In this series of papers, we explore moments of derivatives of L-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials P in 𝔽q[T]. When the cardinality q of the ground field is fixed and the degree of P gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of L-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.

We establish finite analogues of the identities known as the Aoki–Ohno relation and the Le–Murakami relation in the theory of multiple zeta values. We use an explicit form of a generating series given by Aoki and Ohno.

The standard twist $F(s,\unicode[STIX]{x1D6FC})$ of $L$-functions $F(s)$ in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where $F(s)$ satisfies a functional equation with the same $\unicode[STIX]{x1D6E4}$-factor of the $L$-functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such $L$-functions. We show that the standard twist $F(s,\unicode[STIX]{x1D6FC})$ satisfies a functional equation reflecting $s$ to $1-s$, whose shape is not far from a Riemann-type functional equation of degree 2 and may be regarded as a degree 2 analog of the Hurwitz–Lerch functional equation. We also deduce some results on the growth on vertical strips and on the distribution of zeros of $F(s,\unicode[STIX]{x1D6FC})$.

We give bounds on the error in the asymptotic approximation of the log-Gamma function $\ln \unicode[STIX]{x1D6E4}(z)$ for complex $z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer [Theorie der analytischen Funktionen einer komplexen Veränderlichen, 2nd edn (Springer, Berlin, 1962)], Spira [‘Calculation of the Gamma function by Stirling’s formula’, Math. Comp.25 (1971), 317–322], and Hare [‘Computing the principal branch of log-Gamma’, J. Algorithms25 (1997), 221–236]. We show that $|R_{k+1}(z)/T_{k}(z)|<\sqrt{\unicode[STIX]{x1D70B}k}$ for nonzero $z$ in the right half-plane, where $T_{k}(z)$ is the $k$th term in the asymptotic series, and $R_{k+1}(z)$ is the error incurred in truncating the series after $k$ terms. We deduce similar bounds for asymptotic approximation of the Riemann–Siegel theta function $\unicode[STIX]{x1D717}(t)$. We show that the accuracy of a well-known approximation to $\unicode[STIX]{x1D717}(t)$ can be improved by including an exponentially small term in the approximation. This improves the attainable accuracy for real $t>0$ from $O(\exp (-\unicode[STIX]{x1D70B}t))$ to $O(\exp (-2\unicode[STIX]{x1D70B}t))$. We discuss a similar example due to Olver [‘Error bounds for asymptotic expansions, with an application to cylinder functions of large argument’, in: Asymptotic Solutions of Differential Equations and Their Applications (ed. C. H. Wilcox) (Wiley, New York, 1964), 16–18], and a connection with the Stokes phenomenon.

We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.

Halász’s theorem gives an upper bound for the mean value of a multiplicative function $f$. The bound is sharp for general such $f$, and, in particular, it implies that a multiplicative function with $|f(n)|\leqslant 1$ has either mean value $0$, or is ‘close to’ $n^{it}$ for some fixed $t$. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to treat short intervals and arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel’s theorem), and that there are always primes near to the start of an arithmetic progression (Linnik’s theorem).

We finalize the analysis of the trace formula initiated in S. A. Altuğ [Beyond endoscopy via the trace formula-I: Poisson summation and isolation of special representations, Compos. Math.151(10) (2015), 1791–1820] and developed in S. A. Altuğ [Beyond endoscopy via the trace formula-II: asymptotic expansions of Fourier transforms and bounds toward the Ramanujan conjecture. Submitted, preprint, 2015, Available at: arXiv:1506.08911.pdf], and calculate the asymptotic expansion of the beyond endoscopic averages for the standard $L$-functions attached to weight $k\geqslant 3$ cusp forms on $\mathit{GL}(2)$ (cf. Theorem 1.1). This, in particular, constitutes the first example of beyond endoscopy executed via the Arthur–Selberg trace formula, as originally proposed in R. P. Langlands [Beyond endoscopy, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 611–698 (The Johns Hopkins University Press, Baltimore, MD, 2004), chapter 22]. As an application we also give a new proof of the analytic continuation of the $L$-function attached to Ramanujan’s $\unicode[STIX]{x1D6E5}$-function.

We study the values of finite multiple harmonic $q$-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetric multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we prove the duality formula for these values, as an example of linear relations, which induce those among FMZVs and SMZVs simultaneously. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic $q$-series at a primitive root of unity of sufficiently large degree.

