## 1 Introduction

Let p be a prime, let $$q=p$$ if p is odd and $$q=4$$ if $$p=2$$, and let $$\chi$$ be a Dirichlet character of conductor f. A p-adic L-function $$L_p(s,\chi )$$ for a Dirichlet character $$\chi$$ is a p-adic meromorphic function and an analogue of the complex L-function. For powers of the Teichmüller character $$\omega$$ of conductor q, one obtains the p-adic zeta functions $$\zeta _{p,i}=L_p(s,\omega ^{1-i})$$, where $$i=0,\, 1,\, \dots ,\, p-2$$ ($$i=0, 1$$ if $$p=2$$). It is well known that $$L_p(s,\chi )$$ is identically zero for odd $$\chi$$. p-adic L-functions have a long history and the primary constructions going back to Kubota-Leopoldt [6] and Iwasawa [3] are via the interpolation of special values of complex L-functions.

It can also be shown that p-adic L-functions are in fact Iwasawa functions.

It is well known that for $$Re(s) > 0$$,

\begin{aligned} (1-2^{1-s}) \zeta (s) = \sum _{n=1}^{\infty } \frac{(-1)^{n+1}}{n^s} \end{aligned}

and, more generally, if $$c\ge 2$$ is an integer,

\begin{aligned} (1-\chi (c)c^{1-s})L(s, \chi ) = \sum _{n=1}^{\infty } \chi (n) \frac{a_{c,n}}{n^s}, \end{aligned}

where $$a_{c,n} = 1-c$$ if $$n\equiv 0 \text { mod}\ c$$ and $$a_{c,n} = 1$$ if $$n\not \equiv 0 \text { mod}\ c$$. In the following, we derive similar, but slightly different, expansions for p-adic L-functions.

An explicit formula for $$L_p(s,\chi )$$ is given in [9] (Theorem 5.11): let F be any multiple of q and f. Then $$L_p(s,\chi )$$ is a meromorphic function (analytic if $$\chi \ne 1)$$ on $$\{ s \in {\mathbb {C}}_p\ |\ |s| < qp^{-1/(p-1)} \}$$ such that

$$L_{p} (s,\chi ) = \frac{1}{F}\frac{1}{{s - 1}}\sum\limits_{\substack{\mathop {a=1}\\ {p\not\mid a}}}^F \chi (a)\langle a\rangle ^{{1 - s}} \sum\limits_{{j = 0}}^{\infty } {\left( {\begin{array}{*{20}c} {1 - s} \\ j \\ \end{array} } \right)} \left( {\frac{F}{a}} \right)^{j} B_{j} .{\text{ }}$$
(1)

In Sect. 2, we will use formula (1) to derive a Dirichlet series expansion of $$L_p(s,\chi )$$.

p-adic L-functions can be also be defined using distributions and measures. Let $$\chi$$ have conductor $$f=dp^m$$ with $$(d,p)=1$$. Choose an integer $$c \ge 2$$, where $$(c,dp)=1$$. Then there is a measure $$E_{1,c}$$ on $$({\mathbb {Z}}/d {\mathbb {Z}})^{\times } \times \, {\mathbb {Z}}_p^{\times }$$ (the regularized Bernoulli distribution) such that

\begin{aligned} -(1- \chi (c) \langle c \rangle ^{1-s}) L_p(s, \chi ) = \int _{({\mathbb {Z}}/d {\mathbb {Z}})^{\times } \times \, {\mathbb {Z}}_p^{\times }} \chi \omega ^{-1}(a)\langle a\rangle ^{-s} \ dE_{1,c} \end{aligned}
(2)

(see [9] Theorem 12.2). In Sect. 3, we give an explicit formula for the values of $$E_{1,c}$$ and derive the Dirichlet series expansion from (2).

The expansion is particularly simple for $$c=2$$, and this parameter can be used for $$p \ne 2$$ and Dirichlet characters with odd conductor. For this case we obtain similar results as in [1, 2], and [4]. In Sect. 4, we provide examples for different parameters c.

