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Let f be a holomorphic Hecke eigenform of S L 2 (Z) with weight k. Denote by L(s, f) the automorphic L-function attached to f, and S (t , f) = 1 π arg L (1 2 + i t , f) , where the argument is obtained by continuous variation along the straight line { s ∈ C | ℜ s ≥ 1 2 , ℑ s = t } , starting with the value 0 at infinity. Here, a new zero density estimate of L(s, f) in short intervals is achieved. As an application, the integral moment of S(t, f) is obtained, i.e., for l ∈ Z + and sufficiently large T, ∫ T T + H | S (t , f) | 2 l d t = (2 l) ! l ! (2 π) 2 l H (log log T) l + O (H (log log T) l - 1 2 ) holds for T 15 16 + ε ≤ H ≤ T , which improves the previous result. [ABSTRACT FROM AUTHOR]

Let Φ 1 , Φ 2 be Maass forms for S L (2 , Z) with Fourier coefficients C 1 (n) , C 2 (n) . For a positive integer h the meromorphic continuation and growth in s ∈ C (away from poles) of the shifted convolution L-function L h (s , Φ 1 , Φ 2) : = ∑ n ≠ 0 , - h C 1 (n) C 2 (n + h) · | n (n + h) | - 1 2 s is obtained. For Re (s) > 0 it is shown that the only poles are possible simple poles at 1 2 ± i r k , where 1 4 + r k 2 are eigenvalues of the Laplacian. As an application we obtain, for T → ∞ , the asymptotic formula ∑ | n (n + h) | < T n ≠ 0 , - h C 1 (n) C 2 (n + h) l o g ( T | n (n + h) | ) 3 2 + ε = f r 1 , r 2 , h , ε (T) · T 1 2 + O h 1 - ε T ε + h 1 + ε T - 2 - 2 ε , where the function f r 1 , r 2 , h , ε (T) is given as an explicit spectral sum that satisfies the bound f r 1 , r 2 , h , ε (T) ≪ h θ + ε . We also obtain a sharp bound for the above shifted convolution sum with sharp cutoff, i.e., without the smoothing weight log (∗) 3 2 + ε with uniformity in the h aspect. Specifically, we show that for h < x 1 2 - ε , ∑ | n (n + h) | < x C 1 (n) C 2 (n + h) ≪ h 2 3 θ + ε x 2 3 (1 + θ) + ε + h 1 2 + ε x 1 2 + 2 θ + ε. [ABSTRACT FROM AUTHOR]

Let F be a holomorphic cuspidal Hecke eigenform for of weight k that is a Saito–Kurokawa lift. Assuming the Generalized Riemann Hypothesis (GRH), we prove that the mass of F equidistributes on the Siegel modular variety as k⟶∞. As a corollary, we show under GRH that the zero divisors of Saito–Kurokawa lifts equidistribute as their weights tend to infinity. [ABSTRACT FROM AUTHOR]

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in Λ ⊂ N ( gcd (Λ) = 1 ) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if Λ is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree n of so (5) . We also study the Witten zeta function ζ so (5) , which is of independent interest. [ABSTRACT FROM AUTHOR]

