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Inspired by two entries published in Ramanujan's lost notebook on p. 355, Berndt, Kim, and Zaharescu presented Riesz sum identities for the twisted divisor sums. Subsequently, Kim derived analogous results by replacing twisted divisor sums with twisted sums of divisor functions. In a recent work, the authors of the present paper deduced Cohen-type identities and Voronoï summation formulas associated with these twisted sums of divisor functions. The present paper aims to derive an equivalent version of the results offered in the previous article in terms of identities involving finite sums of trigonometric functions and the doubly infinite series. As an application, the authors provide an identity for r 6 (n) , which is analogous to Hardy's famous result where r 6 (n) denotes the number of representations of natural number n as a sum of six squares. [ABSTRACT FROM AUTHOR]

Suppose that x is a sufficiently large number and j ≥ 2 is any integer. Let L (s , sym j f) be the j-th symmetric power L-function associated with the primitive holomorphic cusp form f of weight k for the full modular group SL 2 (Z) . Also, let λ sym j f (n) be the n-th normalized Dirichlet coefficient of L (s , sym j f) . In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients λ sym j f (n) over two sparse sequences of positive integers, which improves previous results. [ABSTRACT FROM AUTHOR]

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya–Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess’ estimate for short character sums, and upper bounds for L (1 , χ) and L (1 + i t , χ) ) are more-or-less “equivalent”. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions. [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 [4], the first and second authors studied the signatures of Dirichlet characters and elliptic curves. In this paper we improve their lower bound estimates in the case of elliptic curves. We also introduce and investigate integral and interval d-constant signatures for cusp forms, and prove lower bound results for the number of twisted cusp forms having positive or negative signatures. [ABSTRACT FROM AUTHOR]

For every positive integer n, Sita Ramaiah's identity states that ∑ a 1 , a 2 , a 1 + a 2 ∈ (Z / n Z) ∗ gcd (a 1 + a 2 - 1 , n) = ϕ 2 (n) σ 0 (n) where ϕ 2 (n) = ∑ a 1 , a 2 , a 1 + a 2 ∈ (Z / n Z) ∗ 1 , where (Z / n Z) ∗ is the multiplicative group of units of the ring Z / n Z and σ s (n) = ∑ d ∣ n d s . This identity can also be viewed as a generalization of Menon's identity. In this article, we generalize this identity to an algebraic number field K involving a Dirichlet character χ . Our result is a further generalization of recent results in Ji and Wang (Ramanujan J 53:585–594, 2020) and Sury (Rend Circ Mat Palermo 58:99–108, 2009). [ABSTRACT FROM AUTHOR]

Let q ≥ 2 be an integer and let s q (n) be the sum-of-digitsfunction of n in base q. The function s q (n) has been studied in many directions and many properties have been obtained on the distribution of s q (n) and s q (P (n)) , where P is a suitable polynomial. In this paper we derive the generatingfunctions of s p (n d mod p k) for d ≥ 2 and prime p ≥ 2 by using the properties of Dirichlet character, and study the correlation properties of the sequences ((- 1) n 2 mod p k ) n < p k . [ABSTRACT FROM AUTHOR]

This paper studies Menon–Sury's identity in a general case, i.e., the Menon–Sury's identity involving Dirichlet characters in residually finite Dedekind domains. By using the filtration of the ring D / n and its unit group U (D / n) , we explicitly compute the following two summations: ∑ a ∈ U (D / n) b 1 , ... , b r ∈ D / n N (⟨ a - 1 , b 1 , b 2 , ... , b r ⟩ + n) χ (a) and ∑ a 1 , ... , a s ∈ U (D / n) b 1 , ... , b r ∈ D / n N (⟨ a 1 - 1 , ... , a s - 1 , b 1 , b 2 , ... , b r ⟩ + n) χ 1 (a 1) ⋯ χ s (a s) , where D is a residually finite Dedekind domain and n is a nonzero ideal of D , N (n) is the cardinality of quotient ring D / n , χ i (1 ≤ i ≤ s) are Dirichlet characters mod n with conductor d i . [ABSTRACT FROM AUTHOR]

Let χ be a nonprincipal Dirichlet character modulo a prime number p ≽ 3 and let a X : = 1 2 (1 − X (− 1)) . Define the mean value ℳ p (− s , X) : = 2 p − 1 ∑ ψ (mod p) ψ (− 1) = − 1 L (1 , ψ) L (− s , X ψ ¯) (σ : = ℜ s > 0). We give an identity for ℳ p (− s , X) which, in particular, shows that ℳ p (− s , X) = L (1 − s , X) + a X 2 p s L (1 , X) ζ (− s) + o (1) (p → ∞) for fixed 0 < σ < 1 2 and | t : = J s | = o (p (1 − 2 σ) / (3 + 2 σ) ) . [ABSTRACT FROM AUTHOR]

