Martin, Mossinghoff, and Trudgian [19] recently introduced a family of arithmetic functions called “fake $\mu $’s,” which are multiplicative functions for which there is a $\{-1,0,1\}$-valued sequence $(\varepsilon _j)_{j=1}^{\infty }$ such that $f(p^j) = \varepsilon _j$ for all primes p. They investigated comparative number-theoretic results for fake $\mu $’s and, in particular, proved oscillation results at scale $\sqrt {x}$ for the summatory functions of fake $\mu $’s with $\varepsilon _1=-1$ and $\varepsilon _2=1$. In this article, we establish new oscillation results for the summatory functions of all nontrivial fake $\mu $’s at scales $x^{1/2\ell }$ where $\ell $ is a positive integer (the “critical index”) depending on f; for $\ell =1$ this recovers the oscillation results in [19]. Our work also recovers results on the indicator functions of powerfree and powerfull numbers; we generalize techniques applied to each of these examples to extend to all fake $\mu $’s.

We prove the central limit theorem (CLT), the first-order Edgeworth expansion and a mixing local central limit theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise $C^2$ expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on $\mathbb {R}$. The class of observables in the CLT and the MLCLT on $\mathbb {R}$ include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber [Sampling the Lindelöf hypothesis with the Cauchy random walk. Proc. Lond. Math. Soc. (3) 98 (2009), 241–270] and Steuding [Sampling the Lindelöf hypothesis with an ergodic transformation. RIMS Kôkyûroku Bessatsu B34 (2012), 361–381] who have proven the strong law of large numbers for sampling the Lindelöf hypothesis.

Let $\ell $ be an odd prime. We investigate the enumeration of cyclic extensions of degree $\ell $ over $\mathbb {Q}$ subject to specified local conditions. By ordering these extensions according to their conductors, we derive an asymptotic count with a power-saving error term. As a consequence of our results, we analyze the distribution of values of L-functions associated with these extensions in the critical strip.

We will give a precise and explicit asymptotic estimate for the characteristic of the Riemann zeta function $\zeta $ with an error term of order $O(\frac {\log r}{r})$ and a corresponding asymptotic estimate for the number of fixed points of $\zeta $.

In this paper, we investigate Frobenius eigenvalues of the compactly supported rigid cohomology of a variety defined over a finite field with q elements, using Dwork’s method. Our study yields several arithmetic consequences. First, we establish that the zeta functions of a set of related affine varieties can reveal all Frobenius eigenvalues of the rigid cohomology of the variety up to a Tate twist. This result does not seem to be known for the $\ell$-adic cohomology. As a second application, we provide several q-divisibility lower bounds for the Frobenius eigenvalues of the rigid cohomology of the variety, in terms of the dimension and multi-degrees of the defining equations. These divisibility bounds for rigid cohomology are generally better than what is suggested from the best known divisibility bounds in $\ell$-adic cohomology, both before and after the middle cohomological degree.

A real number is simply normal to base b if its base-b expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency at which the digit $1$ appears in the binary expansion of $2^{p/q}$ and a mean value of the Riemann zeta function on vertical arithmetic progressions. In particular, we show that

$$\begin{align*}\lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } \zeta\left(\frac{2 n\pi i}{\log 2}\right) \frac{e^{2n\pi i p/q} }{n} =0 \end{align*}$$

$2^{p/q}$

$2$

Let $\pi$ be an irreducible cuspidal automorphic representation of ${\mathrm{GL}}_n(\mathbb{A}_{\mathbb{Q}})$ with associated L-function $L(s, \pi)$. We study the behaviour of the partial Euler product of $L(s, \pi)$ at the centre of the critical strip. Under the assumption of the Generalised Riemann Hypothesis for $L(s, \pi)$ in conjunction with the Ramanujan–Petersson conjecture as necessary, we establish an asymptotic, off a set of finite logarithmic measure, for the partial Euler product at the central point, which confirms a conjecture of Kurokawa (2012). As an application, we obtain results towards Chebyshev’s bias in the recently proposed framework of Aoki–Koyama (2023).

We prove an asymptotic formula for the second moment of central values of Dirichlet L-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulo d inside the full group of characters modulo q. Suppose that $\nu _p(d) \geq \nu _p(q)/2$ for all primes p dividing q. In this range, we obtain an asymptotic formula with a power-saving error term; curiously, there is a secondary main term of rough size $q^{1/2}$ here which is not predicted by the integral moments conjecture of Conrey, Farmer, Keating, Rubinstein, and Snaith. The lower-order main term does not appear in the second moment of the Riemann zeta function, so this feature is not anticipated from the analogous archimedean moment problem.

We also obtain an asymptotic result for smaller d, with $\nu _p(q)/3 \leq \nu _p(d) \leq \nu _p(q)/2$, with a power-saving error term for d larger than $q^{2/5}$. In this more difficult range, the secondary main term somewhat changes its form and may have size roughly d, which is only slightly smaller than the diagonal main term.

In this study, we introduce multiple zeta functions with structures similar to those of symmetric functions such as the Schur P-, Schur Q-, symplectic and orthogonal functions in representation theory. Their basic properties, such as the domain of absolute convergence, are first considered. Then, by restricting ourselves to the truncated multiple zeta functions, we derive the Pfaffian expression of the Schur Q-multiple zeta functions, the sum formula for Schur P- and Schur Q-multiple zeta functions, the determinant expressions of symplectic and orthogonal Schur multiple zeta functions by making an assumption on variables. Finally, we generalize those to the quasi-symmetric functions.

