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 give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula of the latter is then proved for quadratic extensions in the lowest order case. We are also able to reproduce our moments conjecture in the case of quadratic extensions by using a modified version of the moments recipe of Conrey et al. Generalising our methods, we then provide a conjecture for moments of non-primitive L-functions, which is supported by some calculations based on Selberg’s conjectures.

One of the approaches to the Riemann Hypothesis is the Nyman–Beurling criterion. Cotangent sums play a significant role in this criterion. Here we investigate the values of these cotangent sums for various shifts of the argument.

In this paper we prove some one-level density results for the low-lying zeros of families of quadratic and quartic Hecke $L$-functions of the Gaussian field. As corollaries, we deduce that at least 94.27% and 5%, respectively, of the members of the quadratic family and the quartic family do not vanish at the central point.

We study various families of Artin $L$-functions attached to geometric parametrizations of number fields. In each case we find the Sato–Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros.

We prove a new linear relation for multiple zeta values. This is a natural generalisation of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.

Four classes of multiple series, which can be considered as multifold counterparts of four classical summation formulae related to the Riemann zeta series, are evaluated in closed form.

We prove that the complete $L$-function associated to any cuspidal automorphic representation of $\operatorname{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$ has infinitely many simple zeros.

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\unicode[STIX]{x1D700})(\log s)/(\text{log}\log s)}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\unicode[STIX]{x1D700}$ is any positive real number and $s$ is large enough in terms of $\unicode[STIX]{x1D700}$. This lower bound is asymptotically larger than any power of $\log s$; it improves on the bound $(1-\unicode[STIX]{x1D700})(\log s)/(1+\log 2)$ that follows from the Ball–Rivoal theorem. The proof is based on construction of several linear forms in odd zeta values with related coefficients.

It is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2m + 1) and ζ(2Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2m + 1 − 1/N). Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function η(z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ.

By combining classical techniques together with two novel asymptotic identities derived in recent work by Lenells and one of the authors, we analyse certain single sums of Riemann-zeta type. In addition, we analyse Euler-Zagier double exponential sums for particular values of Re{u} and Re{v} and for a variety of sets of summation, as well as particular cases of Mordell-Tornheim double sums.

We prove an upper bound for the fifth moment of Hecke L-functions associated to holomorphic Hecke cusp forms of full level and weight k in a dyadic interval K ≤ k ≤2K, as K → ∞. The bound is sharp on Selberg’s eigenvalue conjecture.

Let $\unicode[STIX]{x1D701}(s)$ be the Riemann zeta function. In 1929, Hardy and Littlewood proved the approximate functional equation for $\unicode[STIX]{x1D701}^{2}(s)$ with error term $O(x^{1/2-\unicode[STIX]{x1D70E}}((x+y)/|t|)^{1/4}\log |t|)$, where $-1/2<\unicode[STIX]{x1D70E}<3/2,x,y\geqslant 1,xy=(|t|/2\unicode[STIX]{x1D70B})^{2}$. Later, in 1938, Titchmarsh improved the error term by removing the factor $((x+y)/|t|)^{1/4}$. In 1999, Hall showed the approximate functional equations for $\unicode[STIX]{x1D701}^{\prime }(s)^{2},\unicode[STIX]{x1D701}(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$, and $\unicode[STIX]{x1D701}^{\prime }(s)\unicode[STIX]{x1D701}^{\prime \prime }(s)$ (in the range $0<\unicode[STIX]{x1D70E}<1$) whose error terms contain the factor $((x+y)/|t|)^{1/4}$. In this paper we remove this factor from these three error terms by using the method of Titchmarsh.

We show that if the zeros of an automorphic $L$-function are weighted by the central value of the $L$-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the $L$-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted $L$-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct $p$-adic $L$-functions for non-critical slope rational modular forms, the theory has been extended to construct $p$-adic $L$-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the $L$-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the $p$-adic $L$-function of the eigenform to be this distribution.

Katz and Sarnak predicted that the one level density of the zeros of a family of L-functions would fall into one of five categories. In this paper, we show that the one level density for L-functions attached to cubic Galois number fields falls into the category associated with unitary matrices.

We generalize a method of Conrey and Ghosh [Simple zeros of the Ramanujan $\unicode[STIX]{x1D70F}$-Dirichlet series. Invent. Math. 94(2) (1988), 403–419] to prove quantitative estimates for simple zeros of modular form $L$-functions of arbitrary conductor.

We aim to re-prove a theorem conjectured by Gauss, namely there are exactly nine imaginary quadratic fields $\mathbf{Q}(\sqrt{-q})$ with class number one: specifically the list is $q\in \{3,4,7,8,11,19,43,67,163\}$. Our method initially follows an idea of Goldfeld, but rather than using an elliptic curve of analytic rank three (provided by the Gross–Zagier theorem), we instead use an elliptic curve of analytic rank two, where this $L$-function vanishing can be proven by modular symbols rather than a difficult height formula. It is already clear that Goldfeld’s work yields a constant lower bound for the class number by such means, but unfortunately it seems that even for the best choice of elliptic curve this numerical constant is less than 1, unless one can show non-trivial cancellation in the $L$-function coefficients restricted to values taken by quadratic forms. To show the latter, we consider a specific analytic rank-two elliptic curve with complex multiplication by $\mathbf{Q}(\sqrt{-1})$, and then by adapting a result of Hooley’s regarding equi-distrbution of roots of a quadratic polynomial to varying moduli, are able to show that there is indeed sufficient coefficient cancellation, giving an effective resolution of class number one. As we use various aspects of the principal form, our proof seems inapplicable for larger class numbers. We also comment on the possibility of using spectral techniques (following Templier and Tsimerman) to show the desired coefficient cancellation, though postpone the details of this to elsewhere.