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We explain the exact meaning of a statement we made in a previous paper on invariants, namely that a complex-valued function of the data of the functional equation of an L-function is an invariant if and only if it is stable under the multiplication and factorial formulae. To this end, we show that every invariant has a so-called rational extension, having some desired invariance properties. The existence of such an extension, which enables to express formally the above heuristic concept, is not obvious and its construction is the main novelty of the paper. Mathematics Subject Classification (2010): 11M41

We prove an explicit formula, analogous to the classical explicit formula for $\psi(x)$, for the Ces\`aro-Riesz mean of any order $k>0$ of the number of representations of $n$ as a sum of two primes. Our approach is based on a double Mellin transform and the analytic continuation of certain functions arising therein., Comment: 16 pages

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 $F(s,\unicode[STIX]{x1D6FC})$.

A nonlinear twist F(s; f) of a function F(s) from the extended Selberg class S# is called internal if it belongs to S#. In a previous paper (2014) we showed that, inside a rather general class of nonlinear twists, the internal twists occur only in very special cases; moreover, we gave a first characterization of such twists. Here we complete our previous work by giving a fully detailed description of such internal twists.

Let [Formula: see text] be a Dirichlet character. The main goal of this paper is to study oscillations of the difference [Formula: see text] where [Formula: see text] denotes the twisted Dedekind function. We prove that for infinitely many odd characters [Formula: see text] called “good”, we have [Formula: see text], and [Formula: see text] when [Formula: see text] is real. We give a necessary and sufficient condition for [Formula: see text] to be good, and in particular we prove that all odd primitive characters are good. We show also that there are infinitely many moduli [Formula: see text], including all prime powers [Formula: see text], for which all odd characters [Formula: see text] are good.

The main goal of this paper is to initiate study of analytic monoids as a general framework for quantitative theory of factorization. So far the latter subject was developed either in concrete settings, for instance in orders of number fields, or abstractly, in an axiomatic way. Some of the abstract approaches are too general to address delicate problems concerning oscillatory nature of the related counting functions, or are too restrictive in the sense that they suffer from the lack of examples except classical ones i.e. the Hilbert monoids of algebraic integers and their products. The notion of an analytic monoid is enough flexible to allow constructions of many other examples, and also ensures sufficiently rich analytic structure. In particular, we construct examples of such monoids with the associated L-functions being products of classical Dirichlet L-functions and L-functions of twisted irreducible unitary cuspidal automorphic representations of $$GL_d({\mathbb {A}}_{\mathbb {Q}})$$ satisfying the Ramanujan conjecture and having real coefficients. Finally, to illustrate how a typical problem from the quantitative theory of factorization can be studied in the framework of analytic monoids, we formulate several results concerning oscillations of the remainder term in the asymptotic formula for the number of irreducible elements with norms less or equal x, as x tends to infinity.

We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\leq 1/d$ of the L-functions of any degree $d \geq 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases., 34 pages

Bezpieczeństwo asymetrycznych systemów kryptologicznych opiera się na założeniu, że istnieją funkcje jednokierunkowe. Fakt ten nie został do tej pory ści- śle udowodniony. Nie mniej jednak pewne trudne obliczeniowo problemy teorii liczb, takie jak na przykład problem faktoryzacji, czy też problem obliczania logarytmu dyskretnego w skończonych grupach abelowych, mogą być podstawą konstrukcji funkcji uważanych za jednkierunkowe. Idea wykorzystania w tym konstekście funkcji typu L (elementów klasy Selberga) pojawiła się po raz pierwszy w pracy M. Anshela i D. Goldfelda z 1997 roku. Ich przydatność ilustrujemy na przykladzie protokołu uwierzytelnienia przy użyciu współczynników Dirichleta funkcji L oraz eliptycznego generatora pseudolosowego. Na zakończenie przedstawiamy propozycję innego typu, a mianowicie opis protokołu rzutu monetą przez telefon opartego na wykorzystaniu nietrywialnych zer funkcji L.

We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L -functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L -functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L -functions and shifted L -functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new.

