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Let f be an algebraic valued periodic arithmetical function and L ( s , f ) , defined as L ( s , f ) : = ∑ n = 1 ∞ f ( n ) / n s for ℜ ( s ) > 1 , be the associated Dirichlet series. In this paper, we study the vanishing and arithmetic nature of the special values L ( k , f ) when k > 1 is a positive integer. We prove a generalization of the Baker-Birch-Wirsing theorem conditional on the Polylog conjecture. Adopting a new approach, we define an induction operator on the space of periodic arithmetic functions, which makes precise the notion of an “imprimitive” arithmetic function. This enables us to obtain an analog of Okada's criterion for L ( 1 , f ) = 0 and derive a natural decomposition of the vector space O k ( N ) = { f : Z → Q | f ( n + N ) = f ( n ) for all n ∈ Z , L ( k , f ) = 0 } .

Let η ( s ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n − s be the alternating zeta function. For a real number τ we define certain complex numbers b M , m ( τ ) and consider finite Dirichlet series υ M ( τ , s ) = ∑ m = 1 M b M , m ( τ ) m − s and η N ( τ , s ) = ∑ M = 1 N υ M ( τ , s ) . Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far. First, numerical data show that η N ( τ , s ) approximates η ( s ) with high accuracy for s in the vicinity of 1 / 2 + i τ ; this allows one to surmise that (*) η ( s ) = ∑ M = 1 ∞ υ M ( τ , s ) . Moreover, it looks that lim M → ∞ ⁡ m υ M ( τ , 1 − σ + i t ) ‾ m υ M ( τ , σ + i t ) = η ( σ + i t ) m η ( 1 − σ + i t ) ‾ m ; in other words, the individual summands in expected expansion ( ⁎ ) satisfy with an increasing accuracy a counterpart of the classical functional equation. Let ϒ M ( τ , σ + i t ) = υ M ( τ , σ + i t ) / η ( σ + i t ) . When M, τ, and either σ or t are fixed, and the fourth parameter varies, the plot of ϒ M ( τ , σ + i t ) on the complex plane contains numerous almost ideally circular arcs with geometrical parameters closely related to the non-trivial zeros of the zeta function.

For 0 ≤ α ≤ 1 , let L α ( x ) = ∑ n ≤ x λ ( n ) n α , where λ ( n ) is the Liouville function. Then famous criteria of Polya and Turan claim that the eventual sign constancy of each of L 0 ( x ) and L 1 ( x ) alone implies the Riemann hypothesis. However, Haselgrove disproved the eventual sign constancy hypothesis. As a remedy for this, we show that the eventual negativity of each of L α ( x ) , L α ′ ( x ) , L α ″ ( x ) for all 1 / 2 α 1 is equivalent to the Riemann hypothesis. Analogous equivalent conditions for the Riemann hypothesis are formulated in terms of partial sums of the Mobius function as well. Our criteria indicate a new tendency concerning the values of the Liouville and Mobius functions in stark contrast with their random behavior. Further generalizations are given in two essential directions, namely that equivalent criteria are developed for any quasi-Riemann hypothesis, and we allow the partial sums to be supported on semigroups after sifting out integers divisible by members of a set of prime numbers whose size is under control. Our negativity conditions for the quasi-Riemann hypothesis exhibit a curious rigidity in two respects, first it doesn't matter which particular primes we use to sift out, but only the size of the set, second the sign of the partial sums is unaltered with respect to two successive derivatives. Finally, other competing necessary or sufficient conditions for a quasi-Riemann hypothesis are studied regarding the monotonicity, zeros and sign changes of partial sums of Dirichlet series.

We introduce a new type of summation formula for central values of GL ( 4 ) × GL ( 2 ) L-functions, when varied over Maas forms. By rewriting such a sum in terms of GL ( 4 ) × GL ( 1 ) L-functions and applying a new “balanced” Voronoi formula, the sum can be shown to be equal to a differently-weighted average of the same quantities. By controlling the support of the spectral weighting functions on both sides, this reciprocity formula gives estimates on spectral sums that were previously obtainable only for lower rank groups. The “balanced” Voronoi formula has Kloosterman sums on both sides, and can be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL ( 4 ) Hecke eigenvalues. As an application we show that for any self-dual cusp form Π for SL ( 4 , Z ) , there exist infinitely many Maas forms π for SL ( 2 , Z ) such that L ( 1 / 2 , Π × π ) ≠ 0 .

