Let $L(s,\unicode[STIX]{x1D712})$ be the Dirichlet $L$ -function associated to a non-principal primitive character $\unicode[STIX]{x1D712}$ modulo $q$ with $3\leqslant q\leqslant 400\,000$ . We prove a new explicit zero-free region for $L(s,\unicode[STIX]{x1D712})$ : $L(s,\unicode[STIX]{x1D712})$ does not vanish in the region $\mathfrak{Re}\,s\geqslant 1-1/(R\log (q\max (1,|\mathfrak{Im}\,s|)))$ with $R=5.60$ . This improves a result of McCurley where $9.65$ was shown to be an admissible value for $R$ .

We prove asymptotic formulas for the number of integers at most $x$ that can be written as the product of $k~({\geqslant}2)$ distinct primes $p_{1}\cdots p_{k}$ with each prime factor in an arithmetic progression $p_{j}\equiv a_{j}\hspace{0.2em}{\rm mod}\hspace{0.2em}q$ , $(a_{j},q)=1$   $(q\geqslant 3,1\leqslant j\leqslant k)$ . For any $A>0$ , our result is uniform for $2\leqslant k\leqslant A\log \log x$ . Moreover, we show that there are large biases toward certain arithmetic progressions $(a_{1}\hspace{0.2em}{\rm mod}\hspace{0.2em}q,\ldots ,a_{k}\hspace{0.2em}{\rm mod}\hspace{0.2em}q)$ , and such biases have connections with Mertens’ theorem and the least prime in arithmetic progressions.

While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form

$$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D70C}}R(\unicode[STIX]{x1D70C})x^{\unicode[STIX]{x1D70C}},\end{eqnarray}$$

$\unicode[STIX]{x1D70C}$

$\unicode[STIX]{x1D701}(s)$

$R(x)\in \overline{\mathbb{Q}}(x)$

$x>0$

$$\begin{eqnarray}f(x)=\mathop{\sum }_{\unicode[STIX]{x1D70C}}\frac{x^{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}}\end{eqnarray}$$

$(1,\infty )$

$f(x)$

$x<1$

$x>1$

$x<1$

$d\geqslant 1$

$f(\unicode[STIX]{x1D70B}\sqrt{d}x)$

$(0,1/\unicode[STIX]{x1D70B}\sqrt{d})$

$$\begin{eqnarray}L^{\prime }(1,\unicode[STIX]{x1D712}_{-d})\neq 0,\end{eqnarray}$$

$\unicode[STIX]{x1D712}_{-d}$

$K:=\mathbb{Q}(\sqrt{-d})$

We investigate the approximation of quadratic Dirichlet $L$ -functions over function fields by truncations of their Euler products. We first establish representations for such $L$ -functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an $L$ -function away from its zeros and that, when the length of the product tends to infinity, we recover the original $L$ -function. We also obtain explicit expressions for the arguments of quadratic Dirichlet $L$ -functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet $L$ -function over a function field, an auxiliary function based on the approximate functional equation that equals the $L$ -function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions and prove that the Riemann hypothesis holds for them and that their zeros are related to those of the associated $L$ -function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated $L$ -function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple.

We study a family of mixed Tate motives over  $\mathbb{Z}$ whose periods are linear forms in the zeta values  $\unicode[STIX]{x1D701}(n)$ . They naturally include the Beukers–Rhin–Viola integrals for  $\unicode[STIX]{x1D701}(2)$ and the Ball–Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over  $\mathbb{Z}$ which contains all the non-trivial extensions between simple objects. In a joint appendix with Don Zagier, we prove the compatibility between the structure of the motives considered here and the representations of their periods as sums of series.

Inspired by the recent work of M. Nakasuji, O. Phuksuwan and Y. Yamasaki, we combine interpolated multiple zeta values and Schur multiple zeta values into one object, which we call interpolated Schur multiple zeta values. Our main result will be a Jacobi–Trudi formula for a certain class of these new objects. This generalizes an analogous result for Schur multiple zeta values and implies algebraic relations between interpolated multiple zeta values.

We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at nonpositive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the $\unicode[STIX]{x1D709}$-function defined by Arakawa and Kaneko. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.

We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by revealing an explicit connection between the Fourier expansion of multiple Eisenstein series and the Goncharov co-product on Hopf algebras of iterated integrals.

Conditionally on the generalized Lindelöf hypothesis, we obtain an asymptotic for the fourth moment of Hecke–Maass cusp forms of large Laplacian eigenvalue for the full modular group. This lends support to the random wave conjecture.

