Non-principal Dirichlet Character Research Articles

L-functions with Riemann’s functional equation and the Riemann hypothesis

Abstract Let χ4 be the non-principal Dirichlet character mod 4 and $L(s,\chi_4)$ be the Dirichlet L-function associated with χ4 and put $R(s):= s 4^{s} L(s+1,\chi_4) + \pi L(s-1,\chi_4)$. In the present paper, we show that the function R(s) has the Riemann’s functional equation and its zeros only at the non-positive even integers and complex numbers with real part $1/2$. We also give other L-functions that have the same property.

Numerical estimates on the Landau-Siegel zero and other related quantities

Let q be an odd prime, χ be a non-principal Dirichlet character modq and L(s,χ) be the associated Dirichlet L-function. For every odd prime q≤107, we show that L(1,χ□)>c1log⁡q and β<1−c2log⁡q, where c1=0.0124862668…, c2=0.0091904477…, χ□ is the quadratic Dirichlet character modq and β∈(0,1) is the Landau-Siegel zero, if it exists, of the set of such Dirichlet L-functions. As a by-product of the computations here performed, we also obtained some information about Littlewood's and Joshi's bounds on L(1,χ□) and on the class number of the imaginary quadratic field Q(−q).

Riemann Hypothesis and Random Walks: The Zeta Case

In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line ℜ(s)&gt;12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.

Small Values of |L'/L(1,χ)|

In this article, we investigate the quantity , as over the primes, where is the Dirichlet L-function attached to a non-principal Dirichlet character modulo q. Our main result shows that . We also compute mq for every odd prime q up to 10. As a consequence we numerically verified that for every prime q, , we have , with . In particular, this shows that for every non-principal Dirichlet character χ mod q where is prime, answering a question of Gun, Murty and Rath in this range. We also provide some statistics and scatter plots regarding the mq-values. The programs used and the computational results described here are available at the following web address: https://www.math.unipd.it/~languasc/smallvalues.html.

A note on prime number races and zero free regions for L functions

Let [Formula: see text] be a real and non-principal Dirichlet character, [Formula: see text] its Dirichlet [Formula: see text]-function and let [Formula: see text] be a generic prime number. We prove the following result: If for some [Formula: see text] the partial sums [Formula: see text] change sign only for a finite number of integers [Formula: see text], then there exists [Formula: see text] such that [Formula: see text] has no zeros in the half plane [Formula: see text].

Asymptotics of Generalized Partial Theta Functions with a Dirichlet Character

We prove asymptotic expansions of generalized partial theta functions with a nonprincipal Dirichlet character and relate these expansions to certain L-series.

The twisted moments and distribution of values of Dirichlet L-functions at 1

In this paper, we seek to evaluate the most general form of moments of L(1,χ) twisted by character value χ(c) as χ ranges over non-principal Dirichlet characters mod f. And we use this result to model the joint distribution for χ(c) and L(1,χ).

The frequency and the structure of large character sums

Let $M(\chi)$ denote the maximum of $|\sum\_{n\le N}\chi(n)|$ for a given non-principal Dirichlet character $\chi$ modulo $q$, and let $N\_\chi$ denote a point at which the maximum is attained. In this article we study the distribution of $M(\chi)/\sqrt{q}$ as one varies over characters modulo $q$, where $q$ is prime, and investigate the location of $N\_\chi$. We show that the distribution of $M(\chi)/\sqrt{q}$ converges weakly to a universal distribution $\Phi$, uniformly throughout most of the possible range, and get (doubly exponential decay) estimates for $\Phi$'s tail. Almost all $\chi$ for which $M(\chi)$ is large are odd characters that are 1-pretentious. Now, $M(\chi)\ge |\sum\_{n\le q/2}\chi(n)| = \frac{|2-\chi(2)|}\pi \sqrt{q} |L(1,\chi)|$, and one knows how often the latter expression is large, which has been how earlier lower bounds on $\Phi$ were mostly proved. We show, though, that for most $\chi$ with $M(\chi)$ large, $N\_\chi$ is bounded away from $q/2$, and the value of $M(\chi)$ is little bit larger than $\frac{\sqrt{q}}{\pi} |L(1,\chi)|$.

Shifted character sums with multiplicative coefficients, II

Let f(n) be a multiplicative function with |f(n)|≤1,q be a prime number and a be an integer with (a,q)=1, χ be a non-principal Dirichlet character modulo q. Let ε be a sufficiently small positive constant, A be a large constant, q12+ε≪N≪qA. In this paper, we shall prove that∑n≤Nf(n)χ(n+a)≪Nlog⁡log⁡qlog⁡q and that∑n≤Nf(n)χ(n+a1)⋯χ(n+at)≪Nlog⁡log⁡qlog⁡q, where t≥2,a1,…,at are distinct integers modulo q.

A One-Formula Proof of the Nonvanishing of L-Functions of Real Characters at 1

We present a simple analytic proof that L-functions of real nonprincipal Dirichlet characters are nonzero at 1.