We consider Dirichlet series attached to a quasimodular form, study its analytic properties, and generalize Hecke's converse theorem to quasimodular forms of any weight and depth over SL 2 ( Z ) . Then we discuss some applications of our results to a certain q-series and to sign changes of the Fourier coefficients of quasimodular forms.

We initiate the study of interpolation of the Hardy spaces of Dirichlet series. We prove that there are no variants of the classical interpolation theorems in the setting of Hardy spaces on the countably infinite polytorus T ∞ as in the case of the Hardy spaces on the finite dimensional torus. As a byproduct of our results we get a similar conclusion for the Hardy spaces of Dirichlet series by using the famous Bohr lift. Our proofs rely on a general method based on interpolation estimates for finite-dimensional Banach spaces. This method is applied to interpolation of injective tensor products and also of spaces of m-homogeneous polynomials.

We consider the space H ( c e s p ) of all Dirichlet series whose coefficients belong to the Cesaro sequence space c e s p , consisting of all complex sequences whose absolute Cesaro means are in l p , for 1 p ∞ . It is a Banach space of analytic functions, for which we study the maximal domain of analyticity and the boundedness of point evaluations. We identify the algebra of analytic multipliers on H ( c e s p ) as the Wiener algebra of Dirichlet series shifted to the vertical half-plane C 1 / q : = { s ∈ C : ℜ s > 1 / q } , where 1 / p + 1 / q = 1 .

We settle a conjecture of Cerone and Dragomir on the concavity of the reciprocal of the Riemann zeta function on ( 1 , ∞ ) . It is further shown in general that reciprocals of a family of zeta functions arising from semigroups of integers are also concave on ( 1 , ∞ ) , thereby giving a positive answer to a question posed by Cerone and Dragomir on the existence of such zeta functions. As a consequence of our approach, weighted type Mertens sums over semigroups of integers are seen to be biased in favor of square-free integers with an odd number of prime factors. To strengthen the already known log-convexity property of Dirichlet series with positive coefficients, the geometric convexity of a large class of zeta functions is obtained and this in turn leads to generalizations of certain inequalities on the values of these functions due to Alzer, Cerone and Dragomir.

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the truncated Dirichlet series. In this setting, we obtain new Hal\'asz-type results for the logarithmic mean value of $f$. More precisely, we prove estimates for the sum $\sum_{n=1}^x f(n)/n$ in terms of the size of $|F(1+1/\log x)|$ and show that these estimates are sharp. As a consequence of our mean value estimates, we establish non-trivial zero-free regions for these partial sums $F_N(s)$. In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field $K$. More precisely, we give some improved results for the number of zeros up to height $T$ as well as new zero density results for the number of zeros up to height $T$, lying to the right of $\Re(s) =\sigma$, where $\sigma > 1/2$., Comment: 27 pages

In this paper we study spaces of multiple Dirichlet series and their properties. We set the ground of the theory of multiple Dirichlet series and define the spaces H ∞ ( C + k ) , k ∈ N , of convergent and bounded multiple Dirichlet series on C + k . We give a representation for these Banach spaces and prove that they are all isometrically isomorphic, independently of the dimension. The analogous result for A ( C + k ) , k ∈ N , which are the spaces of multiple Dirichlet series that are convergent on C + k and define uniformly continuous functions, is obtained.

For a positive irrational number $\alpha,$ we study the ordinary Dirichlet series $\zeta_\alpha(s) = \sum\limits_{n\geq1} \lfloor\alpha n\rfloor^{-s}$ and $S_\alpha(s) = \sum\limits_{n\geq1} (\left\lceil\alpha n\right\rceil - \left\lceil \alpha (n-1)\right\rceil){n^{-s}}.$ We prove relations between them and $J_{\boldsymbol{\alpha}}(s)=\sum\limits_{n\geq1}\left(\lbrace\alpha n\rbrace-\frac{1}{2}\right)n^{-s}.$ Motivated by the previous work of Hardy and Littlewood, Hecke and others regarding $J_{\boldsymbol{\alpha}},$ we show that $\zeta_\alpha$ and $S_\alpha$ can be continued analytically beyond the imaginary axis except for a simple pole at $s=1.$ Based on the latter results, we also prove that the series $\zeta_{\alpha}(s;\beta)=\sum\limits_{n\geq0}\left(\lfloor\alpha n\rfloor+\beta\right)^{-s}$ can be continued analytically beyond the imaginary axis except for a simple pole at $s=1.$, Comment: 17 pages