## 2 Expansions of p-adic L-functions

First, we derive an approximation of $$L_p(s,\chi )$$ that is close to the original definition of Kubota-Leopoldt (see [6]).

For $$r \in {\mathbb {C}}_p^{\times }$$ we write $$\delta (r)$$ for a term with p-adic absolute value $$\le |r|$$.

### Proposition 2.1

Let p be a prime number, $$\chi$$ an even Dirichlet character of conductor f, and F a multiple of q and f. For $$s \in {\mathbb {C}}_p$$ with $$|s| < qp^{-1/(p-1)}$$, we have

$$L_{p} (s,\chi ) = \frac{1}{F}\frac{1}{{s - 1}}\sum\limits_{\substack{\mathop {a=1}\\{p\not\mid a}}}^F \chi (a)\langle a\rangle ^{{1 - s}} + \delta (F/qp).$$
(3)

### Proof

We use formula (1) above and look at the series $$\sum _{j=0}^{\infty } \left( {\begin{array}{c}1-s\\ j\end{array}}\right) \left( \frac{F}{a} \right) ^j B_j$$. The first two terms are $$1+(1-s) \frac{-F}{2a}$$. We claim that the p-adic absolute value of the other terms ($$j \ge 2$$) is less than or equal to $$| (s-1)F^2/qp |$$. To this end, we note that $$|1/j!|\le p^{(j-1)/(p-1)}$$ and

\begin{aligned} \left| \left( {\begin{array}{c}1-s\\ j\end{array}}\right) \right| \le |1-s|\, p^{(j-1)/(p-1)}(qp^{-1/(p-1)})^{j-1}= |1-s|\, q^{j-1} \end{aligned}

since we assumed that $$|s| < qp^{-1/(p-1)}$$. Since $$|F|\le \frac{1}{q}$$, $$|a|=1$$, and $$|B_j| \le p$$, we obtain

\begin{aligned} \left| \left( {\begin{array}{c}1-s\\ j\end{array}}\right) \left( \frac{F}{a} \right) ^j B_j \right| \le |1-s| \, q^{j-1} q^{2-j} |F|^2\, p = |1-s| \, |F|^2\, q p. \end{aligned}

Then (1) implies

$$L_{p} (s,\chi ) = \frac{1}{F}\frac{1}{{s - 1}}\sum\limits_{\substack{\mathop {a=1}\\ {p\not\mid a}}}^F \chi (a)\langle a\rangle ^{{1 - s}} + \frac{1}{2}\sum\limits_{\substack{\mathop{a = 1} \\ {p\not\mid a}}}^{F} \chi \omega ^{{ - 1}} (a)\langle a\rangle ^{{ - s}} + \delta \left( {F/qp} \right).$$

It remains to show that the second sum can be absorbed into $$\delta (F/qp)$$. We have

\begin{aligned} \sum\limits_{\substack{\mathop {a=1}\\ {p\not\mid a}}}^F \chi \omega ^{{ - 1}} (a)\left\langle a \right\rangle ^{{ - s}} & = \sum\limits_{\substack{\mathop {b=1}\\ {p\not\mid b}}}^F \chi \omega ^{{ - 1}} (F - b)\left\langle {F - b} \right\rangle ^{{ - s}} \\ & = - \sum\limits_{\substack{\mathop {b=1}\\{p\not\mid b}}}^F \chi \omega ^{{ - 1}} (b)\left\langle {b - F} \right\rangle ^{{ - s}} \\ & = - \sum\limits_{\substack{\mathop {b=1}\\ {p\not\mid b}}}^F \chi \omega ^{{ - 1}} (b)\left\langle b \right\rangle ^{{ - s}} + \delta (F/qp^{{ - 1/(p - 1)}} ). \end{aligned}

The last step can be justified by noting that

\begin{aligned} \frac{\langle b-F\rangle ^{-s}}{\langle b\rangle ^{-s}} = \left( 1-\frac{F}{b}\right) ^{-s} = 1 +\sum _{j=1}^{\infty } \left( {\begin{array}{c}-s\\ j\end{array}}\right) \left( \frac{-F}{b}\right) ^j = 1 + \delta (F/qp^{-1/(p-1)}), \end{aligned}

since $$|s|< qp^{-1/(p-1)}$$ (this is the same estimate as earlier, without the presence of the Bernoulli number). This proves the proposition. $$\square$$

### Remark 2.2

For $$F=fp^n$$ and $$n \rightarrow \infty$$, formula (3) gives the original definition of $$L_p(s,\chi )$$ by Kubota and Leopoldt (see [6]).

