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We make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near σ = 1 " for Dirichlet L-functions. We use this estimate and recent work of Green to prove that if N ≥ 2 is an integer, A ⊆ { 1 , ... , N } , and for all primes p no two elements in A differ by p - 1 , then | A | ≪ N 1 - 10 - 18 . This strengthens a theorem of Sárközy. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is utilizing the analytic methods to study Hybrid mean value including the character, the general trigonometric sums, and the general quadratic Gauss sums, then obtain several mean value asymptotic formulae. [ABSTRACT FROM AUTHOR]

We study non-vanishing of Dirichlet L-functions at the central point under the unlikely assumption that there exists an exceptional Dirichlet character. In particular we prove that if ψ is a real primitive character modulo D ∈ N with L (1 , ψ) ≪ (log D) - 25 - ε , then, for any prime q ∈ [ D 300 , D O (1) ] , one has L (1 / 2 , χ) ≠ 0 for almost all Dirichlet characters χ (mod q) . [ABSTRACT FROM AUTHOR]

We obtain new bounds, pointwise and on average, for Dedekind sums s (λ , p) modulo a prime p with λ of small multiplicative order d modulo p. Assuming the infinitude of Mersenne primes, the range of our results is optimal. Moreover, we relate high moments of L (1 , χ) over subgroups of characters to some correlations of Dedekind sums, and use recent results of the second and third author to study these correlations. [ABSTRACT FROM AUTHOR]

In this work and its sister paper (Friedlander and Iwaniec (2023)), we give a new proof of the famous Linnik theorem bounding the least prime in an arithmetic progression. Using sieve machinery in both papers, we are able to dispense with the log-free zero density bounds and the repulsion property of exceptional zeros, two deep innovations begun by Linnik and relied on in earlier proofs. [ABSTRACT FROM AUTHOR]

We consider Dirichlet L-functions L (s , χ) where χ is a non-principal quadratic character to the modulus q. We make explicit a result due to Pintz and Stephens by showing that | L (1 , χ) | ⩽ 1 2 log q for all q ⩾ 2 · 10 23 and | L (1 , χ) | ⩽ 9 20 log q for all q ⩾ 5 · 10 50 . [ABSTRACT FROM AUTHOR]

As a consequence of their work, Bruce C. Berndt, Ronald J. Evans, Larry Joel Goldstein and Michael Razar obtained a formula for the square of the class number of an imaginary quadratic number field in terms of Dedekind sums. We give a short proof of it and also express the relative class numbers of imaginary abelian number fields in terms of Dedekind sums. [ABSTRACT FROM AUTHOR]

In this paper, we compute the one-level density of low-lying zeros of Dirichlet L-functions in a family weighted by special values of Dirichlet L-functions at a fixed s ∈ [ 1 / 2 , 1) . We verify both Fazzari's conjecture and the first author's conjecture on the weighted one-level density for our family of L-functions. [ABSTRACT FROM AUTHOR]

In this paper, under a weakened version of Hardy—Littlewood Conjecture on the number of representations in Goldbach problem, we shall prove bounds for the Siegel zeros of real primitive Dirichlet characters for composite moduli. [ABSTRACT FROM AUTHOR]

In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2. [ABSTRACT FROM AUTHOR]

We show that for at least 5 13 of the primitive Dirichlet characters χ of large prime modulus, the central value L (1 2 , χ) does not vanish, improving on the previous best known result of 3 8 . [ABSTRACT FROM AUTHOR]

Let p be a prime, χ denote a Dirichlet character modulo p. For any integer x ( 1 ≤ x ≤ p - 1 ), x ¯ denotes the integer inverse of x such that x x ¯ ≡ 1 (mod p) , we study the following mean value of a kind of character sums with generalized Dirichlet L-functions ∑ χ (- 1) = 1 χ ≠ χ 0 ∑ x = 1 p - 1 χ (x + x ¯) 2 | L (1 , χ , a) | 2 , where χ 0 is the principal character modulo p, and L (1 , χ , a) is the generalized Dirichlet L-functions. In this paper, we will use the analytic method and get a sharp asymptotic formula. [ABSTRACT FROM AUTHOR]

In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$ , was considered by Selberg ['An elementary proof of the prime-number theorem', Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$ , it was studied by Bombieri ['The asymptotic sieve', Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec ['On Bombieri's asymptotic sieve', Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve. Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$ , where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$ -free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$ , $(a,q)=1$ , uniformly on $q$. When $j\geq 2$ , we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function. [ABSTRACT FROM AUTHOR]