A comprehensive study of the generalized Lambert series $\sum _{n=1}^{\infty }\frac{n^{N-2h}\text{exp}(-an^{N}x)}{1-\text{exp}(-n^{N}x)},0<a\leqslant 1,~x>0$, $N\in \mathbb{N}$ and $h\in \mathbb{Z}$, is undertaken. Several new transformations of this series are derived using a deep result on Raabe’s cosine transform that we obtain here. Three of these transformations lead to two-parameter generalizations of Ramanujan’s famous formula for $\unicode[STIX]{x1D701}(2m+1)$ for $m>0$, the transformation formula for the logarithm of the Dedekind eta function and Wigert’s formula for $\unicode[STIX]{x1D701}(1/N),N$ even. Numerous important special cases of our transformations are derived, for example, a result generalizing the modular relation between the Eisenstein series $E_{2}(z)$ and $E_{2}(-1/z)$. An identity relating $\unicode[STIX]{x1D701}(2N+1),\unicode[STIX]{x1D701}(4N+1),\ldots ,\unicode[STIX]{x1D701}(2Nm+1)$ is obtained for $N$ odd and $m\in \mathbb{N}$. In particular, this gives a beautiful relation between $\unicode[STIX]{x1D701}(3),\unicode[STIX]{x1D701}(5),\unicode[STIX]{x1D701}(7),\unicode[STIX]{x1D701}(9)$ and $\unicode[STIX]{x1D701}(11)$. New results involving infinite series of hyperbolic functions with $n^{2}$ in their arguments, which are analogous to those of Ramanujan and Klusch, are obtained.

We prove the reciprocity law for the twisted second moments of Dirichlet $L$-functions over rational function fields, corresponding to two irreducible polynomials. This formula is the analogue of the formulas for Dirichlet $L$-functions over $\mathbb{Q}$ obtained by Conrey [‘The mean-square of Dirichlet $L$-functions’, arXiv:0708.2699 [math.NT] (2007)] and Young [‘The reciprocity law for the twisted second moment of Dirichlet $L$-functions’, Forum Math. 23(6) (2011), 1323–1337].

Assuming a conjecture on distinct zeros of Dirichlet $L$-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the location of zeros of $L$-functions. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.

We construct a random model to study the distribution of class numbers in special families of real quadratic fields ${\open Q}(\sqrt d )$ arising from continued fractions. These families are obtained by considering continued fraction expansions of the form $\sqrt {D(n)} = [f(n),\overline {u_1,u_2, \ldots ,u_{s-1} ,2f(n)]} $ with fixed coefficients u1, …, us−1 and generalize well-known families such as Chowla's 4n2 + 1, for which analogous results were recently proved by Dahl and Lamzouri [‘The distribution of class numbers in a special family of real quadratic fields’, Trans. Amer. Math. Soc. (2018), 6331–6356].

Putnam [‘On the non-periodicity of the zeros of the Riemann zeta-function’, Amer. J. Math.76 (1954), 97–99] proved that the sequence of consecutive positive zeros of $\unicode[STIX]{x1D701}(\frac{1}{2}+it)$ does not contain any infinite arithmetic progression. We extend this result to a certain class of zeta functions.

We prove new integral representations of the approximation forms in zeta values.

In this paper we follow the approach of Bertrand–Beukers (and of Bertrand’s later work), based on differential Galois theory, to prove a very general version of Shidlovsky’s lemma that applies to Padé-approximation problems at several points, both at functional and numerical levels (that is, before and after evaluating at a specific point). This allows us to obtain a new proof of the Ball–Rivoal theorem on irrationality of infinitely many values of the Riemann zeta function at odd integers, inspired by the proof of the Siegel–Shidlovsky theorem on values of $E$-functions: Shidlovsky’s lemma is used to replace Nesterenko’s linear independence criterion with Siegel’s, so that no lower bound is needed on the linear forms in zeta values. The same strategy provides a new proof, and a refinement, of Nishimoto’s theorem on values of $L$-functions of Dirichlet characters.

This paper [1], which was published online on 1 June 2011, has been retracted by agreement between the authors, the journal’s Editor-in-Chief Derek Holt, the London Mathematical Society and Cambridge University Press. The retraction was agreed to prevent other authors from using incorrect mathematical results. (In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no. 1, 50–58; J. Number Theory 130 (2010) no. 4, 1109–1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.)

Subconvexity bounds on the critical line are proved for general Epstein zeta-functions of $k$-ary quadratic forms. This is related to sup-norm bounds for unitary Eisenstein series on $\text{GL}(k)$ associated with the maximal parabolic of type $(k-1,1)$, and the exact sup-norm exponent is determined to be $(k-2)/8$ for $k\geqslant 4$. In particular, if $k$ is odd, this exponent is not in $\frac{1}{4}\mathbb{Z}$, which is relevant in the context of Sarnak’s purity conjecture and shows that it can in general not directly be generalized to Eisenstein series.

In [31] we defined a regularized analytic torsion for quotients of the symmetric space $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$ by arithmetic lattices. In this paper we study the limiting behavior of the analytic torsion as the lattices run through sequences of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for principal congruence subgroups and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the $L^{2}$-analytic torsion.

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.