### Remark 2.3

Suppose that $$p\ne 2$$. Then the error term in the above Proposition (as well as in the following Theorem 2.4) can be improved to $$\delta (F/p^{2-(p-2)/(p-1)})$$. First we note that $$B_j=0$$ for odd $$j \ge 3$$. By the von Staudt–Clausen Theorem (see [9] 5.10), we have for even $$j \ge 2$$: $$|B_j| =p$$ iff $$(p-1) \mid j$$, and otherwise $$|B_j| \le 1$$. Furthermore, $$|1/j!| = p^{(j-S_j)/(p-1)}$$, where $$S_j$$ is the sum of the digits of j, written to the base p (see [5]). Since $$j \equiv S_j \text { mod}\ (p-1)$$, $$j \equiv 0 \text { mod}\ (p-1)$$ is equivalent to $$S_j \equiv 0 \text { mod}\ (p-1)$$. We conclude that $$|B_j |=p$$ yields $$S_j \ge p-1$$ and $$|1/j!| \le p^{(j-1)/(p-1) } p^{-(p-2)/(p-1)}$$. This implies the above error term. We also see that this error term cannot be further improved.$$\lozenge$$

Now we give the Dirichlet expansion of $$L_p(s,\chi )$$. For $$m \in {\mathbb {N}}$$, we denote by $$\{x\}_{m}$$ the unique representative of $$x \text { mod}\ m{\mathbb {Z}}$$ between 0 and $$m-1$$.

### Theorem 2.4

Let p be a prime number, $$\chi$$ be an even Dirichlet character of conductor f, and F a multiple of q and f. Let $$c>1$$ be an integer satisfying $$(c,F)=1$$. For $$a \in {\mathbb {Z}}$$, define

\begin{aligned} \epsilon _{a,c,F} = \frac{c-1}{2} - \{ -a F^{-1} \}_{c} \, \in \left\{ - \frac{c-1}{2} ,\, - \frac{c-1}{2} + 1,\, \dots ,\ \frac{c-1}{2}\right\} . \end{aligned}

Then we have for $$s \in {\mathbb {C}}_p$$ with $$|s| < qp^{-1/(p-1)}$$ the formula

\begin{aligned} -(1- \chi (c) \langle c \rangle ^{1-s}) L_p(s, \chi ) = \sum\limits_{\substack{\mathop {a=1} \\ {p\not\mid a}}}^F \chi \omega ^{-1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,F} + \delta (F/qp) . \end{aligned}

### Proof

Use (3) with cF in place of F, and subtract $$\chi (c) \langle c \rangle ^{1-s}$$ times (3) with F, to obtain

\begin{aligned} (1 - \chi (c)\langle c\rangle ^{{1 - s}} )L_{p} (s,\chi ) & = \frac{1}{{cF}}\frac{1}{{s - 1}}\sum\limits_{\substack{\mathop {a=1}\\{p\not\mid a}}}^{cF} \chi (a)\langle a\rangle ^{{1 - s}} \\ & - \frac{1}{F}\frac{1}{{s - 1}}\sum\limits_{\substack{\mathop {a=1}\\ {p\not\mid a}}}^F \chi (ac)\langle ac\rangle ^{{1 - s}} + \delta (F/qp). \end{aligned}
(4)

