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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]