We explore the connections between the Complex Dimensions and the cohomological properties of a fractal object. In the case of the Weierstrass Curve, we define the corresponding fractal cohomology and show that the functions belonging to the cohomology groups associated to the Curve (and its prefractal approximations) are obtained, by induction, as (finite or countably infinite) sums indexed by the underlying Complex Dimensions. The functions that constitute the cohomology groups also satisfy the same discrete local Hölder conditions as the Weierstrass function W itself. More precisely, for any natural integer m, the m th cohomology group is obtained as the set of continuous functions f on the Curve such that, for any point M located in the m th finite graph of the prefractal approximation to the Curve, f(M) has an expansion which might be interpreted as a generalized Taylor expansion, with fractional derivatives of orders the underlying Complex Dimensions. Those expansions are then compared to the fractal expansion of the function which is the most naturally defined on the Curve, namely, the Weierstrass function W itself. An important new result comes from the fact that, contrary to fractal tube formulas, which are obtained for small values of a positive parameter ε , the aforementioned fractal expansions are only valid for the values of the (multi-scales) cohomology infinitesimal ε associated to the scaling relationship obeyed by the Weierstrass Curve. As a consequence, it makes it possible to express, in a very precise way, the relations satisfied by the functions which belong to the cohomology groups, as well as to completely characterize the elements of these groups. We also obtain a suitable counterpart of Poincaré Duality in this context. Our results shed new light on the theory and the interpretation of Complex Fractal Dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we show that any Dirichlet series ∑ n = 1 ∞ f (n) n s , where s = σ + i t ∈ C , | f (n) | = 1 for all n ≥ 1 , and f (1) = 1 , has the property that a n n > - log 2 for n > 1 , where a n = inf ℜ s : S n (s) = 0 and S n (s) : = ∑ k = 1 n f (k) k s . Furthermore, if f is completely multiplicative, then - log 2 = lim n → ∞ a n n . [ABSTRACT FROM AUTHOR]

We present a new random approximation method that yields the existence of a discrete Beurling prime system P = { p 1 , p 2 , ⋯ } which is very close in a certain precise sense to a given non-decreasing, right-continuous, nonnegative, and unbounded function F. This discretization procedure improves an earlier discrete random approximation method due to Diamond et al. (Math Ann 334:1–36, 2006), and refined by Zhang (Math Ann 337:671–704, 2007). We obtain several applications. Our new method is applied to a question posed by Balazard concerning Dirichlet series with a unique zero in their half plane of convergence, to construct examples of very well-behaved generalized number systems that solve a recent open question raised by Hilberdink and Neamah (Int J Number Theory 16(05):1005–1011, 2020), and to improve the main result from (Adv Math 370:Article 107240, 2020), where a Beurling prime system with regular primes but extremely irregular integers was constructed. [ABSTRACT FROM AUTHOR]

In this article we define and study a zeta function ζ G —similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζ G (k) - 1 at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζ G is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζ G . We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro- C groups, where C is a class of finite groups with prescribed composition factors. We prove that every real number a ≥ 1 is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζ G are rational functions in p - s if G is virtually abelian. For finite groups G we calculate ζ G using the rational representation theory of G. [ABSTRACT FROM AUTHOR]

We develop the L-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic twist of modular L-functions using multiple Dirichlet series under the generalized Riemann hypothesis. [ABSTRACT FROM AUTHOR]

In this paper, we consider universality in short intervals for the zeta-function attached to a normalized Hecke-eigen cusp form with respect to the modular group. For this, we apply a conjecture for the mean square in short interval on the critical strip for that zeta-function. The proof of the obtained universality theorem is based on a probabilistic limit theorem in the space of analytic functions. [ABSTRACT FROM AUTHOR]

We give a conditional lower bound on the number of non-trivial simple zeros for the Dedekind zeta function ζ K (s) , where K is a quadratic number field. The conditional result is given by assuming a Lindelöf condition on average (in the L 6 sense) for both ζ (s) and L (s , χ) , which can be seen as a stronger version of Conrey–Gonek–Ghosh's (Invent Math 86:563–576, 1986) conditional result. This improves upon the work of Wu and Zhao (Mathematika 65:851–861, 2019), who had a similar result. [ABSTRACT FROM AUTHOR]

Let π be a Hecke-Maass cusp form for SL 3 (Z) and let g be a holomorphic or Maass cusp form for SL 2 (Z) . Let χ be a primitive Dirichlet character of modulus M = M 1 M 2 with M i prime, i = 1 , 2 . Suppose that M 1 / 2 + 2 η < M 1 < M 1 - 2 η with 0 < η < 1 / 8 . Then we have L 1 2 , π ⊗ g ⊗ χ ≪ π , g , ε M 3 / 2 - η + ε. [ABSTRACT FROM AUTHOR]