Let q be an odd integer, let τ be the order of 2 modulo q , and let a be coprime with q . Finally, let s ( a / q ) : = ∑ r = 1 τ e ( a 2 r / q ) . We prove that | s ( a / q ) | can be as large as τ − c ′ for a suitable constant c ′ for infinitely many q , but that max ( a , q ) = 1 | s ( a / q ) | is always bounded from above by τ − c for a suitable positive constant c whose value is considerably larger that any previously known. An upper bound for min ( a , q ) = 1 | s ( a / q ) | is also discussed.

We study an asymptotic formula for the sum of values of the Euler φ-function twisted by a real Dirichlet character. The error term is split into the arithmetic and the analytic part. The former is studied with minimal use of analytic tools in contrast to the latter, where the analysis depends heavily on the distribution of the non-trivial zeros of the corresponding Dirichlet L-function. The results of the present paper are an extension of a recent work by the authors, where the case of the classical Euler φ-function has been studied. The present, more general situation invites new technical difficulties. Not all of them can be successfully overcome. For instance, satisfactory omega results for the analytic part are proved in the case of an even Dirichlet character only. Nevertheless, a method providing good omega estimates for the arithmetic part as well as for the complete error term is developed. Moreover, it is noted that the Riemann Hypothesis for the involved Dirichlet L-function is equivalent to a sufficiently sharp estimation of the analytic part. This shows in particular that the arithmetic part can be much larger than the corresponding analytic part.

We split the remainder term in the asymptotic formula for the mean of the Euler phi function into two summands called the arithmetic and the analytic part respectively. We show that the arithmetic part can be studied with a mild use of the complex analytic tools, whereas the study of the analytic part heavily depends on the properties of the Riemann zeta function and on the distribution of its non-trivial zeros in particular.

In 1983, S. M. Voronin obtain an analytic continuation to the entire complex plane for some twists of L-functions associated with holomorphic modular forms on SL(2, Z). In this paper, we extend Voronin’s result to L-functions of degree 2 from the extended Selberg class.

We study large values of the remainder term EK (x) in the asymptotic formula for the number of irreducible integers in an algebraic number field K. We show that EK (x) = Ω± (√(x)(log x)) for certain positive constant BK, improving in that way the previously best known estimate EK (x) = Ω± (x(1/2)-e) for every e > 0, due to A. Perelli and the present author. Assuming that no entire L-function from the Selberg class vanishes on the vertical line σ = 1, we show that EK (x) = Ω± (√(x)(log log x)D (K)-1(log x)-1), supporting a conjecture raised recently by the author. In particular, it follows that the last omega estimate is a consequence of the Selberg Orthonormality Conjecture (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In many cases known methods of detecting oscillations of arithmetic error terms involve certain smoothing pro-cedures. Usually an application of the smoothing operator does not change significantly the order of magnitude of the error under consideration. This is so for instance in the case of the classical error terms known in the prime number theory. The main purpose of this paper is to show that the situation for primes is not general. Considering the error term in the asymptotic formula for the Euler totient function we show that just one application of an integral smoothing operator changes situation dramatically: the order of magnitude of drops from x to √x (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

It is known that the space of solutions (in a suitable class of Dirichlet series with continuation over C) of the functional equation of a DirichletL-functionL.s; / has dimension 2 as soon as the conductor q of is at least 4. Hence the Dirichlet L-functions are not characterized by their functional equation for q 4. Here we characterize the conductors q such that for every primitive character (mod q), L.s; / is the only solution with an Euler product in the above space. It turns out that such conductors are of the form q D 23m with any square-free m coprime to 6 and finitely many a and b. Mathematics Subject Classification (2000). 11M06, 11M41.

A general method of estimating the number of oscillations of a given size of arithmetic error terms is developed. Special attention is paid to the remainder terms in the prime number formula, in the Dirichlet prime number theorem for primes in arithmetic progressions and to the remainder term in the asymptotic formula for the number of square free divisors of an integer.