Let $$0<a_0 < F$$ with $$(a_0,\, p)=1$$. Since we assumed $$(c,\, F)=1$$ and $$p \mid F$$, there is a unique number of the form $$a_0 c$$ with $$0<a_0c < cF$$ and $$(a_0 c,\, p)=1$$ in each congruence class modulo F relatively prime to p. The first sum in (4) can be written as

\begin{aligned} \frac{1}{cF} \frac{1}{s-1} \sum\limits_{\substack{\mathop {a_0=1}\\ {p\not\mid a_0}}}^F \chi (a_0 c) \langle a_0 c \rangle ^{1-s} \left( \sum\limits_{\substack{\mathop {a=1} \\ \mathop{a \equiv a_{0} c{\text{ }}\bmod \;F}}}^{cF} \left\langle 1+ \frac{a-a_0 c}{a_0 c} \right\rangle ^{1-s} \right) \\ \quad = \frac{1}{cF} \frac{1}{s-1} \sum\limits_{\substack{\mathop {a_0=1} \\ {p\not\mid a_0}}}^F \chi (a_0 c) \langle a_0 c \rangle ^{1-s} \left( \sum\limits_{\substack{\mathop{a=1} \\ \mathop {a \equiv a_{0} c{\text{ }}\bmod \;F}}}^{cF} \left( 1+ (1-s) \frac{a-a_0 c}{a_0 c} \right) \right) + \delta (F/q) . \end{aligned}

Note that $$| \frac{a-a_0 c}{a_0 c} | \le |F|$$, so this is the same type of estimate used in the proof of Proposition 2.1. Subtracting the second sum in (4) yields

$$\begin{gathered} (1 - \chi (c)\langle c\rangle ^{{1 - s}} )L_{p} (s,\chi ) \hfill \\ = \frac{{ - 1}}{{cF}}\sum\limits_{\substack{\mathop {a_0=1}\\ {p\not\mid a_0}}}^F \chi (a_{0} c)\langle a_{0} c\rangle ^{{1 - s}} \left( \sum\limits_{\substack{\mathop{a=1} \\ {a \equiv a_{0} c{\text{ }}\bmod \;F}}}^{cF} \frac{a-a_0 c}{a_0 c} \right) + \delta (F/qp) \hfill \\ = \frac{{ - 1}}{c} \sum\limits_{\substack{\mathop {a_0=1}\\{p\not\mid a_0}}}^F \chi \omega ^{{ - 1}} (a_{0} c)\langle a_{0} c\rangle ^{{ - s}} \left( \sum\limits_{\substack{\mathop{a=1} \\ \mathop {a \equiv a_{0} c{\text{ }}\bmod \;F}}}^{cF} \frac{a-a_0 c}{F} \right) + \delta (F/qp) . \hfill \\ \end{gathered}$$

We compute the inner sum. Let $$b=\{a_0 c\}_F$$. Then $$a_0 c = b + \{ -F^{-1} b \}_c\, F$$, since the latter sum is congruent to b modulo F and congruent to 0 modulo c. If a satisfies $$a \equiv a_0 c \text { mod}\ F$$ and $$0<a<cF$$, then $$a=b+jF$$ with $$0\le j <c$$. Hence

\begin{aligned} \sum\limits_{\substack{\mathop{a=1} \\ \mathop {a \equiv a_{0} c{\text{ }}\bmod \;F}}}^{cF} \frac{a-a_0 c}{F} = \sum _{j=0}^{c-1} (j - \{ - F^{-1} b \}_c) = c\ \epsilon _{b,c,F}\ . \end{aligned}

Since $$b \equiv a_0 c \text { mod}\ F$$, we have $$\chi \omega ^{-1}(b) \langle b \rangle ^{-s} = \chi \omega ^{-1}(a_0 c) \langle a_0 c \rangle ^{-s} + \delta (F/q)$$ by the same estimate as earlier, so

$$- (1 - \chi (c)\langle c\rangle ^{{1 - s}} )L_{p} (s,\chi ) = \sum\limits_{\substack{\mathop {b=1}\\ {p\not\mid b}}}^F \chi \omega ^{{ - 1}} (b)\langle b\rangle ^{{ - s}} \epsilon _{{b,c,F}} + \delta (F/qp).{\text{ }}$$

This completes the proof. $$\square$$

We can take the limit of $$F=fp^n$$ as $$n \rightarrow \infty$$ and obtain:

### Corollary 2.5

Let p be a prime number, $$\chi$$ an even Dirichlet character of conductor f, and $$c>1$$ an integer satisfying $$(c,pf)=1$$. Then we have for $$s \in {\mathbb {C}}_p$$ with $$|s| < qp^{-1/(p-1)}$$,

\begin{aligned} -(1- \chi (c) \langle c \rangle ^{1-s}) L_p(s, \chi ) = \lim _{n \rightarrow \infty } \sum\limits_{\substack{\mathop {a=1} \\ {p\not\mid a}}}^{fp^n} \chi \omega ^{-1}(a) \frac{ \epsilon _{a,c,fp^n} }{ \langle a \rangle ^{s}}. \end{aligned}

The next Theorem shows that a finite number of Euler factors can be factored off in a similar way as in [8], where a weak Euler product was obtained. The main statement is that the remaining Dirichlet series has the expected form, similar to the complex case.

### Theorem 2.6

Let p be a prime number and let $$\chi$$ be an even Dirichlet character of conductor f. Let S be any finite (or empty) set of primes not containing p and set $$S^+ = S \cup \{p\}$$. Let F be a multiple of q, f and all primes in S. Let $$c>1$$ be an integer satisfying $$(c,F)=1$$. Then we have for $$s \in {\mathbb {C}}_p$$ with $$|s| < qp^{-1/(p-1)}$$ the formula

$$- (1 - \chi (c)\langle c\rangle ^{{1 - s}} ) \cdot \prod\limits_{{l \in S}} {(1 - \chi \omega ^{{ - 1}} (l)\langle l\rangle ^{{ - s}} )} \; \cdot \;L_{p} (s,\chi ) = \sum\limits_{\substack{\mathop {a=1}\\{ \left( {a,S^{ + } } \right) = 1 }}}^F\chi \omega ^{{ - 1}} (a)\frac{{ \epsilon_{{a,c,F}} }}{{\langle a\rangle ^{s} }}\; + \;\delta (F/qp).$$

### Proof

We prove the statement by induction on |S|. By Theorem 2.4, the formula is true for $$S=\varnothing$$. Now assume the formula is true for S, and $$l \ne p$$ is a prime with $$l \notin S$$ and $$(c,l)=1$$. It suffices to prove the following formula:

\begin{aligned} \begin{aligned} (1-\chi \omega ^{-1}(l) \langle l \rangle ^{-s} ) \sum\limits_{\substack{\mathop {a=1} \\ \left( {a,S^{ + } } \right) = 1 }}^F \chi \omega ^{-1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,F}\ = \\ \sum\limits_{\substack{\mathop {a=1} \\ \left( {a,S^{ + } \cup \left\{ l \right\}} \right) = 1 }}^{lF} \chi \omega ^{-1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,lF}\ + \ \delta (F/qp) . \end{aligned} \end{aligned}
(5)

Note that $$|1-\chi \omega ^{-1}(l) \langle l \rangle ^{-s} | \le 1$$ and $$|lF|=|F|$$, so we can keep the error term. We can use lF in place of F and write the left side of (5) as

\begin{aligned} \sum\limits_{\substack{\mathop {a=1} \\ \left( {a,S^{ + } } \right) = 1 }}^{lF} \chi \omega ^{-1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,lF} \ - \sum\limits_{\substack{ \mathop {a=1} \\ \left( {a,S^{ + } } \right) = 1 }}^{F} \chi \omega ^{-1}(la) \langle la \rangle ^{-s} \epsilon _{a,c,F} + \delta (F/qp) . \end{aligned}
(6)

Now we have

\begin{aligned} \epsilon _{la,c,lF} = \frac{c-1}{2} - \{ -la (lF)^{-1} \}_{c} = \frac{c-1}{2} - \{ -a F^{-1} \}_{c} = \epsilon _{a,c,F}. \end{aligned}