Let f and g be holomorphic or Maass cusp forms for S L 2 (Z) and let χ be a primitive Dirichlet character of prime power conductor R = p κ with p an odd prime and κ > 12 . A subconvex bound for the central values of the Rankin–Selberg L-functions L (s , f ⊗ g ⊗ χ) is proved in the depth aspect L 1 2 , f ⊗ g ⊗ χ ≪ f , g , ε p 3 / 4 R 15 / 16 + ε. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over Q . We conjecture asymptotic estimates for the number of vanishings of L (E , 1 , χ) as χ varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis on E. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order ℓ characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order ℓ twisted elliptic curve L-functions, with ℓ even and composite, is a mixed family with both unitary and orthogonal aspects. [ABSTRACT FROM AUTHOR]

Given a number field, the Euler–Kronecker constant is defined as the constant term in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function at the point s = 1 . In the case of real and imaginary quadratic fields, a closed-form expression for the Euler–Kronecker constants can be obtained with the help of suitable Kronecker limit formulas. In this article, we avoid the use of Kronecker limit formulas and derive an explicit series representation of these constants with the help of an asymptotic series representation of the coefficients appearing in the Laurent series expansion of the Dedekind zeta function at s = 1 . As a result, the expressions obtained do not require evaluation of the special functions appearing in the Kronecker limit formulas. [ABSTRACT FROM AUTHOR]

We develop the L-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of quadratic Hecke L-functions in the Gaussian field using multiple Dirichlet series under the generalized Riemann hypothesis. We also obtain an asymptotical formula for the first moment of central values of the same family of L-functions, obtaining an error term of size O (X 1 / 2 + ε) . [ABSTRACT FROM AUTHOR]

Completely multiplicative arithmetic functions with sum zero (in short CMO functions) were initially introduced by Kahane and Saïas (Expo Math 35(4):364–389, 2017). These functions were recently generalized to Beurling generalized prime number systems and denoted as C M O P by Neamah (Res Number Theory 6:45, 2020). In this article, we generalize C M O P to multiplicative functions and refer to them as M O P functions. Thereafter, we examine some of these functions’ properties and provide examples of such functions. The aim was to investigate how small the partial sums of this class of functions can be. The results of this study indicate that for all M O P functions with an abscissa of convergence 1, we have ∑ n ≤ x n ∈ N f (n) = Ω (1 x ). We also discuss the consequences of this statement. [ABSTRACT FROM AUTHOR]

In this article, using the arithmetic of Hurwitz zeta functions and Gauss sums, we investigate the arithmetic nature of ζ K (2 k + 1) ζ K + (2 k + 1) , where K is any totally imaginary abelian extension with maximal totally real subfield K + and k ≥ 1 is an integer. As a consequence, we derive an elementary proof of the elegant result of Murty and Pathak about irrationality of Dedekind zeta values associated to imaginary quadratic fields at odd integers. This alternate line of action allows us to link these questions to a question of Chowla and Milnor, resulting in some new results about Dedekind zeta values. [ABSTRACT FROM AUTHOR]

Let L/K be a Galois extension of number fields with Galois group G. We show that if the density of prime ideals in K that split totally in L tends to 1/∣G∣ with a power saving error term, then the density of prime ideals in K whose Frobenius is a given conjugacy class C ⊂ G tends to ∣C∣/∣G∣ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros of ζL(s)/ζK(s). [ABSTRACT FROM AUTHOR]

In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration [9]. In particular, there are additional relations in the depth graded algebra coming from period polynomials of cusp forms for $\operatorname {\mathrm {SL}}_2({\mathbb {Z}})$ . In contrast, a simple combinatorial filtration, the block filtration [13, 28] is known to agree with the coradical filtration, and so there is no similar defect in the associated graded. However, via an explicit evaluation of $\zeta (2,\ldots ,2,4,2,\ldots ,2)$ as a polynomial in double zeta values, we derive these period polynomial relations as a consequence of an intrinsic symmetry of block graded multiple zeta values in block degree 2. In deriving this evaluation, we find a Galois descent of certain alternating double zeta values to classical double zeta values, which we then apply to give an evaluation of the multiple t values [22] $t(2\ell ,2k)$ in terms of classical double zeta values.