We study distribution of irreducible algebraic integers in short intervals and prove that if the class number of an algebraic number field K exceeds 2, every interval of the form (x, x + x θ) with a fixed θ > 1/2 contains absolute value of the norm of an irreducible algebraic integer from K provided x ≥ x 0. The constant x 0 effectively depends on K and θ.

We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this is always the case when h = 2.

The degree conjecture for the Selberg class Open image in new window of L-functions states that the degree dF of every F ∈ Open image in new window is an integer. Moreover, it is expected that every F ∈ Open image in new window has polynomial Euler product, and that the degree ∂F of such an Euler product coincides with dF. In this note we prove that a suitable continuity assumption on the degree dF implies that ∂F = dF for all F ∈ Open image in new window with polynomial Euler product.

In this paper we prove the unique factorization of certain subsemigroups of the extended Selberg class of L-functions. These semigroups are generated by the functions of degree 0 and 1. We also consider extensions of such results and applications to classical L-functions.

The main aim of this paper is to present a general method of proving Ω-estimates for a class of arithmetic error terms. We assume that error terms in question are boundary values of harmonic functions on the upper half-plane satisfying certain subsidiary conditions. We prove a general theorem for an axiomatically defined class of such functions and then we show how this result can be used to give statements in concrete situations. As examples we treat the classical case of the remainder term in the prime number formula obtaining a new proof of the well-known result of J. E. Littlewood, and the case of the remainder term in the asymptotic formula for the summatory function of the square-free divisor function. In the latter case our result is new.

First we show that the abscissae of uniform and absolute convergence of Dirichlet series coincide in the case of $L$-functions from the Selberg class $\mathcal{S}$. We also study the latter abscissa inside the extended Selberg class, indicating a different behavior in the two classes. Next we address two questions about majorants of functions in $\mathcal{S}$, showing links with the distribution of the zeros and with independence results., 10 pages

We continue our investigations of the analytic properties of nonlinear twists of L-functions developed in [4],[5] and [7]. Given an L-function of degree d, we first extend the transformation formula in [5], relating a twist with leading exponent > 1/d to its dual twist. Then we combine the results in [7] with such a transformation formula to obtain the analytic properties of new classes of nonlinear twists. This allows to detect several new cases of resonance of the classical L-functions., 19 pages

We show that certain twists of Bessel type of a given L-function have meromorphic continuation over \(\mathbb{C}\).

The principal problem considered in this paper can be formulated as follows: given an L-function satisfying the Riemann Hypothesis or at least a non-trivial density estimate, is it true that it has an Euler product expansion? A positive answer would mean that arithmetic is necessary for proving the Riemann Hypothesis or a non-trivial density estimate, respectively. The paper contains a solution for degree one L-functions from the extended Selberg class. The main tools in the proof are: explicit description of the structure of the extended Selberg class in degree one due to A. Perelli and the first named author (see [7]) and a “hybrid” type joint universality theorem for Dirichlet L-functions. The latter result seems to be of an independent interest, embodying in one statement a special case of the classical Kronecker-Weyl theorem on diophantine approximations and Voronin’s joint universality theorem for Dirichlet L-functions.

Shifts of universal Dirichlet series are jointly universal, that means, roughly speaking, they approximate simultaneously any given family of non-vanishing analytic functions. In this note we study consequences of this remarkable property on the value-distribution.

Let S denote the Selberg class of L -functions. We prove the strong multiplicity one property for the subclass of functions F∈ S with polynomial Euler product.

The Selberg class S is a rather general class of Dirichlet series with functional equation and Euler product and can be regarded as an axiomatic model for the global L-functions arising from number theory and automorphic representations. One of the main problems of the Selberg class theory is to classify the elements of S. Such a classication is based on a real-valued invariant d called degree, and the degree conjecture asserts that d 2 N for every L-function in S. The degree conjecture has been proved for d < 5=3, and in this paper we extend its validity to d < 2. The proof requires several new ingredients, in particular a rather precise description of the properties of certain nonlinear twists associated with the L-functions in S.

A uniform version of Stirling's formula, suitable for future applications, is obtained.