Thus (6) is equal to

\begin{aligned} \sum\limits_{\substack{\mathop {a=1}\\ \left( {a,S^{ + } } \right) = 1 }}^{lF} \chi \omega ^{- 1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,lF} \ - \sum\limits_{\substack{\mathop {a=1}\\ \left( {a,S^{ + } } \right) = 1 }}^F \chi \omega ^{-1}(la) \langle la \rangle ^{-s} \epsilon _{la,c,lF} + \ \delta (F/qp) \\ & = \sum\limits_{\substack{\mathop{a = 1} \\ \left( {a,S^{ + } } \right) = 1 \\ l\not\mid a} }^{{lF}} {} \chi \omega ^{-1}(a) \langle a \rangle ^{-s} \epsilon _{a,c,lF}\ + \ \delta (F/qp), \end{aligned}

which shows equation (5).

$$\square$$

### Remark 2.7

What happens if S contains more and more primes? It is well known that the Euler product does not converge p-adically (see [2]), since the factors $$(1-\chi \omega ^{-1}(l) \langle l \rangle ^{-s} )$$ have absolute value $$\le 1$$ and do not converge to 1 as $$l \rightarrow \infty$$. Furthermore, there are infinitely many primes l with $$\chi \omega ^{-1} (l) = 1$$ and $$(1 - \langle l \rangle ^{-s})^{-1}$$ has a pole at $$s=0$$. We have for $$l \ne p$$ and $$|s| < qp^{-1/(p-1)}$$,

\begin{aligned} 1 - \langle l \rangle ^{-s} = - \sum _{j=1}^{\infty } \left( {\begin{array}{c}-s\\ j\end{array}}\right) ( \langle l \rangle -1 )^j . \end{aligned}

The p-adic absolute value of each term of the above series is less than

\begin{aligned} (qp^{-1/(p-1)})^j p^{(j-1)/(p-1)} q^{-j} = p^{-1/(p-1)} < 1 . \end{aligned}

Hence the product $$\prod _{l \in S} (1-\chi \omega ^{-1}(l) \langle l \rangle ^{-s} )$$ approaches 0 as S expands to include all primes.

## 3 Regularized Bernoulli distributions

Let p be a prime number and let d be a positive integer with $$(d,p)=1$$. Define $$X_n = ({\mathbb {Z}}/dp^n {\mathbb {Z}})$$ and $$X = \varprojlim X_n \cong {\mathbb {Z}}/d {\mathbb {Z}}\times {\mathbb {Z}}_p$$. Let $$k \ge 1$$ be an integer. Then the Bernoulli distribution $$E_k$$ on X is defined by

\begin{aligned} E_k( a + dp^ n X) = (dp^n)^{k-1} \frac{1}{k} B_k \left( \frac{ \{ a \}_{d p^n} }{d p^n} \right) , \end{aligned}

where $$B_k(x)$$ is the k-th Bernoulli polynomial and $$B_k=B_k(0)$$ are the Bernoulli numbers (see [5, 7]). For $$k=1$$, one has $$B_1(x)= x - \frac{1}{2}$$. Choose $$c \in {\mathbb {Z}}$$ with $$c \ne 1$$ and $$(c,dp)=1$$. Then the regularization $$E_{k,c}$$ of $$E_k$$ is defined by

\begin{aligned} E_{k,c} ( a + dp^n X ) = E_k ( a + dp^n X) - c^k E_k \left( \left\{ \frac{a}{c} \right\} _{d p^n} + d p^n X \right) . \end{aligned}

One shows that the regularized Bernoulli distributions $$E_{k,c}$$ are measures (see [7]). In the following, we consider only $$k=1$$; the cases $$k \ge 2$$ are similar.

### Theorem 3.1

Let p be a prime, $$c,\, d \in {\mathbb {N}}$$, and $$c \ge 2$$ such that $$(c,dp)=1$$. Let X be as above, and let $$E_{1,c}$$ be the regularized Bernoulli distribution on X. For $$a \in \{0,1,\dots , dp^n-1\}$$, we have

\begin{aligned} E_{1,c} ( a + dp^n X) = \frac{c-1}{2} - \{ -a (dp^n)^{-1} \}_{c} = \epsilon _{a,c,dp^n}. \end{aligned}