In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic L-function L (s , π) on the critical line averaged over characters and conductors, where π denotes an irreducible tempered cuspidal automorphic representation of GL 4 (A Q) with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation. [ABSTRACT FROM AUTHOR]

In 2011, Li [A uniqueness theorem for Dirichlet series satisfying a Riemann type functional equation. Adv Math. 2011;226(5):4198–4211] proved that if two L-functions L 1 and L 2 satisfy the same functional equation with a (1) = 1 and L 1 − 1 (c j) = L 2 − 1 (c j) for two distinct finite complex numbers c 1 and c 2 , then L 1 = L 2. We prove that if two L-functions L 1 and L 2 have positive degree and satisfy the same functional equation with a (1) = 1 and E L 1 (S) = E L 2 (S) for a finite set S = { c 1 , c 2 , c 3 } , where c 1 , c 2 and c 3 are three distinct finite complex values, then L 1 = L 2. The main results of this paper are concerning some questions posed by Gross [Factorization of meromorphic functions and some open problems. Complex Analysis, Kentucky 1976 (Proc.Conf.); Berlin: Springer-Verlag; 1977. p. 51–69. (Lecture Notes in Mathematics; 599)]. [ABSTRACT FROM AUTHOR]

We shall establish an explicit formula for the Davenport series in terms of trivial zeros of the Riemann zeta-function, where by the Davenport series we mean an infinite series involving a PNT (Prime Number Theorem) related to arithmetic function an with the periodic Bernoulli polynomial weight and PNT arithmetic functions include the von Mangoldt function, Möbius function and Liouville function, etc. The Riesz sum of order 0 or 1 gives the well-known explicit formula for respectively the partial sum or the Riesz sum of order 1 of PNT functions. Then we may reveal the genesis of the Popov explicit formula as the integrated Davenport series with the Riesz sum of order 1 subtracted. The Fourier expansion of the Davenport series is proved to be a consequence of the functional equation, which is referred to as the Davenport expansion. By the explicit formula for the Davenport series, we also prove that the Davenport expansion for the von Mangoldt function is equivalent to the Kummer's Fourier series up to a formula of Ramanujan and a fortiori is equivalent to the functional equation for the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

Let f be a Hecke–Maass or holomorphic primitive cusp form of full level for S L (2 , Z) with normalized Fourier coefficients λ f (n) . Let χ be a primitive Dirichlet character of modulus p, a prime. In this article, we shorten the range of cancellation for N in the twisted GL(2) short character sum. Here, we consider the problem of cancellation in short character sum of the form S f , χ (N) : = ∑ n ∈ Z λ f (n) χ (n) W (n N). We show that, for 0 < θ < 1 10 , S f , χ (N) ≪ f , ϵ N 3 / 4 + θ / 2 p 1 / 6 (p N) ϵ + N 1 - θ (p N) ϵ , which is non-trivial if N ≥ p 2 / 3 + α + ϵ where α = = 4 θ 1 - 6 θ . Previously, such a bound was known for N ≥ p 3 / 4 + ϵ. [ABSTRACT FROM AUTHOR]

We show a non-vanishing result for the derivatives of L-functions associated to cuspidal Hecke eigenforms of half-integral weight in plus space. In particular, we show that for large weights, ∑ j = 1 d 1 ⟨ f k , j , f k , j ⟩ d n d s n [ L ∗ (f k , j | W 4 , s) ] does not vanish at any point s = σ + i t 0 with t = t 0 , k / 2 - 1 / 4 < σ < k / 2 + 3 / 4 . [ABSTRACT FROM AUTHOR]