Chains of the CME Group Time and Sales E-mini S&P 500 futures tick prices and their a-b-c-d-increments are studied. A discrete probability distribution based on the Hurwitz Zeta function and Dirichlet series is suggested for the price increments. The randomness of the ticks is discussed using the notions of typicalness, chaoticness, stochasticness introduced by Kolmogorov and Uspenskii, and developed by predecessors, them, and pupils. They define randomness in terms of the theory of algorithms.

Inspired by the work of Hedenmalm, Lindqvist and Seip, we consider different properties of dilations systems of a fixed function φ ∈ L 2 ( 0 , 1 ) . More precisely, we study when the system { φ ( n x ) } n is a Bessel sequence, a Riesz sequence, or it satisfies the lower frame bound. We are able to characterize these properties in terms of multipliers of the Hardy space H 2 of Dirichlet series and, also, in terms of Hardy spaces on the infinite polytorus. We also address the multivariate case.

I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta Function is the critical line. The methods and results of this paper are based on well-known theorems on the number of zeros for complex value functions (Jensen, Titchmarsh, Rouche theorems), with the Riemann Mapping Theorem acting as a bridge between the Unit Disk on the complex plane and the critical strip. By primarily relying on well-known theorems of complex analysis our approach makes this paper accessible to a relatively wide audience permitting a fast check of its validity. Both proofs do not use any functional equation of the Riemann Zeta Function, except leveraging its implied symmetry for non-trivial zeros on the critical strip., Subj-Class: Number Theory; submitted to the Annals of Mathematics for publication

If the hypothesis is not true, you can construct a sequence of numbers tending to zero this function, so that at these points the partial sum of this function multiplied by the number in the power of the argument reaches in the limit a predetermined any number from the extended complex plane, and the remainder of the multiplications by the number in the power argument remains uniformly bounded in any closed region inside the region of convergence of the Dirichlet series of this function.

We study when the spaces of general Dirichlet series bounded on a half plane are Banach spaces, and show that some of those classes are isometrically isomorphic between themselves. In a precise way, let { λ n } be a strictly increasing sequence of positive real numbers such that lim n → ∞ ⁡ λ n = ∞ . We denote by H ∞ ( λ n ) the complex normed space of all Dirichlet series D ( s ) = ∑ n b n λ n − s , which are convergent and bounded on the half plane [ Re s > 0 ] , endowed with the norm ‖ D ‖ ∞ = sup Re s > 0 ⁡ | D ( s ) | . If (⁎) there exists q > 0 such that inf n ⁡ ( λ n + 1 q − λ n q ) > 0 , then H ∞ ( λ n ) is a Banach space. Further, if there exists a strictly increasing sequence { r n } of positive numbers such that the sequence { log ⁡ r n } is Q -linearly independent, μ n = r α for n = p α , and { λ n } is the increasing rearrangement of the sequence { μ n } , then H ∞ ( λ n ) is isometrically isomorphic to H ∞ ( B c 0 ) . With this condition (⁎) we explain more explicitly the optimal cases of the difference among the abscissas σ c , σ b , σ u and σ a .

The Mellin transform and several Dirichlet series related with the Riemann zeta function are used to deduce some identities similar to the classical Muntz formula [4] . These formulas are derived in the critical strip and in the half-plane Re ( s ) 0 . As particular cases, integral representations for products of the gamma and zeta functions are exhibited.

In this paper, we study weighted composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The Hermitianness, Fredholmness and invertibility of such operators are characterized, and the spectra of compact and invertible weighted composition operators are also described.

The paper concerns the problem of algebraic differential independence between the gamma function and the function in a certain class F , which contains Dirichlet L -functions, L -functions in the extended Selberg class and some periodic functions. It is proved that the gamma function and the function in F cannot satisfy a class of algebraic differential equations with meromorphic coefficients ϕ having Nevanlinna characteristic satisfying T ( r , ϕ ) = o ( r ) as r → ∞ .