### Proof

By definition,

\begin{aligned} E_{1,c}(a+dp^n X) = E_1(a + dp^n X) - c E_1(c^{-1}a + dp^n X) = \frac{a}{dp^n} - \frac{1}{2} - c \left( \frac{ \{c^{-1} a \}_{dp^n}}{dp^n} \right) + \frac{c}{2}. \end{aligned}

We give the standard representative of $$c^{-1} a \text { mod}\ dp^n$$:

\begin{aligned} \{c^{-1} a \}_{dp^n} = \frac{ \{ -a (dp^n)^{-1} \}_{c}\ dp^n + a}{c} \end{aligned}

Note that the numerator is divisible by c, since $$\{ -a (dp^n)^{-1} \}_{c}\ dp^n \equiv -a \text { mod}\ c$$. Hence the quotient is an integer between 0 and $$dp^n-1$$. Furthermore, the numerator is congruent to a modulo $$dp^n$$, and so the quotient has the desired property. We obtain

\begin{aligned} E_{1,c}(a+dp^n X) = \frac{a}{dp^n} + \frac{c-1}{2} - \frac{ \{ -a (dp^n)^{-1} \}_{c}\ dp^n + a}{dp^n} = \frac{c-1}{2} - \{ -a (dp^n)^{-1} \}_{c} \end{aligned}

which is the assertion. $$\square$$

Now the Dirichlet series expansion in Corollary 2.5 follows from Theorem 3.1 and the integral formula (2).

## 4 Expansions for different regularization parameters

We look at the coefficients $$\epsilon _{a,c,dp^n}$$ for different parameters c and the resulting Dirichlet series expansions. The following observation follows directly from the definition.

### Remark 4.1

The sequence of values $$E_{1,c} ( a + dp^n X) = \epsilon _{a,c,dp^n}$$ for $$a=$$ 0,1, 2, $$\dots$$, $$dp^n-1$$ is periodic with period c. The sequence begins with $$\frac{c-1}{2}$$ and continues with a permutation of $$\frac{c-3}{2}, \dots , - \frac{c-1}{2}$$. If we restrict to values of n such that $$dp^n$$ lies in a fixed congruence class modulo c, then the values do not change as $$n \rightarrow \infty$$. $$\lozenge$$

The measure $$E_{1,c}$$ and the Dirichlet series expansion are particularly simple for $$c=2$$. Note that we assumed that d and p are odd in this case. If a is even, then $$\{ -a (dp^n)^{-1} \}_{2} = 0$$ and

\begin{aligned} E_{1,2}(a+dp^n X) = \epsilon _{a,2,dp^n} = \frac{1}{2}. \end{aligned}

If a is odd, then $$-a(dp^n)^{-1}$$ is odd, $$\{ -a (dp^n)^{-1} \}_{2} = 1$$ and

\begin{aligned} E_{1,2}(a+dp^n X) = \epsilon _{a,2,dp^n} = - \frac{1}{2}. \end{aligned}

Hence $$E_{1,2}$$ is up to the factor $$\frac{1}{2}$$ equal to the following simple measure:

### Definition 4.2

Let $$p \ne 2$$ be a prime, and let $$X \cong {\mathbb {Z}}/d{\mathbb {Z}}\times {\mathbb {Z}}_p$$ be as above. Then

\begin{aligned} \mu (a + dp^n X) = (-1)^{\{a\}_{dp^n}} \end{aligned}

defines a measure on X. We call $$\mu$$ the alternating measure, since the measure of all clopen balls is $$\pm 1$$.$$\lozenge$$

The corresponding integral is also called the fermionic p-adic integral (see [4]).

Now we obtain the following Dirichlet series expansion from Corollary 2.5.