We establish sharp upper bounds for shifted moments of quadratic Dirichlet L-function under the generalized Riemann hypothesis. Our result is then used to prove bounds for moments of quadratic Dirichlet character sums.

In this paper, we study the twisted Ruelle zeta function associated with the geodesic flow of a compact, hyperbolic, odd-dimensional manifold X. The twisted Ruelle zeta function is associated with an acyclic representation $\chi \colon \pi _{1}(X) \rightarrow \operatorname {\mathrm {GL}}_{n}(\mathbb {C})$, which is close enough to an acyclic, unitary representation. In this case, the twisted Ruelle zeta function is regular at zero and equals the square of the refined analytic torsion, as it is introduced by Braverman and Kappeler in [6], multiplied by an exponential, which involves the eta invariant of the even part of the odd-signature operator, associated with $\chi $.

In his “lost notebook,” Ramanujan used iterated derivatives of two theta functions to define sequences of q-series $\{U_{2t}(q)\}$ and $\{V_{2t}(q)\}$ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of “partition Eisenstein series,” extensions of the classical Eisenstein series $E_{2k}(q),$ defined by

$$ \begin{align*}\lambda=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \longmapsto \ \ \ E_{\lambda}(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}. \end{align*} $$

For functions $\phi : \mathcal {P}\mapsto {\mathbb C}$ on partitions, the weight $2n$ partition Eisenstein trace is

$$ \begin{align*}\operatorname{\mathrm{Tr}}_n(\phi;q):=\sum_{\lambda \vdash n} \phi(\lambda)E_{\lambda}(q). \end{align*} $$

For all t, we prove that $U_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _U;q)$ and $V_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _V;q),$ where $\phi _U$ and $\phi _V$ are natural partition weights, giving the first explicit quasimodular formulas for these series.

This paper contains a method to prove the existence of smooth curves in positive characteristic whose Jacobians have unusual Newton polygons. Using this method, I give a new proof that there exist supersingular curves of genus $4$ in every prime characteristic. More generally, the main result of the paper is that, for every $g \geq 4$ and prime p, every Newton polygon whose p-rank is at least $g-4$ occurs for a smooth curve of genus g in characteristic p. In addition, this method resolves some cases of Oort’s conjecture about Newton polygons of curves.

To date, the bestmethodsfor estimating the growth of mean values of arithmetic functions rely on the Voronoï summation formula. By noticing a general pattern in the proof of his summation formula, Voronoï postulated that analogous summation formulas for $\sum a(n)f(n)$ can be obtained with ‘nice’ test functions f(n), provided a(n) is an ‘arithmetic function’. These arithmetic functions a(n) are called so because they are expected to appear as coefficients of some L-functions satisfying certain properties. It has been well-known that the functional equation for a general L-function can be used to derive a Voronoï-type summation identity for that L-function. In this article, we show that such a Voronoï-typesummation identity in fact endows the L-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.

Multiples zeta values and alternating multiple zeta values in positive characteristic were introduced by Thakur and Harada as analogues of classical multiple zeta values of Euler and Euler sums. In this paper, we determine all linear relations between alternating multiple zeta values and settle the main goals of these theories. As a consequence, we completely establish Zagier–Hoffman’s conjectures in positive characteristic formulated by Todd and Thakur which predict the dimension and an explicit basis of the span of multiple zeta values of Thakur of fixed weight.

Let $\zeta _K(s)$ denote the Dedekind zeta-function associated to a number field K. We give an effective upper bound for the height of the first nontrivial zero other than $1/2$ of $\zeta _K(s)$ under the generalised Riemann hypothesis. This is a refinement of the earlier bound obtained by Sami [‘Majoration du premier zéro de la fonction zêta de Dedekind’, Acta Arith. 99(1) (2000), 61–65].

We investigate the joint distribution of L-functions on the line $ \sigma= {1}/{2} + {1}/{G(T)}$ and $ t \in [ T, 2T]$, where $ \log \log T \leq G(T) \leq { \log T}/{ ( \log \log T)^2 } $. We obtain an upper bound on the discrepancy between the joint distribution of L-functions and that of their random models. As an application we prove an asymptotic expansion of a multi-dimensional version of Selberg’s central limit theorem for L-functions on $ \sigma= 1/2 + 1/{G(T)}$ and $ t \in [ T, 2T]$, where $ ( \log T)^\varepsilon \leq G(T) \leq { \log T}/{ ( \log \log T)^{2+\varepsilon } } $ for $ \varepsilon > 0$.

For an elliptic curve E defined over a number field K and $L/K$ a Galois extension, we study the possibilities of the Galois group Gal$(L/K)$, when the Mordell–Weil rank of $E(L)$ increases from that of $E(K)$ by a small amount (namely 1, 2, and 3). In relation with the vanishing of corresponding L-functions at $s=1$, we prove several elliptic analogues of classical theorems related to Artin’s holomorphy conjecture. We then apply these to study the analytic minimal subfield, first introduced by Akbary and Murty, for the case when order of vanishing is 2. We also investigate how the order of vanishing changes as rank increases by 1 and vice versa, generalizing a theorem of Kolyvagin.