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that send nowhere-vanishing functions to nowhere-vanishing functions. We also investigate possible extensions to weighted Dirichlet spaces with superharmonic weights. As part of our investigation, we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane.

Let N be the linear space of functions ∑ k = 1 n a k ρ ( θ k / x ) with a condition ∑ k = 1 n a k θ k = 0 for 0 θ k ≤ 1 . Here ρ ( x ) denotes the fractional part of x. Beurling pointed out that the problem of how well a constant function can be approximated by functions in N is closely related to the zero-free region of the Riemann zeta function. More precisely, Baez-Duarte gave a zero-free region related to a L p -norm estimation of a constant function by using the Dirichlet series for the zeta function. In this paper, we consider the L ∞ -norm estimation of a constant function and give a wider zero-free region than that of the Baez-Duarte result.

It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let B denote an integral square matrix and 〈 B 〉 denote the subring of the full matrix ring generated by B. Then 〈 B 〉 is a free Z -module of finite rank, which guarantees that there are only finitely many ideals of 〈 B 〉 with given finite index. Thus, the formal Dirichlet series ζ 〈 B 〉 ( s ) = ∑ n ≥ 1 a n n − s is well-defined where a n is the number of ideals of 〈 B 〉 with index n. In this article we aim to find an explicit form of ζ 〈 B 〉 ( s ) when B has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, 〈 B 〉 is isomorphic to Z [ x ] / m ( x ) Z [ x ] where m ( x ) is the minimal polynomial of B over Q , and Z [ x ] / m ( x ) Z [ x ] is isomorphic to Z [ x ] / m ( x + γ ) Z [ x ] for each γ ∈ Z . Thus, the problem is reduced to counting the number of ideals of Z [ x ] / x ( x − α ) ( x − β ) Z [ x ] with given finite index where 0 , α and β are distinct integers.

Completely multiplicative functions whose sum is zero ($CMO$). The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors. The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem~3). The same idea applies also in a more general context (theorem~4). After general statements of that sort, including examples of $CMO$ (theorem~5), the paper is devoted to ``small'' functions, that is, functions of the form $\frac{f(n)}{n}$, where the $f(n)$ are bounded. Here is a typical result : if $|f(p)|\le 1$ and $Re\, f(p)\le0$ for all $p$, a necessary and sufficient condition for $\big(\frac{f(n)}{n}\big)$ to be $CMO$ is $\sum \, Re\, f(p)/p=-\infty$ (theorem~8). Another necessary and sufficient condition is given under the assumption that $|1+f(p)|\le 1$ and $f(2)\not=-2$ (theorem~7). A third result gives only a sufficient condition (theorem~9). The three results apply to the particular case $f(p)=-1$, the historical example of Euler. Theorems 7 and 8 need auxiliary results, coming either from the existing literature (Hal\'asz, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Ska\l ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the multiplicative convolution (theorems~10~and~11).

We give an asymptotic formula for the divisor sum $\sum_{c, Comment: 16 pages

There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a new flavour: we give the Dirichlet series representation to solution $f=f(s,w)$ of the functional equation $L(s-wf)=\exp(f)$, where $L(s)$ is the L-function corresponding to a completely multiplicative function. Our result seems to be a Dirichlet series analogue of the well known Lagrange-B\"urmann formula for power series. The proof is probabilistic in nature and is based on Kendall's identity, which arises in the fluctuation theory of L\'evy processes., Comment: 12 pages, 1 figure

In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.

On the Hilbert space of Dirichlet series with square summable coefficients, there are no non-normal complex symmetric composition operators induced by non-constant symbols. This is in sharp contrast with the phenomenon on the classical Hardy space over the unit disk.

In 1997, Y. Ohno empirically stumbled on an astoundingly simple identity relating the number of cubic rings h ( Δ ) of a given discriminant Δ, over the integers, to the number of cubic rings h ˆ ( Δ ) of discriminant − 27 Δ in which every element has trace divisible by 3: (1) h ˆ ( Δ ) = { 3 h ( Δ ) if Δ > 0 h ( Δ ) if Δ 0 , where in each case, rings are weighted by the reciprocal of their number of automorphisms. This allows the functional equations governing the analytic continuation of the Shintani zeta functions (the Dirichlet series built from the functions h and h ˆ ) to be put in self-reflective form. In 1998, J. Nakagawa verified (1) . We present a new proof of (1) that uses the main ingredients of Nakagawa's proof (binary cubic forms, recursions, and class field theory), as well as one of Bhargava's celebrated higher composition laws, while aiming to stay true to the stark elegance of the identity.