We obtain the asymptotic main term of moments of arbitrary derivatives of L-functions in the function field setting. Specifically, we obtain the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus Q ∈ F q [ T ] , and the asymptotic limit is as deg Q ⟶ ∞ . This extends the work of Tamam who obtained the asymptotic main term of low moments of L-functions, without derivatives, in the function field setting. It is also the function field q-analogue of the work of Conrey, who obtained the fourth moment of derivatives of the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

Menon's identity is a classical identity involving gcd sums and the Euler totient function ϕ . In a recent paper, Zhao and Cao (Int. J. Number Theory13(9) (2017) 2373–2379) derived the Menon-type identity ∑ k = 1 n (k - 1 , n) χ (k) = ϕ (n) τ (n d) , where χ is a Dirichlet character mod n with conductor d. We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao–Cao identity can be improved if one uses the method we employ here. [ABSTRACT FROM AUTHOR]

Let n be a positive integer. Arai-Carlitz's identity is ∑ a , b , a + b ∈ (Z / n Z) × gcd (a + b - 1 , n) = X (n) τ (n) , where X (n) = ∑ a , b , a + b ∈ (Z / n Z) × 1 and τ (n) is the divisor function. In this paper, we generalize Arai-Carlitz's identity to the ring of algebraic integers concerning Dirichlet characters. In particular, let n be a positive integer and χ a Dirichlet character modulo n with the conductor d. Then ∑ a , b , a + b ∈ (Z / n Z) × gcd (a + b - 1 , n) χ (a) = μ (d) φ (n 0 2 / d) X (n / n 0) τ (n / d) , where n 0 | n such that n 0 has the same prime factors with d and gcd (n 0 , n n 0) = 1 , φ (n) is the Euler's totient function and μ (n) is the Möbius function. [ABSTRACT FROM AUTHOR]

In this article, we prove an asymptotic formula for the fourth power mean of a general 4-dimensional Kloosterman sum. We use a result of P. Deligne, which counts the number of F p -points on the surface (x - 1) (y - 1) (z - 1) (1 - x y z) - u x y z = 0 , u ≠ 0 , and then take an average of the error terms over u to prove the asymptotic formula. We also find the number of solutions of certain congruence equations mod p which are used to prove our main result. [ABSTRACT FROM AUTHOR]

Let X be a sufficiently great real number and M denote the set of natural numbers not exceeding X which cannot be written as a sum of a prime and a fixed degree of a prime number from the arithmetical progression with difference d. Let Ed(X) = cardM. We obtain a new numerical degree estimate for the set Ed(X) and an estimate from below for the number of presentations of n ∉ M in the specified type. The proven estimates refine the generalization for an arithmetical progression of results earlier got by V.A. Plaksin. [ABSTRACT FROM AUTHOR]

In this paper we establish why the p-adic zeta function has a Dirichlet series expansion. We compute an improved expansion, which allows us to express it as a power-series modulo pn. Using this expansion, we compute all the zeros of Lp(s, χωj) for those quadratic characters χ of conductor < 200. For the calculation we use a PARI-GP Program. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the analytic methods, the solutions of the congruence equation mod p and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 3-dimensional Kloostermann sums mod p, and give a sharp asymptotic formula for it. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 2-dimensional Kloostermann sums mod p, and give an exact computational formula for it. [ABSTRACT FROM AUTHOR]

Structural theorems for spaces of modular forms with respect to congruence subgroups are proved. The Dedekind η-function plays an important role in our study. [ABSTRACT FROM AUTHOR]

A limit theorem involving an increasing modulus of the character is obtained for twists with the Dirichlet character of L-functions of elliptic curves. [ABSTRACT FROM AUTHOR]

This note gives a simple proof that certain values of Artin's L-function, for a representation ρ with character χ, are stable under twisting by an even Dirichlet character χ, up to the dim( ρ)th power of the Gauss sum τ(χ) and an element generated over $${\mathbb{Q}}$$ by the values of χ and χ. This extends a result due to J. Coates and S. Lichtenbaum. [ABSTRACT FROM AUTHOR]

In this paper we consider an approach of Dobrowolski and Williams which leads to a generalization of the Pólya-Vinogradov inequality. We show how the Dobrowolski-Williams approach is related to the classical proof of Pólya-Vinogradov using Fourier analysis. Our results improve upon the earlier work of Bachman and Rachakonda (Ramanujan J. 5:65-71, ). In passing, we also obtain sharper explicit versions of the Pólya-Vinogradov inequality. [ABSTRACT FROM AUTHOR]

The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan's sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers m, n, k, q, with k ⩾ 1 and q ⩾ 3, and Dirichlet characters χ, χ̄ modulo q we define a mixed exponential sum , with Dirichlet character χ and general Gauss sum G( a, χ̄) as coefficient, where Σ′ denotes the summation over all a such that ( a, q) = 1, aā ≡ 1 mod q and e( y) = e. We mean value of , and give an exact computational formula for it. [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]

Let q be a sufficiently large integer and χ be a Dirichlet character modulo q. In this paper, we extend the product Π L(1, χ) with prime q, arising from the Kummer conjecture, to the products of some general Dirichlet series, and give some meaningful estimates for them. [ABSTRACT FROM AUTHOR]