On the size of L(1,χ) and S. Chowla's hypothesis implying that L(1,χ)&gt;0 for s&gt;0 and for real characters χ

We give explicit constants κ such that if χ is a real non-principal Dirichlet character for which L(1,χ)≤κ, then Chowla's hypothesis is not satisfied and we cannot use Chowla's method for proving that L(s,χ)>0 for s>0. These constants are larger than the previous ones κ=1−log2=0.306… and κ=0.367… we obtained elsewhere.

On the second smallest prime non-residue

Let χ be a non-principal Dirichlet character modulo a prime p . Let q 1 < q 2 denote the two smallest prime non-residues of χ . We give explicit upper bounds on q 2 that improve upon all known results. We also provide a good upper estimate on the product q 1 q 2 which has an upcoming application to the study of norm-Euclidean Galois fields.

The average least character non-residue and further variations on a theme of Erdős

For each nonprincipal Dirichlet character $\chi$, let $n_\chi$ be the least $n$ with $\chi(n) \notin \{0,1\}$. We show that as the average of $n_\chi$ over all nonprincipal characters $\chi$ modulo $q$ is $\ell(q) + o(1)$, where $\ell(q)$ denotes the least prime not dividing $q$. Moreover, if one averages over all nonprincipal characters of modulus at most $x$, the average approaches a particular limiting value 2.5350541804. We also prove a result of this type for cubic number fields: If one averages over all cubic fields $K$, ordered by the absolute value of their discriminant, then the mean value of the least rational prime that does not split completely in $K$ is another particular constant 2.1211027269.

Erratum to: On the pairs of multiplicative functions satisfying some relations

This assertion is incorrect, as the following example shows: Let F (n) = (−1)n−1 (n = 1, 2, . . .), G(2n + 1) = χ4(2n + 1), where χ4 is the nonprincipal Dirichlet character (mod 4). Then F (n) and G(n) are multiplicative functions (G(n) can be arbitrary on the powers of 2), furthermore G(2n+ 1) = −F (n) (n ∈ N). (1.2) The failure comes from an incorrect assertion made in the proof of Lemma 8. Let H(n) = f(n) g(n) . Following the argument of the proof of Lemma 8, we obtain that H(3) = H(Q) if Q ≡ 3, 11 (mod 12), thus H(3) = H (3(4n+ 1)) = H(3)H(4n + 1) if (3, 4n + 1) = 1. For Q = 11 we have H(3) = H(11). Since ν, μ ≡ 3 (mod 4), (ν, μ) = 1 implies that H(νμ) = H(ν)H(μ) = 1, if (νμ, 3) = 1, it can occur only if H(n) = constant if n runs

Sums with convolutions of Dirichlet characters to cube-free modulus

We find estimates for short sums of the form ∑ n m ⩽ X χ 1 ( n ) χ 2 ( m ) \sum _{nm \leqslant X} \chi _1 (n) \chi _2 (m) , where χ 1 \chi _1 and χ 2 \chi _2 are non-principal Dirichlet characters to modulus q q , a cube-free integer, and X X can be taken as small as q 1 2 + ϵ q^{\frac 12+\epsilon } .

BOUNDS FOR A MEAN VALUE OF CHARACTER SUMS

We obtain new upper bounds for the mean squared modulus of sums ∑n∈ℕAnχ(n), where the sequence (An) is fixed and the variable χ belongs to the set of non-principal Dirichlet characters for some modulus q. It is assumed that, for some M, some complex sequence (cm) satisfying cm= 0 for m ∉ (M/2,M], and some α(x) and β(y) (smooth functions with compact support), one has An= ∑uvm = nα(u)β(v)cm(n ∈ ℕ). There is a natural analogy between the bounds obtained and bounds on mean values of Dirichlet polynomials previously obtained by Deshouillers and Iwaniec. Our proofs make use of results from the spectral theory of automorphic functions, including the bound of Kim and Sarnak for the eigenvalues of Hecke operators acting on certain spaces of Maass cusp forms. The results depend on the size of P, the largest prime factor of q, and improve as logq(P) is diminished. In separate work, Harman has given an application of our results to the theory of Carmichael numbers.

Weighted trigonometric sums over a half-period

We derive formulas for evaluating weighted sums of trigonometric functions over evenly-spaced angles in the first quadrant. These results generalize those of a previous paper, where we considered trigonometric sums weighted by real, primitive, non-principal Dirichlet characters.

Note on a hypothesis implying the non-vanishing of Dirichlet L-series L(s,χ) for s&gt;0 and real characters χ

We prove that if is a real non-principal Dirichlet character for which L(1; ) 1 log 2, then Chowla's hypothesis is not satised and we cannot use Chowla's method for proving that L(s; ) > 0 for s > 0.

Transcendental infinite sums

We show that it follows from results on linear forms in logarithms of algebraic numbers such as ▪ where χ is any non-principal Dirichlet character and ( F n ∞ n=0 the Fibonacci sequence, are transcendental.

Partial Gaussian sums III

For integers a, N and H &gt; 0 writewhere ϰ denotes a non-principal Dirichlet character modulo the positive integer k and e(y) denotes e2πiy. By a well-known generalisation of the Póya–Vinogradov inequality