Let $f$ be a weight $k$ holomorphic cusp form of level one, and let $S_f(n)$ denote the sum of the first $n$ Fourier coefficients of $f$. In analogy with Dirichlet's divisor problem, it is conjectured that $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}$. Understanding and bounding $S_f(X)$ has been a very active area of research. The current best bound for individual $S_f(X)$ is $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{3}} (\log X)^{-0.1185}$ from Wu. Chandrasekharan and Narasimhan showed that the Classical Conjecture for $S_f(X)$ holds on average over intervals of length $X$. Jutila improved this result to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length $X^{\frac{3}{4} + \epsilon}$. Building on the results and analytic information about $\sum \lvert S_f(n) \rvert^2 n^{-(s + k - 1)}$ from our recent work, we further improve these results to show that the Classical Conjecture for $S_f(X)$ holds on average over short intervals of length $X^{\frac{2}{3}}(\log X)^{\frac{1}{6}}$., Comment: To Appear in the Journal of Number Theory

We develop sufficient analytic conditions for conservativeness of non-sectorial perturbations of symmetric Dirichlet forms which can be represented through a carr\'e du champ on a locally compact separable metric space. These form an important subclass of generalized Dirichlet forms which were introduced in \cite{St1}. In case there exists an associated strong Feller process, the analytic conditions imply conservativeness, i.e. non-explosion of the associated process in the classical probabilistic sense. As an application of our general results on locally compact separable metric state spaces, we consider a generalized Dirichlet form given on a closed or open subset of $\mathbb{R}^d$ which is given as a divergence free first order perturbation of a symmetric energy form. Then using volume growth conditions of the carr\'e du champ and the non-sectorial first order part, we derive an explicit criterion for conservativeness. We present several concrete examples which relate our results to previous ones obtained by different authors. In particular, we show that conservativeness can hold for a cubic variance if the drift is strong enough to compensate it. This work continues our previous work on transience and recurrence of generalized Dirichlet forms., Comment: Additional minor corrections

We study the degree of compactness of composition operators C φ acting on weighted Hilbert spaces of entire functions, which include (i) the space of entire Dirichlet series, (ii) the space of entire power series, and (iii) the Fock space (we must have φ ( z ) = a z + b , and it is known that C φ is compact if and only if | a | 1 ). More precisely, the sequence ( a n ) of approximation numbers of C φ is investigated: for (i), we give the exact formula for ( a n ) , while for (ii) and (iii) we give upper and lower estimates for a n , showing that a n behaves like | a | n up to a subexponential factor. In particular, C φ belongs to all Schatten classes S p , p > 0 as soon as it is compact.

Let $f$ and $g$ be weight $k$ holomorphic cusp forms and let $S_f(n)$ and $S_g(n)$ denote the sums of their first $n$ Fourier coefficients. Hafner and Ivic [HI], building on Chandrasekharan and Narasimhan [CN], proved asymptotics for $\sum_{n \leq X} \lvert S_f(n) \rvert^2$ and proved that the Classical Conjecture, that $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + \epsilon}$, holds on average over long intervals. In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series $D(s, S_f \times S_g) = \sum S_f(n)\overline{S_g(n)} n^{-(s+k-1)}$ and $D(s, S_f \times \overline{S_g}) = \sum_n S_f(n)S_g(n) n^{-(s + k - 1)}$. Using these meromorphic continuations, we prove asymptotics for the smoothed second moment sums $\sum S_f(n)\overline{S_g(n)} e^{-n/X}$, proving a smoothed generalization of [HI]. We also attain asymptotics for analogous smoothed second moment sums of normalized Fourier coefficients, proving smoothed generalizations of what would be attainable from [CN]. Our methodology extends to a wide variety of weights and levels, and comparison with [CN] indicates very general cancellation between the Rankin-Selberg $L$-function $L(s, f\times g)$ and shifted convolution sums of the coefficients of $f$ and $g$. In forthcoming works, the authors apply the results of this paper to prove the Classical Conjecture on $\lvert S_f(n) \rvert^2$ is true on short intervals, and to prove sign change results on $\{S_f(n)\}_{n \in \mathbb{N}}$., Comment: To appear in the Journal of Number Theory

A sharp multiple convolution inequality with respect to Dirichlet probability measure on the standard simplex is presented. Its discrete version in terms of the negative binomial coefficients is proved as well. The new bounds for the Dirichlet distribution and iterated convolutions are obtained as the consequences of the main result. Also some binomial, exponential, and generalized hypergeometric applications are discussed.

