srinivas [‘On the uniform distribution of certain sequences’, Ramanujan J.7 (2003), 185–192].
We prove that the Riemann hypothesis is equivalent to the condition
$\int _{2}^x\left (\pi (t)-\operatorname {\textrm {li}}(t)\right )\textrm {d}t<0$
for all
$x>2$
. Here,
$\pi (t)$
is the prime-counting function and
$\operatorname {\textrm {li}}(t)$
is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function
$\theta (t)$
and discuss the extent to which one can make related claims unconditionally.
In this paper, we study lower-order terms of the one-level density of low-lying zeros of quadratic Hecke L-functions in the Gaussian field. Assuming the generalized Riemann hypothesis, our result is valid for even test functions whose Fourier transforms are supported in $(-2, 2)$. Moreover, we apply the ratios conjecture of L-functions to derive these lower-order terms as well. Up to the first lower-order term, we show that our results are consistent with each other when the Fourier transforms of the test functions are supported in $(-2, 2)$.
The aim of this article is to establish the behaviour of partial Euler products for Dirichlet L-functions under the generalised Riemann hypothesis (GRH) via Ramanujan’s work. To understand the behaviour of Euler products on the critical line, we invoke the deep Riemann hypothesis (DRH). This work clarifies the relation between GRH and DRH.
In this paper, we obtain a precise formula for the one-level density of L-functions attached to non-Galois cubic Dedekind zeta functions. We find a secondary term which is unique to this context, in the sense that no lower-order term of this shape has appeared in previously studied families. The presence of this new term allows us to deduce an omega result for cubic field counting functions, under the assumption of the Generalised Riemann Hypothesis. We also investigate the associated L-functions Ratios Conjecture and find that it does not predict this new lower-order term. Taking into account the secondary term in Roberts’s conjecture, we refine the Ratios Conjecture to one which captures this new term. Finally, we show that any improvement in the exponent of the error term of the recent Bhargava–Taniguchi–Thorne cubic field counting estimate would imply that the best possible error term in the refined Ratios Conjecture is
$O_\varepsilon (X^{-\frac 13+\varepsilon })$
. This is in opposition with all previously studied families in which the expected error in the Ratios Conjecture prediction for the one-level density is
$O_\varepsilon (X^{-\frac 12+\varepsilon })$
.
A new reciprocity formula for Dirichlet L-functions associated to an arbitrary primitive Dirichlet character of prime modulus q is established. We find an identity relating the fourth moment of individual Dirichlet L-functions in the t-aspect to the cubic moment of central L-values of Hecke–Maaß newforms of level at most
$q^{2}$
and primitive central character
$\psi ^{2}$
averaged over all primitive nonquadratic characters
$\psi $
modulo q. Our formula can be thought of as a reverse version of recent work of Petrow–Young. Direct corollaries involve a variant of Iwaniec’s short interval fourth moment bound and the twelfth moment bound for Dirichlet L-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms.
We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms
$u_j$
with spectral parameter
$t_j$
, where the second moment is a sum over
$t_j$
in a short interval. At the central point
$s=1/2$
of the L-function, our interval is smaller than previous known results. More specifically, for
$\left \lvert t_j\right \rvert $
of size T, our interval is of size
$T^{1/5}$
, whereas the previous best was
$T^{1/3}$
, from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at
$s=1/2+it$
provided
$\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $
for any fixed
$\delta>0$
. Since
$\lvert t\rvert $
can be taken significantly smaller than
$\left \lvert t_j\right \rvert $
, this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at
$s=1/2$
.
In this note, by introducing a new variant of the resonator function, we give an explicit version of the lower bound for
$\log |L(\sigma ,\chi )|$
in the strip
$1/2<\sigma <1$
, which improves the result of Aistleitner et al. [‘On large values of
$L(\sigma ,\chi )$
’, Q. J. Math.70 (2019), 831–848].
We prove a new generalization of Davenport's Fourier expansion of the infinite series involving the fractional part function over arithmetic functions. A new Mellin transform related to the Riemann zeta function is also established.
The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$, for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).
En s’appuyant sur la notion d’équivalence au sens de Bohr entre polynômes de Dirichlet et sur le fait que sur un corps quadratique la fonction zeta de Dedekind peut s’écrire comme produit de la fonction zeta de Riemann et d’une fonction L, nous montrons que, pour certaines valeurs du discriminant du corps quadratique, les sommes partielles de la fonction zeta de Dedekind ont leurs zéros dans des bandes verticales du plan complexe appelées bandes critiques et que les parties réelles de leurs zéros y sont denses.
The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.
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.