We estimate the number of primes represented by a general quadratic polynomial with discriminant Δ , assuming that the corresponding real character is exceptional. [ABSTRACT FROM AUTHOR]

Let ψ be a real primitive character modulo D. If the L-function L (s , ψ) has a real zero close to s = 1 , known as a Landau–Siegel zero, then we say the character ψ is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values L (1 / 2 , χ) of the Dirichlet L-functions L (s , χ) are nonzero, where χ ranges over primitive characters modulo q and q is a large prime of size D O (1) . Under the same hypothesis we also show that, for almost all χ , the function L (s , χ) has at most a simple zero at s = 1 / 2 . [ABSTRACT FROM AUTHOR]

We prove a nontrivial estimate for the size of the least rational prime that splits completely in the ring of integers of certain families of nonabelian Galois number fields. Our result complements results of Linnik and Vinogradov and of Pollack who studied this problem in the quadratic and abelian number field settings, respectively. [ABSTRACT FROM AUTHOR]

Let $$\mathcal {F}(h)$$ be the number of imaginary quadratic fields with class number h. In this note, we improve the error term in Soundararajan's asymptotic formula for the average of $$\mathcal {F}(h)$$ . Our argument leads to a similar refinement of the asymptotic for the average of $$\mathcal {F}(h)$$ over odd h, which was recently obtained by Holmin, Jones, Kurlberg, McLeman and Petersen. [ABSTRACT FROM AUTHOR]

We use suitably Page's theorem to get effective results for interesting problems, by avoiding the ineffective Siegel's theorem. [ABSTRACT FROM AUTHOR]

We assume a weak Hardy–Littlewood conjecture and derive an upper bound for a real exceptional zero associated with a prime modulus. [ABSTRACT FROM AUTHOR]

Text De Bruijn and Newman introduced a deformation of the completed Riemann zeta function ζ , and proved there is a real constant Λ which encodes the movement of the nontrivial zeros of ζ under the deformation. The Riemann hypothesis is equivalent to the assertion that Λ ≤ 0 . Newman, however, conjectured that Λ ≥ 0 , remarking, “the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so”. Andrade, Chang and Miller extended the machinery developed by Newman and Pólya to L -functions for function fields. In this setting we must consider a modified Newman's conjecture: sup f ∈ F ⁡ Λ f ≥ 0 , for F a family of L -functions. We extend their results by proving this modified Newman's conjecture for several families of L -functions. In contrast with previous work, we are able to exhibit specific L -functions for which Λ D = 0 , and thereby prove a stronger statement: max L ∈ F ⁡ Λ L = 0 . Using geometric techniques, we show a certain deformed L -function must have a double root, which implies Λ = 0 . For a different family, we construct particular elliptic curves with p + 1 points over F p . By the Weil conjectures, this has either the maximum or minimum possible number of points over F p 2 n . The fact that # E ( F p 2 n ) attains the bound tells us that the associated L -function satisfies Λ = 0 . Video For a video summary of this paper, please visit http://youtu.be/hM6-pjq7Gi0 . [ABSTRACT FROM AUTHOR]

For any periodic function f : N → C with period q , we study the Dirichlet series L ( s , f ) : = ∑ n ≥ 1 f ( n ) / n s . It is well-known that this admits an analytic continuation to the entire complex plane except at s = 1 , where it has a simple pole with residue ρ : = q − 1 ∑ 1 ≤ a ≤ q f ( a ) . Thus, the function is analytic at s = 1 when ρ = 0 and in this case, we study its non-vanishing using the theory of linear forms in logarithms and Dirichlet L -series. In this way, we give new proofs of an old criterion of Okada for the non-vanishing of L ( 1 , f ) as well as a classical theorem of Baker, Birch and Wirsing. We also give some new necessary and sufficient conditions for the non-vanishing of L ( 1 , f ) . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the estimation of the character sums to study the 2k-th power mean of L-functions with the weight of a character sums, and give an interesting asymptotic formula for it. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce new sums that are analogous to Dedekind sums. Using analysis and properties of Dirichlet $$L$$ -functions, we study mean values for these new sums, and give a sharper mean value formula for it. [ABSTRACT FROM AUTHOR]

Let q ≥ 3 be an integer, and χ be a Dirichlet character modulo q, L( s, χ) denote the Dirichlet L-function corresponding to χ. In this paper, we show some early information on the mean value of the Dirichlet L-functions and give some new identities for [Figure not available: see fulltext.] where a = 2, 3, or 4. Then we give general identities for the case that the integer a divides q t- 1. [ABSTRACT FROM AUTHOR]