### Corollary 4.3

Let $$p \ne 2$$ be a prime number, and let $$\chi$$ be an even Dirichlet character of odd conductor f. Then we have for $$s \in {\mathbb {C}}_p$$ with $$|s| < p^{(p-2)/(p-1)}$$,

\begin{aligned} (1- \chi (2) \langle 2 \rangle ^{1-s}) L_p(s, \chi ) = \lim _{n \rightarrow \infty } \frac{1}{2} \sum\limits_{\substack{\mathop {a=1}\\{p\not\mid a}}}^{fp^{n}} (-1)^{a+1} \chi \omega ^{-1}(a) \frac{1}{\langle a \rangle ^{s}} . \end{aligned}

For $$\chi =\omega ^{1-i}$$ and odd $$i=1,\, \dots ,\, p-2$$, we obtain the branches of the p-adic zeta function:

\begin{aligned} \zeta _{p,i}(s) = L_p(s,\, \omega ^{1-i}) = \frac{1}{1-\omega (2)^{1-i} \langle 2 \rangle ^{1- s}} \cdot \lim _{n \rightarrow \infty } \frac{1}{2}\sum\limits_{\substack{\mathop {a=1} \\ {p\not\mid a}}}^{p^{n}} (-1)^{a+1}\omega (a)^{-i}\frac{1}{ \langle a \rangle ^{s}} \end{aligned}

### Remark 4.4

Dirichlet series expansions of p-adic L-functions were studied by D. Delbourgo in [1] and [2]. He considers Dirichlet characters $$\chi$$ satisfying $$(p,\, 2 f \phi (f))=1$$ and their Teichmüller twists. We obtain the same expansion for $$c=2$$ and $$\chi = \omega ^{1-i}$$. However, we require $$(c,\, fp)=1$$ and use other methods for the proof.

Similar expansions for a slightly different p-adic L-function using a fermionic p-adic integral (i.e., $$c=2$$) were also obtained by M.-S. Kim and S. Hu (see [4]).

### Example 4.5

We look at the case $$c=3$$. The sequence of values $$\epsilon _{a,3,dp^n}$$ is periodic with period 3. If $$dp^n \equiv 1 \text { mod}\ 3$$, then the sequence is $$1,\ -1,\ 0,\, \dots$$ . If $$dp^n \equiv 2 \text { mod}\ 3$$, then we obtain the sequence $$1,\ 0,\ -1,\, \dots$$ .

### Corollary 4.6

Let p be a prime number, and let $$\chi$$ be an even Dirichlet character of conductor $$f=dp^m$$ such that $$(3,\, dp)=1$$. If $$d \equiv 1 \text { mod}\ 3$$, then define a sequence $$\epsilon _0=1$$, $$\epsilon _1=-1$$, $$\epsilon _2=0$$, $$\dots$$ with period 3. Otherwise, set $$\epsilon _0=1$$, $$\epsilon _1=0$$, $$\epsilon _2=-1$$ and extend it with period 3. Then we have for $$s \in {\mathbb {C}}_p$$ with $$|s| < qp^{-1/(p-1)}$$,

\begin{aligned} -(1- \chi (3) \langle 3 \rangle ^{1-s}) L_p(s, \chi ) = \lim _{n \rightarrow \infty } \sum\limits_{\substack{\mathop {a=1} \\ {p\not\mid a}}}^{ dp^{{2n}} } \chi \omega ^{-1}(a)\frac{\epsilon _a}{ \langle a \rangle ^{s} }. \end{aligned}

### Example 4.7

For $$c=5$$, we get a periodic sequence with period 5 and we have $$\epsilon _{a,5,dp^n}=2$$ for $$a \equiv 0 \text { mod}\ 5$$. The next four coefficients are a permutation of the values $$-2$$, $$-1$$, 0 and 1, depending on the class of $$dp^n \text { mod}\ 5$$.

### Example 4.8

Let $$c=7$$. Then $$\epsilon _{0,7,dp^n} = 3$$. Now suppose, for example, that $$dp^n \equiv 3 \text { mod}\ 7$$. Then $$(dp^n)^{-1} \equiv 5 \text { mod}\ 7$$. This yields the values

\begin{aligned} \epsilon _{1,7,dp^n} = 1,\ \epsilon _{2,7,dp^n} = -1,\ \epsilon _{3,7,dp^n} = -3,\ \epsilon _{4,7,dp^n} = 2,\ \epsilon _{5,7,dp^n} = 0,\ \epsilon _{6,7,dp^n} = -2, \end{aligned}

and these are extended with period 7.