The article investigates the growth of multiple Dirichlet series. The lower order and the linear order of n-tuple Dirichlet series in ℂn are defined and some relations between them and the coefficients and exponents of n-tuple Dirichlet series are obtained.

We provide a converse theorem for Jacobi–Maass forms, as introduced by Pitale (2009), and give three applications. Firstly, we generalize a converse theorem for holomorphic Jacobi cusp forms due to Martin (1996) to non-cuspidal Jacobi forms. Secondly, we prove the analogue result for skew-holomorphic Jacobi forms, and thirdly, we give a new proof for the existence of an explicit isomorphism between half integral weight Maass forms in Kohnen's plus space of level 4 and Jacobi–Maass forms of index 1, first proved by Pitale (2009).

We deduce an asymptotic formula with error term for the sum $\sum_{n_1,\ldots,n_k \le x} f([n_1,\ldots, n_k])$, where $[n_1,\ldots, n_k]$ stands for the least common multiple of the positive integers $n_1,\ldots, n_k$ ($k\ge 2$) and $f$ belongs to a large class of multiplicative arithmetic functions, including, among others, the functions $f(n)=n^r$, $\varphi(n)^r$, $\sigma(n)^r$ ($r>-1$ real), where $\varphi$ is Euler's totient function and $\sigma$ is the sum-of-divisors function. The proof is by elementary arguments, using the extension of the convolution method for arithmetic functions of several variables, starting with the observation that given a multiplicative function $f$, the function of $k$ variables $f([n_1,\ldots,n_k])$ is multiplicative., Comment: 12 pages

The paper discusses the following contradictions: (1) Contradictory definitions of the Zeta function. These three definitions of the Zeta function contradict each other: the Dirichlet series, Riemann’s Zeta function, and a third definition that is conditionally convergent throughout the critical strip and divergent throughout half-plane Re(s)≤0. If the Dirichlet series definition and one of the other two definitions are both true, then there is a contradiction regarding convergence/divergence of Zeta in the critical strip. If Riemann’s Zeta function and the third definition are both true, then there is a contradiction regarding convergence/divergence of Zeta in half-plane Re(s)≤0. And if only the Dirichlet series definition is true, than all theories that falsely assume that the other definitions are true are rendered logically unsound. (2) The Hankel contour’s contradiction of the preconditions of Cauchy’s integral theorem. The derivation of the Riemann Zeta function uses both of these,but the contradiction between the two renders the Riemann Zeta function invalid. (3) Conditionally convergent series. According to the Riemann series theorem, any conditionally convergent series can be rearranged to be divergent. This contradicts the associative and commutative properties of addition. It also means that all conditionally convergent series are paradoxes, and that any argument that uses a conditionally convergent series (e.g. the third definition of the Zeta function, and the definition of the Euler-Mascheroni constant) violates the Law of Non-Contradiction (LNC) in logic, and triggers Ex Contradictione Quodlibet (ECQ), also called the "Principle of Explosion".

We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4]., 15 pages, 1 figure

We investigate the first moment of the second derivative of quadratic Dirichlet L-functions over the rational function field. We establish an asymptotic formula when the cardinality of the finite field is fixed and the genus of the hyperelliptic curves associated to a family of Dirichlet L-functions over F q ( T ) tends to infinity. As a more general result, we compute the full degree three polynomial in the asymptotic expansion of the first moment of the second derivative of this particular family of L-functions.