Let f be a completely multiplicative function that assumes values inside the unit disc. We show that if $${\sum_{n \leq x}f(n)\ll x/(\rm log x)^A}$$ , $${x \geq 2}$$ , for some A > 2, then either f( p) is small on average or f pretends to be $${\mu(n)n^{it}}$$ for some t. [ABSTRACT FROM AUTHOR]

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions. [ABSTRACT FROM AUTHOR]

The various properties of classical Dedekind sums S( h, q) have been investi-gated by many authors. For example, Yanni Liu and Wenpeng Zhang: A hybrid mean value related to the Dedekind sums and Kloosterman sums, Acta Mathematica Sinica, 27 (2011), 435-440 studied the hybrid mean value properties involving Dedekind sums and generalized Kloosterman sums K( m, n, r; q). The main purpose of this paper, is using the analytic methods and the properties of character sums, to study the computational problem of one kind of hybrid mean value involving Dedekind sums and generalized Kloosterman sums, and give an interesting identity. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new estimate on exponential sums by using the Bombieri-type theorem and the modified Huxley-Hooley contour. We also generalize the famous Goldbach-Vinogradov theorem, via different argument from that of Vinogradov. In particular, our major arcs are quite large and these enlarged major arcs are treated by the estimate we have established. [ABSTRACT FROM AUTHOR]

In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X+ nY. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the mean value formula of Dirichlet L-functions and the analytic methods to study a hybrid mean value problem related to certain Hardy sums and Kloosterman sums, and give some interesting mean value formulae and identities for it. [ABSTRACT FROM AUTHOR]

We prove a Lipschitz type summation formula with periodic coefficients. Using this formula, representations of the values at positive integers of Dirichlet L-functions with periodic coefficients are obtained in terms of Bernoulli numbers and certain sums involving essentially the discrete Fourier transform of the periodic function forming the coefficients. The non-vanishing of these L-functions at s = 1 are then investigated. There are additional applications to the Fourier expansions of Eisenstein series over congruence subgroups of $${SL_2(\mathbb{Z})}$$ and derivatives of such Eisenstein series. Examples of a family of Eisenstein series with a high frequency of vanishing Fourier coefficients are given. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to use the M. Toyoizumi's important work, the properties of the Dedekind sums and the estimates for character sums to study a hybrid mean value of the Dedekind sums, and give a sharper asymptotic formula for it. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums, and give some interesting mean value formulae and identities for it. [ABSTRACT FROM AUTHOR]

Let q ⩾ 3 be an integer, let χ denote a Dirichlet character modulo q. For any real number a ⩾ 0 we define the generalized Dirichlet L-functions where s = σ + it with σ > 1 and t both real. They can be extended to all s by analytic continuation. In this paper we study the mean value properties of the generalized Dirichlet L-functions especially for s = 1 and s = 1/2 + i t, and obtain two sharp asymptotic formulas by using the analytic method and the theory of van der Corput. [ABSTRACT FROM AUTHOR]

We prove that the Eisenstein series E( z, s) have no real zeroes for s ∈ (0, 1) when the value of the imaginary part of z is in the range $$\tfrac{1}{5}$$ < Im z < 4.94. For very large and very small values of the imaginary part of z, E( z, s) have real zeros in (½, 1), i.e. GRH does not hold for the Eisenstein series. Using these properties, we prove that the Rankin-Selberg L-function attached with the Ramanujan τ-function has no real zeros in the critical strip, except at the central point s = ½. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to use estimates for character sums and analytic methods to study the first power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums, and three asymptotic formulae are obtained. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to use the estimate for character sums and the method of trigonometric sums to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of the Gauss sums, and give a sharper asymptotic formula. [ABSTRACT FROM AUTHOR]

Let θ be a constant satisfying . We prove that there exists , such that the number of even integers in the interval which cannot be written as a sum of two primes is . References [ABSTRACT FROM AUTHOR]

Let P be an odd prime, denote by p n ( q n) the nth prime ≠ P with ( p n /P)=1(=−1), d n= q n− p n. We discuss the question whether d n changes sign infinitely often or not. Without using Turán's power sum method the following theorem is proved. Suppose that the L-function L(s, χ), defined by the real primitive character mod P, has no real root σ with 1/2<σ<1. Then the numbers d n change sign infinitely often. The hypothesis is known to be true for all P with 2< P≤227 (J. B. Rosser. J. of Research of the Nat. Bureau of Standards 45, 505–514 (1950)). [ABSTRACT FROM AUTHOR]