For integers m, n, q, k, with q, k≧1 and Dirichlet characters $\chi, \chi' \text {\rm \;(mod}\,q)$ we define a generalized Kloosterman sum with a Dirichlet character and a Gauss sum G( a, χ′) as coefficient, where e( z)= e. The aim of this paper is to study the fourth power mean obtaining explicit formulas for M( q). [ABSTRACT FROM AUTHOR]

Let the set of positive integers be colored in an arbitrary way in finitely many colors (a 'finite coloring'). Is it true that, in this case, there are x, y ∈ ℤ such that x + y, xy, and x have the same color? This well-known problem of the Ramsey theory is still unsolved. In the present paper, we answer this question in the affirmative in the group ℤ/ pℤ, where p is a prime, and obtain an even stronger density result. [ABSTRACT FROM AUTHOR]

By introducing the Rademacher-Menchov device, we prove 'maximal' analogs of principal bounds of character sums. This allows us to present the Burgess method so as to separate the main idea of this method from the technical issues. [ABSTRACT FROM AUTHOR]

In the paper, a Voronin-type joint universality theorem for Lerch zeta functions is obtained under weaker assumptions than those in A. Laurinčikas and K. Matsumoto, 'Joint value distribution theorems on Lerch zeta-functions. III,' in Analytic and Probabilistic Methods in Number Theory (TEV, Vilnius, 2007), pp. 87-98. [ABSTRACT FROM AUTHOR]

Enlarged major arcs in the Waring-Goldbach problem are studied by using large sieve estimates for Dirichlet polynomials and estimates for exponential sums over primes. [ABSTRACT FROM AUTHOR]

We obtain lower bounds for the fractional moments of linear combinations of analogs of the Hardy function. In addition, we apply these estimates to the Karatsuba problem of finding a lower bound for the number of zeros of the linear combination of analogs of Hardy functions on the interval (0, T]. [ABSTRACT FROM AUTHOR]

We obtain a limit theorem for L-functions of elliptic curves twisted by Dirichlet characters with increasing modulus. [ABSTRACT FROM AUTHOR]

We prove functional independence and joint functional independence for a set of Hurwitz zeta functions with periodic coefficients and parameters algebraically independent over the field of rational numbers. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is, using the analytic method, to study the mean value properties of the complete trigonometric sums with Dirichlet characters, and give an exact calculating formula for its fourth power mean. [ABSTRACT FROM AUTHOR]

In this paper we prove that, with at most $$ O{\left( {N^{{\frac{5} {{12}} + \in }} } \right)} $$ exceptions, all positive odd integers n ≤ N with n ≡ 0 or 1(mod 3) can be written as a sum of a prime and two squares of primes. [ABSTRACT FROM AUTHOR]

Suppose that g(n) is an additive real-valued function, W(N) = 4+min λ(λ² + ∑p

Suppose that $g(n)$ is a real-valued additive function and $\backslash tau(n)$ is the number of divisors of $n$. In this paper, we prove that there exists a constant $C$ such that

For a kind of L-function of the real quadratic number fields, we prove a Kronecker limit formula which generalized a result of Hecke. And taking an example we give an interesting identity on a fundamental unit of such a field [ABSTRACT FROM AUTHOR]

We study twists of the Lerch zeta-functions with shifted Dirichlet characters. This generalizes Dirichlet L-functions. These twists turn out to be meromorphic with at most one simple pole and satisfy a functional equation (similar to the functional equation of the Lerch zeta-function) if the Dirichlet character is primitive. We also investigate the distributions of their zeros. In particular, we disprove the analogue of Riemann's hypothesis for some twists. [ABSTRACT FROM AUTHOR]

In [2] E. Dobrowolski and K.S. Williams considered a problem of obtaining estimates for the sum ∑ f(n),for a certain class of functions f. One specific application of their result is a new method for estimating character sums. In particular, they obtain a form of the Pólya-Vinogradov inequality with the constant 1/(2 log 2). In this note we improve their estimates and obtain, in particular, a form of the Pólya-Vinogradov inequality with the constant 1/(3 log 3). A nice feature of our estimate is that it is obtained by a very simple argument. [ABSTRACT FROM AUTHOR]

In this paper we shall make a further study on the “the equivalence problem,” i.e., equivalent conditions to the Riemann hypothesis in terms of Farey series by developing a rather analytical (-arithmetical) method to establish unexpected short interval results, namely $$\frac{1} {5} - ,\frac{1} {6} - $$ results, therewith simultaneously clarifying the underlying reasons for $$\frac{1} {2} - ,\frac{1} {3} - \frac{1} {4} - $$ results obtained in Part I. [ABSTRACT FROM AUTHOR]

In the famous work of Dirichlet on class number formula, L(s, χ) at s = 1 has been expressed as a finite sum, where L(s, χ) is the Dirichlet L-series of a real Dirichlet character. We show that this expression with obvious modification is valid for the general primitive Dirichlet character χ. [ABSTRACT FROM AUTHOR]