We introduce $\mu$-Abstract Elementary Classes ($\mu$-AECs) as a broad framework for model theory that includes complete boolean algebras and Dirichlet series, and begin to develop their classification theory. Moreover, we note that $\mu$-AECs correspond precisely to accessible categories in which all morphisms are monomorphisms, and begin the process of reconciling these divergent perspectives: not least, the preliminary classification-theoretic results for {\mu}-AECs transfer directly to accessible categories with monomorphisms., Comment: 26 pp

The author developed a new method for obtaining formal series solutions to polynomial-like iterative functional equations of the form ? n = 1 N a n f n ( x ) = g ( x ) , where a n ? R , n = 1 , 2 , ? , N , f n is the n-th iterate of an unknown function f and where g ( x ) is a promptered exponential series, namely, the sum of a Dirichlet series and a linear term called prompter. In this method, a formal composition f 1 ? f 2 of two promptered exponential series f 1 and f 2 , where the coefficient of the prompter of f 2 is positive, plays a crucial role. We also solve the equation above where g ( x ) is a promptered trigonometric series.

We define a new kind of divisor function D ( 1 ) ( n ) by the nth coefficient of the Dirichlet series ( ζ ′ ( s ) ) 2 and denote by Δ ( 1 ) ( x ) the error term in the asymptotic formula for ∑ n ≤ x D ( 1 ) ( n ) . Then we compute the first moment of Δ ( 1 ) ( x ) that is ∫ 1 X Δ ( 1 ) ( x ) d x and consider ‘discrete mean values’ ∑ n ≤ x Δ ( 1 ) k ( n ) ( k = 1 , 2 ) and deduce asymptotic formulas which are analogous to the results obtained by Voronoi, Hardy and Furuya.

Counting integral binary quadratic forms with certain restrictions is a classical problem. In this paper, we count binary quadratic forms of fixed discriminant given restrictions on the size of their coefficients. We accomplish this by investigating the analytic properties of a certain double Dirichlet series, which is a shifted convolution sum of certain classical automorphic forms.

Let π be an irreducible cuspidal automorphic representation of GL ( r , A ) where r ≥ 2 and A is the adele ring of Q . Let a π ( n ) denote the n th Dirichlet series coefficient of the L-function associated to π . The main goal of this paper is to obtain strong bounds for the first moment ∑ n ≤ x a π ( n ) as x → ∞ . The bounds we obtain are better than all previously obtained bounds for the higher rank situation when r ≥ 3 and π is not a symmetric power of a GL ( 2 , A ) cuspidal automorphic representation.

We give character analogues of a generalization of a result due to Ramanujan, Hardy and Littlewood, and provide Riesz-type criteria for Riemann Hypotheses for the Riemann zeta function and Dirichlet L -functions. We also provide analogues of the general theta transformation formula and of recent generalizations of the transformation formulas of W.L. Ferrar and G.H. Hardy for real primitive Dirichlet characters.

This note deals with the relationship between the abscissas of simple, uniform and absolute convergence for the Dirichlet series $f(s) = \sum_{n=1}^\infty a_n n^{-s}$, when the coefficients $a_n$ are either multiplicative or completely multiplicative., Comment: Minor revision. Two new references added. This note has been accepted for publication in Expositiones Mathematicae

The analytic continuation of a family of Dirichlet series whose coefficients are partition functions having parts in a finite set is established. The singularities arising from this continuation are shown to be finitely many simple poles, where the residues are computable positive rational numbers. Then we give several applications. First, special values of the continuation are studied in terms of algebraic combinations of the Hurwitz zeta function. Second, a transformation formula is obtained for exponentially decaying series involving these partition functions using special values of the continuation at negative integers. Finally, asymptotic behavior and orthogonality of certain convolutions with the normalization of these partition functions are investigated in the mean value sense with precise error terms.

For any a ∈ ( 0 , ∞ ) , we prove the strict concavity of the function η a ( t ) : = ∑ m = 0 ∞ ( − 1 ) m ( a m + 1 ) t on ( 0 , ∞ ) , and provide fast computations of their derivatives on ( 0 , ∞ ) . We give short proofs mainly based on differentiation formulas concerning the gamma process. In particular, our results apply to Dirichlet's eta and beta functions η 1 ( t ) and η 2 ( t ) , respectively.

We are concerned here with Dirichlet series f ( s ) = 1 + ∑ n = 2 ∞ c ( n ) n s which satisfy a function equation similar to that of the Riemann zeta function, typically of the form f ( s ) = 2 s q 1 / 2 − s π s − 1 Γ ( 1 − s ) ( sin π 2 ( s + κ ) ) f ( 1 − s ) , but for which the Riemann hypothesis is false. Indeed we show that the zeros of such functions are ubiquitous in the complex plane.