We study the $1$ -level density of low-lying zeros of Dirichlet $L$ -functions attached to real primitive characters of conductor at most $X$ . Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$ , which is valid when the support of the Fourier transform of the corresponding even test function $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$ . We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$ , both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$ , and is analogous to a lower order term which was isolated by Rudnick in the function field case.

This paper concerns the problem of algebraic differential independence of the gamma function and ${\mathcal{L}}$ -functions in the extended Selberg class. We prove that the two kinds of functions cannot satisfy a class of algebraic differential equations with functional coefficients that are linked to the zeros of the ${\mathcal{L}}$ -function in a domain $D:=\{z:0<\text{Re}\,z<\unicode[STIX]{x1D70E}_{0}\}$ for a positive constant $\unicode[STIX]{x1D70E}_{0}$ .

We give an explicit upper bound for the number of zeros of Hecke–Landau zeta-functions in a rectangle.

This paper draws connections between the double shuffle equations and structure of associators; Hain and Matsumoto’s universal mixed elliptic motives; and the Rankin–Selberg method for modular forms for $\text{SL}_{2}(\mathbb{Z})$ . We write down explicit formulae for zeta elements $\unicode[STIX]{x1D70E}_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed Tate motives over $\mathbb{Z}$ ) in depths up to four, give applications to the Broadhurst–Kreimer conjecture, and solve the double shuffle equations for multiple zeta values in depths two and three.

We consider an arithmetic function defined independently by John G. Thompson and Greg Simay, with particular attention to its mean value, its maximal size, and the analytic nature of its Dirichlet series generating function.

We introduce the Hurwitz-type spectral zeta functions for the quantum Rabi models, and give their meromorphic continuation to the whole complex plane with only one simple pole at $s=1$ . As an application, we give the Weyl law for the quantum Rabi models. As a byproduct, we also give a rationality of Rabi–Bernoulli polynomials introduced in this paper.

We define a multiple Dirichlet series whose group of functional equations is the Weyl group of the affine Kac–Moody root system $\widetilde{A}_{n}$ , generalizing the theory of multiple Dirichlet series for finite Weyl groups. The construction is over the rational function field $\mathbb{F}_{q}(t)$ , and is based upon four natural axioms from algebraic geometry. We prove that the four axioms yield a unique series with meromorphic continuation to the largest possible domain and the desired infinite group of symmetries.

Two results related to the mixed joint universality for a polynomial Euler product $\unicode[STIX]{x1D711}(s)$ and a periodic Hurwitz zeta function $\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D6FC};\mathfrak{B})$ , when $\unicode[STIX]{x1D6FC}$ is a transcendental parameter, are given. One is the mixed joint functional independence and the other is a generalised universality, which includes several periodic Hurwitz zeta functions.

The Li coefficients $\unicode[STIX]{x1D706}_{F}(n)$ of a zeta or $L$ -function $F$ provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the $\unicode[STIX]{x1D70F}$ -Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport–Heilbronn zeta function. The behavior of the $\unicode[STIX]{x1D70F}$ -Li coefficients varies depending on whether the function in question has any zeros in the half-plane $\text{Re}(z)>\unicode[STIX]{x1D70F}/2.$ We investigate analytically and numerically the behavior of these coefficients for such functions in both the $n$ and $\unicode[STIX]{x1D70F}$ aspects.

Assuming the Riemann Hypothesis, Soundararajan [Ann. of Math. (2) 170 (2009), 981–993] showed that $\int _{0}^{T}|\unicode[STIX]{x1D701}(1/2+\text{i}t)|^{2k}\ll T(\log T)^{k^{2}+\unicode[STIX]{x1D716}}$ . His method was used by Chandee [Q. J. Math.62 (2011), 545–572] to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas of Chandee and Soundararajan, we obtain, conditionally, upper bounds for shifted moments of Dirichlet $L$ -functions which allow us to derive upper bounds for moments of theta functions.

We determine a bound for the valency in a family of dihedrants of twice odd prime orders which guarantees that the Cayley graphs are Ramanujan graphs. We take two families of Cayley graphs with the underlying dihedral group of order $2p$ : one is the family of all Cayley graphs and the other is the family of normal ones. In the normal case, which is easier, we discuss the problem for a wider class of groups, the Frobenius groups. The result for the family of all Cayley graphs is similar to that for circulants: the prime $p$ is ‘exceptional’ if and only if it is represented by one of six specific quadratic polynomials.